Dynamic Spectrum Access: From the Concept to the Implementation

We are today witnessing an explosive growth in the deployment of wireless communication services. At the same time, wireless system designers are facing the continuously increasing demand for capacity and mobility required by the new user applications. The scarcity of the radio spectrum, densely allocated by the regulators, is a major bottleneck in the development of new wireless communications systems. However actual spectrum occupancy measurements show that the frequency band scarcity is not a result of the heavy usage of the spectrum but is rather due to the inefficient static frequency allocation pursued by the regulators. Dynamic spectrum access, also generally referred to as cognitive radios, has been proposed as a new technology to resolve this paradox. Sparse assigned frequency bands are opened to secondary users, provided that interference generated on the primary licensee is negligible. Even if the concept constitutes a real paradigm shift, it is still unclear how the dynamic spectrum access can operate efficiently and how it can be implemented cost-effectively. The goal of this special issue has been to solicit high-quality unpublished research papers on the spectrum sensing and access, the intelligence and learning capability, and implementation aspects of communication systems relying on dynamic spectrum access. Based on the 16 submitted manuscripts, six papers have been accepted, which will be summarized briefly in this paper. Spectrum sensing techniques have been heavily discussed and treated in the literature. Their performance in a noisy environment is measured in terms of the probability of false alarm and the probability of miss detection. A radiometer (also called energy detector) can be used to detect completely unknown signals in a frequency band. Unfortunately when the primary user bandwidth is much smaller than the cognitive radio spectrum sensing frequency range, scanning a wide range of frequencies can be time consuming. In their paper, S. Kandeepan et al. propose to make use of the primary user time-domain spectral occupancy statistics to enhance the performance of the wideband energy detector. They assume that the primary user spectral occupancy can be modeled with a Poisson law and derive analytically the minimum required sensing time for the cognitive radio to detect the primary user. The energy detector is historically the oldest and simplest detector, and it achieves good performance when the signal-to-noise ratio is strong enough. Unfortunately, since it is based on the estimation of the in-band noise power spectral density, it is affected by the …

We are today witnessing an explosive growth in the deployment of wireless communication services. At the same time, wireless system designers are facing the continuously increasing demand for capacity and mobility required by the new user applications. The scarcity of the radio spectrum, densely allocated by the regulators, is a major bottleneck in the development of new wireless communications systems.
However actual spectrum occupancy measurements show that the frequency band scarcity is not a result of the heavy usage of the spectrum but is rather due to the inefficient static frequency allocation pursued by the regulators.
Dynamic spectrum access, also generally referred to as cognitive radios, has been proposed as a new technology to resolve this paradox. Sparse assigned frequency bands are opened to secondary users, provided that interference generated on the primary licensee is negligible. Even if the concept constitutes a real paradigm shift, it is still unclear how the dynamic spectrum access can operate efficiently and how it can be implemented cost-effectively.
The goal of this special issue has been to solicit highquality unpublished research papers on the spectrum sensing and access, the intelligence and learning capability, and implementation aspects of communication systems relying on dynamic spectrum access. Based on the 16 submitted manuscripts, six papers have been accepted, which will be summarized briefly in this paper.
Spectrum sensing techniques have been heavily discussed and treated in the literature. Their performance in a noisy environment is measured in terms of the probability of false alarm and the probability of miss detection.
A radiometer (also called energy detector) can be used to detect completely unknown signals in a frequency band. Unfortunately when the primary user bandwidth is much smaller than the cognitive radio spectrum sensing frequency range, scanning a wide range of frequencies can be time consuming. In their paper, S. Kandeepan et al. propose to make use of the primary user time-domain spectral occupancy statistics to enhance the performance of the wideband energy detector. They assume that the primary user spectral occupancy can be modeled with a Poisson law and derive analytically the minimum required sensing time for the cognitive radio to detect the primary user.
The energy detector is historically the oldest and simplest detector, and it achieves good performance when the signalto-noise ratio is strong enough. Unfortunately, since it is based on the estimation of the in-band noise power spectral density, it is affected by the noise level uncertainty due to measurement errors or a changing environment, especially at low signal-to-noise ratio. Cyclic feature detectors rely on the hidden periodicities such as the carrier frequency, the symbol rate, or the chip rate hidden in man-made communications signals, that can be extracted based on nonlinear operations of the nth order (order equal to or larger than 2). Cyclostationarity detectors perform better than energy detectors in low signal-to-noise ratio environments.
In their paper, D. Noguet et al. analyze the architectural trade-offs that the cognitive radio system designer has to

Introduction
The Cognitive Radio (CR) concept is being under deep consideration to opportunistically utilize the electromagnetic spectrum for efficient radio transmission [1][2][3][4]. The CR basically acts as a secondary user of the spectrum allowing the incumbent (primary) users of the spectrum to have higher priority for spectrum utilization. The notion of efficient spectrum utilization has also attracted the radio spectrum regulatory bodies around the world [5,6] to further investigate the technology. The secondary users therefore need to learn the environment in the presence of any primary users (PUs) and keep track of them to ensure that it does not interfere with the PU. Learning the environment and performing radio scene analysis (RSA) becomes a challenging and an essential task for the CR to successfully perform secondary communication with reduced interference to the PU. Reliably performing the RSA is quite important in order to avoid interfering with the PU's communications and also to satisfy the regulatory requirements. To perform such an RSA, the CR nodes need to sense the spectrum continuously in the time, frequency, and spatial domains. Spectrum sensing therefore, amongst many, is one of the key functionalities of a CR in order to perform (RSA) of the communication environment.

Problem Statement.
In the recent years, Ultra Wideband (UWB) technology has emerged as one of the key candidates for CR based secondary user communications [7,8]. When UWB technology is used as CRs for secondary communications, it is required to scan the entire spectrum from 2.9 GHz-10 GHz (in many cases a significant portion of it) to detect the presence of any PUs in the network. In such situations, scanning a wide range of frequencies (7 GHz) can be a time consuming process and hence a narrow band PU, which has a bandwidth much smaller than the UWB node, can be gone undetected when the UWB-CR node is scanning a large portion of the spectrum. It is obvious to state that such miss detection depends on the PU's transmission statistics, or in other words the Spectral occupancy Statistics (SoS) as well as the spectrum sniffing hardware unit of the UWB-CR node and the corresponding time required to sense the spectrum. Such a problem is considered to be a crucial one to be solved by many researchers and engineers working in this filed, such as in the European Union funded 20 M-Euros project on EUWB [8]. The problem defined here is not specific to UWB based CR systems only, but in general it applies to any CR systems with a bandwidth much larger than the bandwidth of the potential victim services (i.e., the PUs) in the network.
Furthermore, the authors in [21] have used the Poisson traffic model to design an admission controller for the CR node to improve the Quality of Service (QoS) for secondary transmissions. In [22], for Poisson based PU traffic, the authors have studied by means of simulations the tradeoff between sensing time and the achievable throughput by considering the probability of collision. The Poisson traffic model is also used in [23,24] to derive a-priori probability based detection schemes for spectrum sensing and have presented some numerical results on the improvement over the traditional energy-based detection [45] and the classical Maximum Likelihood-(ML-) based detection techniques [46], respectively. A channel selection scheme for CR nodes for a multichannel multiuser environment is also proposedbased on Poisson traffic model in [25], and recently in [26,27] we have studied the performance of shared spectrums sensing for UWB-based CR assuming the Poisson model for the PU. Furthermore, moving away from the Poisson based theoretical model, the authors in [47] have analyzed the performances of dynamic spectrum access techniques based on experimentally measured spectrum occupancy statistics.

Contribution.
In this paper, we study the performance of detecting the PU based on its time domain SoS, by classifying them as light, average or heavy users of the spectrum, together with noisy sensing at the CR nodes. We mainly consider the case where the PU bandwidth is much smaller than the CR's spectrum scanning frequency range. Though the references in [22][23][24][25][26] have considered the Poisson model for the PU channel SoS they have presented mainly some simulation results to analyze the performances of the the CR node for spectrum sensing and channel occupancy efficiency. In our work presented here, we perform some detailed theoretical analysis of the performance of PU detection, based on an envelope-based energy detector, by initially considering the transmission statistics only case and then together with the sensing noise. The theoretical anaylses are also verified by simulations. We also study the minimum required sensing time for the CR node to reliably detect the narrow band PU given its temporal characteristics.

Paper
Organization. The rest of the paper is organized as follows. In Section 2, we provide the model for the CR network, and in Section 3, we derive the Poisson arrival model for the PU's transmission statistics from the fundamentals. In Section 4, we provide the signal envelope based spectrum sensing technique followed by some theoretical analysis on the detection performances for the noiseless case with a constant channel occupancy (hold) time in Section 5. In Section 6, we present a PU detection risk analysis based on the transmission statistics (Poisson process) of the PU. In Section 7, we present the detection performance considering noise, and in Section 8, we extend the analysis for a random channel occupancy (hold) time. In Section 9, we present the sensing time requirements for the CR based on the SoS of the PU, and finally we make some concluding remarks in Section 10.

Cognitive Radio-System Model
The CR network model is presented in this section. We define the following parameters for the wideband sensing CR nodes and the PU in the network. For the PU, we define B w as its transmission bandwidth, Δ as the time duration between transmissions (spectrum idle time) that is calculated from the end to the beginning of two successive transmissions, and τ as the PU's transmission duration (spectrum hold/busy time) per transmission. For the CR node, we define W as the total observation bandwidth to sweep, and T w as the total time to linearly sweep the observation bandwidth W. Linear sweeping is defined by having some constant rate for sweeping the frequency, or in other words the time to sweep a range of frequencies is proportional to the same range of frequencies. Figure 1 depicts the above parameters and also shows the frequency-time observation plot which explains the spectrum scanning process of the wideband sensing CR node. As depicted in the figure, the cognitive radio scans the entire spectrum linearly at a rate of W/T w . Using such a linear frequency sweeping process, given that the PU is transmitting, the CR node can only detect the PU during its mth scan within the time slot (only) from t m 1 = t 0 + f 1 T w /W + (m − 1)T w and t m 2 = t 0 + f 2 T w /W + (m − 1)T w , where t 0 is an arbitrary real constant, f 1  (iv) W-total bandwidth to be scanned by the CR node, (v) T w -average time to scan the frequency band W by the CR node, (vi) t m 1 , t m 2 -start and end times of scanning the PU bandwidth by the CR node, as shown in Figure 1, during the mth iteration.
The presence of a PU in the network is defined by the hypotheses H 0 and H 1 , as described in (1). Given that the PU is detected during the mth scan, the CR decides upon H 1 for the entire period of the scan before reinitializing it back to H 0 for the successive scan (i.e., (m + 1)th scan). The hypothetical decision d m made by the CR node at the end of every scan for the particular PU within the frequency band of f 1 and f 2 in the absence of noise can be mathematically defined as follows: (1) A more generic detection model considering the PU SoS together with the sensing noise at the CR node is provided in Section 4. The PU's transmission is modeled as a Poisson arrival process. Therefore, Δ follows an exponential distribution [30] with a mean time between transmission given by Δ a . We validate the Poisson arrival model for the CR network that we consider here by referring back to Section 1: Literature Review. In other words, the CR network model considered here has a PU delivering Poissonian traffic to the network within its frequency band. Initially, we treat the transmission duration (hold time) τ of the PU as a constant to simplify the analysis and then in Section 8 we extend the analysis to a random transmission duration τ by modeling it as a random process.
Furthermore, based on the values of Δ a and τ we can characterize the spectral occupancy levels of a PUs as light, average, or heavy users. Figure 2 depicts the spectral occupancy levels of a PU based on their traffic characteristics. In the figure, we further see that the pairs {Δ L a , τ L } and {Δ H a , τ H } separate the occupancy level regions of the PUs as light, average, or heavy. Corresponding to the occupancy levels, we also characterize the risk of miss detecting the PU as High Risk, Medium Risk, and Low Risk regions, respectively.

EURASIP Journal on Wireless Communications and Networking
For analytical purposes, the risk regions for miss detecting the PU are defined by (i) high risk region (light user); for 0 < T w /Δ a < T w /Δ L a and 0 < τ/T w < τ L /T w , (ii) medium risk region (average user); for T w /Δ L a < T w /Δ a < T w /Δ H a and τ L /T w < τ/T w < τ H /T w , (iii) low risk region (heavy user); for T w /Δ H a < T w /Δ a and τ H /T w < τ/T w .
In later sections, by using the detection probabilities derived from our theoretical analyses, we present numerical values for Δ L a , Δ H a , τ L , and τ H .

Primary User Spectral Occupancy Statistics
To define the primary user's spectral occupancy statistics based on the Poisson arrival process we state (consider) some axioms. It is important to note that these axioms are the fundamentals in defining the Poisson arrival process in general, and based on these axioms we then analyze the spectrum sensing detection performances considering the spectral occupancy model of the PU. Let N(t) be the number of times that the PU has been present in the network (number of transmissions) up to time t, where t ∈ R, the axioms are then defined as in [46]. Axiom 1. At time t = 0, the PU has got no occupancy of the spectrum at all. That is, N(0) = 0.
The Poisson distribution for the arrival process is then given by where n = 0, 1, 2, . . . and λ = 1/Δ a is the mean arrival (spectral occupancy) rate of the PU. On the other hand, the occupancy time τ of the PU is initially considered to be a constant. Such a model essentially creates an M/D/1 arrival model considering a single CR and single PU system with a Poissonian arrival (M) and a deterministic (D) occupancy time. In later sections, we extend this to an M/M/1 arrival model by considering an exponential distribution and an M/G/1 model considering a Pareto distribution for the random occupancy time for the PU transmissions.

Spectrum Sensing
The spectrum sensing technique considered here is the signal envelope-based method [45] where the envelope of the signal is computed within a given range of frequencies in time and compared against a threshold value μ. We consider the envelope-based detection method over the standard energybased detection method [45,[48][49][50] mainly considering its simplicity in hardware implementation and computation. From the analytical framework we provide in this paper on the detection performance of the envelope-based detector, together with SoS of the PU, it is rather straight forward to derive the corresponding theoretical expressions for the energy-based detector and perform similar analyses to the ones that we present here. The Energy detectors in general including the envelope-based detector have drawbacks [10,11,49] for spectral occupancy detection especially when the noise power is not known, but on the other hand it is the simplest detection method when the CR node has got no knowledge about the PU transmission.
The received baseband signal of bandwidth B w in its complex envelope form received over a time period of t m 1 ≤ t ≤ t m 2 is given by when PU is not present, where, s(t) is the complex envelope of the received signal from the PU without noise, and ν(t) is the additive bandpass and band limited complex noise component associated with the sensing process. The additive noise ν(t) is modeled as a zero mean complex Gaussian random process with a power of 2σ 2 over a bandwidth B w . Note that in (3) we only consider the signal of our interest given within the frequencies f 1 and f 2 (corresponding to the scanning time duration of t m 1 ≤ t ≤ t m 2 ). Since we consider only one PU in our model, therefore r(t) corresponds to the signal received within the PU's transmission bandwidth. We also assume negligible fading associated with s(t) in our model or in other words the amount of fading is small considering the time period for computing the envelope (energy) of the signal, which is a valid assumption as there exist many cases with slow fading scenarios [51]. The envelope of the received signal at t = α over a time of t m 1 ≤ α ≤ t m 2 is used as the test statistic ξ s (α) to detect the PU (transmitting in the frequency band between f 1 and f 2 ), where ξ s (α) is given by f 2 . The test statistic ξ s (α) follows two different distributions under the hypotheses H 0 and H 1 depending on whether the signal s(t) is present or not. From the signal model presented in (3), it is well known that [46] (since ν(t) is complex Gaussian) ξ s follows a Rayleigh distribution under H 0 and a Rice distribution under H 1 , given by where E r is the time domain root mean square value of the signal r(t), and I 0 is the zeroth order modified Bessel function of the first kind. The distributions in (7) are used to analyze the detection performance which we present in the subsequent sections.

Performance Analysis: Noiseless Sensing
The theoretical performance analysis for detecting the PU is performed in three stages. Initially, we study the detection performance considering only the SoS of the PU with a constant hold time with no sensing noise (i.e., SNR ρ = ∞), then we extend the analysis considering the sensing noise, and then we further extend the analysis for random hold time. We also provide simulation results to support our theoretical analysis.

Theoretical Analysis.
In the noiseless case, the detection performance of the CR node is characterized by the SoS of the PU, the bandwidth B w of the PU, and the total time T w for the CR node to scan the entire bandwidth W. As mentioned previously, we assume the CR nodes linearly scan the frequency in time over the desired (wideband) spectrum.

Occupancy Probability.
For the PU, we define the spectral occupancy probability P O as the probability of initiation of at least one transmission by the PU over a time of T o seconds. Therefore; P O is given by (2), we find a closed form expression for the probability of occupancy as From (8), we can compute the spectral occupancy probability of the PU for a single scanning period T w by letting T o = T w .

Detection
Probability. The detection probability, for detecting the PU by the CR node over a single scan duration of T w , in the noiseless case is defined by the probability of initiation of at least one transmission within the time slot of t m 1 − τ ≤ t ≤ t m 2 for the mth scanning iteration (t m 1 and t m 2 are defined in Section 2). The probability of detection is given by Using the incremental independence and stationarity property of the arrival process (Axiom 2), we can rewrite P D as (2) we find a closed form expression given by Note that when τ + Γ = T w , then P D = P O . Therefore, the CR detects all the transmissions from the PU in a single scanning period. At the same time, we also observe from (9) that P D = 1 when λ → ∞.

Miss Detection
Probability. The probability of miss detection is defined by the probability that CR deciding H 0 gives H 1 for a single scan. In other words, miss detection for the noiseless case occurs when there is at least one initiation of transmission occurring for some t outside the interval t / ∈ [t m 1 − τ, t m 2 ], but no initiations (of transmissions) during the interval, in a single scan. Therefore, for a single scan the miss detection probability is given by for some α 1 , α 2 ∈ [t 0 +mT w , t 0 +(m+1)T w ] and, (10) are mutually disjoint, P M can be rewritten as . Again, using the incremental independence and stationarity property of the arrival process with (2), we find a closed form expression as We further verify the analytical expressions for P D and P M , from (9) and (11), since P D = 1 − P M .

False Alarm
Probability. The false alarm probability for the noiseless case is defined by the probability of detecting a transmission given that no transmissions have been initiated by the PU during a single scan. Since we do not consider noise in this case, the probability of false alarm is simply zero. Further, we could verify this by expressing the false alarm probability as and since (N(t 0 +(m+1)T w )−N(t 0 +mT w )) = 0 implies that  counting process by generating a binary random event with a probability of p within a small time duration of T s , as described in [46,Section 21.3], which converges to a Poissonian process as T s → 0 with λ = p/T s . Simulations were performed in Matlab to generate the random arrival process with a deterministic hold time. Figure 3 presents the occupancy probability P O for various time durations T o . From the figure, we observe that P O increases with the arrival rate λ, as expected, and also marginally improves with the time duration T o . The figure shows how the the spectral occupancy probability of the PU (defined over a period of T o = T w ) increases when the scanning duration T w increases for a given arrival rate λ. This observation is quite important, especially when we study the minimum time requirement (minimum T w ) for sensing the PU, which we present in the later sections. Figure 4 depicts the miss detection probability P M for various holding times τ and arrival rate λ. From the figures, we see that the detection performance at the CR node improves when both τ and λ increase, we further observe that the improvement in the detection performance is greater when λ increases than when τ increases, relatively. From the figure we also see a very close match between the theoretical expressions derived in the previous section and the simulated results further verifying our analysis.

Analysis of the Primary User Miss Detection Risk Regions
In this section, we study the risk regions associated with miss detecting the PU depending on their SoS. Continuing from Section 2 and referring back to Figure 2, here we define the risk regions based on the probability of detection P D (for the noiseless case). In other words, we define the values for    can be varied depending on our traffic model (traffic sources) based on the application. The values of P D L and P D H for the risk regions are basically custom defined and chosen to comply with regulatory requirements. For example, if the regulatory requirement for a minimum detection probability (to minimize interference to the PU) is given, then it could be assigned to P D H . This would ensure that the low risk region (defined by P D H ) does satisfy the regulatory requirements, which we explain in Example 1 later. On the other hand, P D L which defines the medium and high risk regions is defined by the CR Network as a benchmark for classifying interference level as medium or high interference, respectively, (i.e., interference from the CR to the PU and vice versa).
Further, for a given set of values B w , T w and W, we can define two theoretical curves R 1 and R 2 for the boundaries of risk regions, which are given by where Γ = T w B w /W (from Section 5.1). The curves R 1 and R 2 and the corresponding risk regions are shown in Figure 6. The figure here ( Figure 6) defines the risk regions more precisely than Figure 2. Note that the risk regions can be custom defined based on the values of P D L and P D H that define the curves R 1 and R 2 , respectively. As observed in the figure, the risk regions become independent of the hold time τ for small values of τ. In practice, this is true since the detection performance only depends on the arrival rate λ(= 1/Δ a ) for small values of τ/Δ a . The values of Δ L a and Δ H a defining the risk regions are then computed from R 1 and R 2 , respectively as For the parameters given in Figure 6, the values of Δ a defining the risk regions are given by Δ L a = 0.0064 sec (for 20% confidence, i.e., P D L = 0.2) and Δ H a = 0.0016 sec (for 60% confidence, i.e., for P D H = 0.6). The values τ L and τ H , on the other hand, can only be defined for a given value of λ, letting λ → 0 to compute τ L and τ H has no practical significance since when λ = 0 there are no transmissions from the PU. Therefore, for a given λ, say λ = (1/Δ a ) = 1/10 −3 , we find the values of τ defining the risk regions by using R 1 and R 2 (or by using Figure 6), as τ L = 223.14 sec and τ H = 916.3 sec.
Example 1 (WiMedia-UWB based Cognitive Radio System with Constant Scanning Time (T w )). We now provide an example of a WiMedia based Ultra-Wideband (UWB) [7] CR system which requires a 90% confidence (P D > 0.9, P D L = P D H = 0.9) in detecting a PU in the network. The PU system that we consider here is the WiMax radio [ In practice the temporal behavior of WiMax transmissions may not be Poisson arrival process, however we consider the Poisson arrival here for analytical purposes). We analyze, for the noiseless case, the values of Δ a that enables the WiMedia based CR node to detect a WiMax PU with the given confidence level (90%). We assume that the WiMedia system has an operational bandwidth of W = 1.5 GHz (from 3 GHz-4.5 GHz) and a hardware that has a spectrum scanning time of T W = 10 msec (assumption only). Then, by using (15), we compute the value of Δ a for small values of τ, that defines whether the requirement of 90% confidence for detecting the WiMax radio can be achieved or not in the noiseless case. Accordingly, we get Δ a = 2.89 × 10 −5 where Γ = (10 × 10 −3 )(10 × 10 6 )/(1.5 × 10 9 ) = 6.67 × 10 −5 . Therefore, given a fixed value T w , the CR can detect the PU with 90% confidence for the noiseless case (or for very high received SNR of the PU signal) provided that the SoS of the PU is such that the mean time between transmissions Δ a satisfies Δ a < 2.89 × 10 −5 sec for small values of τ.

Performance Analysis: Noisy Sensing
The detection performance with noise is of great interest to us. In this section, we analyze the overall probability of false alarm and the overall probability of miss detection for the envelope-based PU detection given the Poisson SoS of the PU for noisy sensing.
From (16), by using (7), we come up with two closed form expressions for P D and P F as where, Q 1 (x 1 , x 2 ) is the Marcum Q-Function defined by, From the expressions we, observe that the probability of false alarm P F does not depend on the transmission statistics of the PU but only depends on the sensing noise. In the following section, we perform some simulations to study the detection performances and also verify the theoretical analysis performed in this section.

Complementary Receiver Operating Characteristic Curves and Simulation
Results. The Complementary Receiver Operating Characteristic Curves (C-ROCs) for the envelopebased detector is presented here considering the PU transmission statistics. The C-ROC curve is the plot between P F and P M by varying the detection threshold μ. Together with the theoretical C-ROC curves, derived from the previous section, we also present some simulation results to verify our theoretical analysis. Monte-Carlo simulations were performed to generate the PU's (Poisson) transmission process and the noisy sensing process with Gaussian noise. Figure 7 shows the C-ROC curves for various SNR values for λ = 25 and τ = 0.01. As expected, we see that the C-ROC curves improve when the SNR is increased, the figure also shows the probability of miss detection (P M ) for the noiseless case as well. Figure 8 shows the C-ROC curves for different values of arrival rate λ with γ = 5 dB and τ = 0.02. From the figure, we observe how the detection performance improves when the arrival rate (spectral occupancy rate) of the PU is increased. Moreover, we see that when λ is increased further the detection performance predominantly depends on the sensing noise. Figure 9 shows the C-ROC curves for various values of spectral hold time τ for λ = 15 and γ = 5 dB. Improvements in the detection performances are observed when τ is increased but not as much as when λ is increased as in Figure 8. Furthermore, we clearly see that the simulation results very closely match with the theoretical results on all the figures, verifying our analysis in the previous section.

Performance Analysis: Random Hold Time
In our analysis so far we have assumed a constant hold time τ. In this section we extend the analysis for the   detection performance with a randomly distributed hold time τ. We consider two random hold time models namely (1) the exponential distribution which is a traditional way for modeling the call hold time describing many real time applications, and (2) the Pareto distribution which is used to model the World Wide Web IP traffic.
The exponential model is a very traditional model used to describe the random call hold time process. It is useful in modeling voice traffic as recommended by the ITU [32]. In  [28], the exponential model is also verified experimentally for aggregated HSDPA data traffic by Telefonica I + D. Based on the type of traffic, the exponential model can also be extended to obtain further traffic models such as Erlange and Phase-type models [32]. However, in our paper, we adopt the exponential model whereas the other models could simply follow similar analytical procedures. We also adopt another model for the random hold time which suits modern teletraffic such as the World Wide Web IP traffic, namely, the Pareto distributed hold time model. The Pareto model is (experimentally) proven to fit Internet traffic as described in [28,37,[53][54][55][56]. Furthermore, in practice, the analyses for the constant τ with the M/D/1 model and the random τ with exponential and Pareto models are all useful depending on the type of traffic generated by the PU.
The exponentially distributed hold time τ is given by the density function f τ (τ) = 1/τ a exp(−τ/τ a ), with a mean hold time of τ a . The Pareto distributed hold time τ is given by the density function f τ (τ) = kτ k min τ −(k+1) for τ > τ min and ∀k > 0, with The parameter k for the Pareto model describes the peaky nature of the density function and τ min describes the minimum hold time that models a typical IP packet oriented service.

Theoretical Analysis.
Using the exponential and Pareto hold time models with Poisson arrival for the PU transmission we rederive the expression for the probability of detection here for random τ. The probability of false alarm P F stays unchanged, as in (17), because P F is independent of the transmission statistics of the PU (i.e., P F is independent of τ). From (17), we derive the new probability of detection for the random hold time by averaging over all possible values of τ. Therefore, the probability of detection is given by (Note that for the Pareto model the integration range in the above is from τ min to ∞.) By solving the integral in (19) for both exponential and Pareto models, we come up with a closed form expression for P D as where P M = ∞ 0 P M f τ (τ)dτ is the probability of miss detection for the random τ in the absence of the sensing noise (i.e., considering only the transmission statistics of the PU), given by for the exponential model, k exp(−λ(τ min + Γ)) (λτ min + k) for the Pareto model.
Furthermore, the probability of detection P D for random τ in the absence of noise is given by P D (τ) = 1 − P M . The curves R 1 and R 2 defining the risk regions (Section 6) can also be redefined for the random holding time case. From Section 8.1 we have two new curves for the exponential hold time model given by The new curves in (22) differ mainly for low values of λ but has the same limit values of Δ L a and Δ H a defining the risk regions, as in (15), when τ a → 0.

Numerical Results.
We compare the differences in the detection performance between the constant τ and the random τ cases. The random τ case obviously relates to reality more than the former. Figure 10 depicts the theoretical curves for the probability of miss detection with respect to the arrival rate λ for the noiseless case. The figure shows both the constant and the random τ cases for the exponential and the Pareto models. As we observe from the figure, the differences between the two arise for higher values of mean hold time and λ. This is due to the fact that for larger values of τ a and τ b , the probability of having smaller hold times is greater in the random case, and hence we observe poorer performances compared to the constant τ case. Figure 11 on the other hand shows the complementary ROC curves for the random and constant τ cases for noisy sensing. In Figure 10, we observed that the differences between the random and the constant τ detection performances arise for higher values of (both) mean hold time and λ, and in Figure 11 we observe that the difference between the two arise only when the SNR is higher, provided that the mean hold time and λ are high.
For lower values of γ we see that the difference is negligible. Furthermore, we observe that Pareto model shows better detection performance compared to the exponential model based on the value of k(= 2) for the same mean hold time. Note that when k increases the, Pareto model approaches the constant hold time case because the density function gets more peakier.

Minimum Required Spectrum Sensing Time
The temporal behavior of the PU influences the required spectrum sensing time for the CR node to reliably detect the PU. From the theoretical analyses performed in the previous sections, we study the minimum required sensing time to reliably detect the narrow band PU by a wideband CR node with a given confidence level. We compute the minimum required time for spectrum sensing based (only) on the temporal characteristics of the PU for both constant and random τ cases. The advantages of knowing the minimum required sensing time T w = T min w here is two fold, (1) To reliably detect the PU with a given confidence level by not underscanning the spectrum (not scanning lower than the minimum required time) and (2) To save power at the CR node by not overscanning the spectrum (not scanning higher than the minimum required time), From (9) and Section 8.1, for a given confidence level of Υ% (i.e., probability of detection is Υ) for detecting the PU, the minimum required spectrum sensing time T min w for the constant and random hold time τ cases are given by The expressions in (23) can then be used by the CR nodes to dynamically adopt the spectrum sensing time T w , given the hardware capability to adopt its time to sense, to reliably detect the PU, and at the same time save power. The temporal characteristics of the PU that is required to compute T min w may be known a priori or be acquired by learning the environment at the CR node.  an example, continuing form Example 1 in Section 6, for a WiMedia-UWB based CR to detect a WiMax terminal with 90% confidence and the minimum required time to sense the spectrum. Note that, again here we assume as explained in Example 1 that the WiMax transmission has a Poisson arrival process. Figure 12 shows the curves of T min w B w /W for 90% confidence in detecting the PU for both constant and random EURASIP Journal on Wireless Communications and Networking  τ considering only the SoS of the PU (without noise). The figure shows the curves with respect to the arrival rate λ for various values of mean hold times. From the results; we observe that the minimum time required to sense the spectrum increases when the arrival rate (λ) is low and as well as when the holding time is low. In the figure, we also show the operational region (for T w > T min w ) and the inoperational region (for T w < T min w ) of the CR nodes to reliably detect the PU (in the absence of noise). We further observe from the figure that the minimum sensing time T min w reduces significantly when the hold time increases. Though the results shown in Figure 12 is for the noiseless case, it holds true for relatively high values of γ. Similar analytical curves also can be generated for the noisy case by following the above procedure.

Conclusion
In this paper, we presented some detailed theoretical analysis on the performance of detecting a narrow band PU by a wideband CR terminal. The performance analyses were based on noisy sensing at the CR node as well as on the temporal characteristics of the PU. Closed form expressions were presented for the probabilities of detection, miss detection, and false alarm at the CR node, and were also verified using simulations. Further, we classify different risk regions for miss detecting the PU based on the temporal characteristics of the PU transmission, and consequently derive the minimum required spectrum sensing time to reliably detect the PU with a given confidence level. The analyses and the results were presented for both constant and random spectrum holding time, as well as random spectrum idle time considering the Poisson arrival process. Further research is to be conducted for optimizing the detection threshold for envelope-(energy-) based detection depending on the temporal characteristics of the PU together with the sensing noise. From the analytical framework that is provided in this paper, application (traffic) specific temporal (empirical) model for PU transmissions can also be used to perform similar analysis.

Introduction
Recently, there has been a growing interest in signal detection in the context of Cognitive Radio [1], and more specifically in that of opportunistic radio, where secondary Cognitive Radio Networks (CRNs) can be operated over frequency bands allocated to some primary system in so far as this primary system is absent or, in a more general case, whenever harmful interference with primary systems can be avoided. In most cases, the presence of the primary system is assessed through direct detection of its communication signal, although beaconing is sometimes considered [2]. Thus, in many situations, the primary system detection problem is transposed to the problem of detecting a communication signal in the presence of noise. Surveys of signal detection in the context of spectrum sensing have been proposed in the literature [3,4]. These detectors operate according to the a priori knowledge they have about the signal and the model of this signal. Telecommunication signals are modulated by sine wave carriers, pulse trains, repeated spreading, hopping sequences, or exhibit cyclic prefixes. Thus, these signals are characterized by the fact that their momentum (mean, autocorrelation, etc.) exhibits periodicity. This builtin periodicity, which of course is not present in noise, can be exploited to detect signals in the presence of noise even at a low Signal-to-Noise Ratio (SNR) [5]. Using this model, the signal detection process becomes a test for presence of cyclostationary characteristics of the tested signal [6][7][8].
Many scenarios have been investigated in the context of CRN over the past years. The two most likely to occur in the short term are, on the one hand, the unlicensed usage of TV bands and, on the other hand, the opportunistic use of unlicensed bands by nonlegacy secondary systems. The first scenario, often referred to as the TV White Space (TVWS) scenario, was made possible by the FCC in the US in 2008, with some restrictions which include high-sensitivity requirements for primary user detection [9]. In the context of this scenario, standardization has been very active, especially under the IEEE802.22 banner [10]. Industry fora, like the White Space Coalition, have given more momentum to this option. The second scenario is, for obvious regulatory reasons, the first that can be practically experimented and used [11].
In this context, implementation of blind cyclostationarity detectors has been proposed. In [12], a detector based on Cyclostationary Spectrum Density (CSD) is suggested. The CSD theoretically makes it possible to explore the presence of cyclic frequencies for any autocorrelation lag at any frequency (also referred to as 2D CSD). However, the comprehensive 2D CSD is never implemented in practice due to its huge implementation cost. To sort out this issue, 1D CSDs are preferred to limit implementation cost. The CSD can be performed on the time domain autocorrelation [13,14], or through the analysis of signal periodicity redundancy in the frequency domain [15]. In both cases however, a large FFT operator (512 to 2048) needs to be implemented, leading to significant hardware complexity. The approach described hereafter goes one step further in narrowing down the CSD domain. Indeed, in both scenarios of interest, the primary systems (which are the ones requiring the highest detection sensitivity) are known. Therefore, analysis of the primary signal nature helps narrow down the CSD search to very specific cyclic frequencies, thereby avoiding implementation of a large FFT.
However, when the CSD is narrowed down, the algorithm becomes more specific to the signal to detect. For this reason, this paper will analyze two different implementation options depending on the aforementioned scenarios. The main reason justifying different types of implementation in the WiFI and TVWS scenarios is the sensitivity level required in each case. In the case of TVWS, the guarantee that secondary CRN will not interfere with licensed systems (TV, microphones) leads to high-sensitivity requirements. On the other hand, unlicensed band networks, such as IEEE802.11x, have lighter coexistence constraints. These specific requirements lead to architectural tradeoffs which are examined in this paper. First, the principle of prefix-based cyclostationarity detection will be recapped. Then, the two aforementioned scenarios will be analyzed by pinpointing their impact on the sensor requirement. Considering these requirements, two hardware implementation architectures will be described and evaluated. These approaches will be compared and discussed before concluding the paper.

Cyclostationarity Detector for OFDM Signal
In both scenarios considered in this paper, in the primary system-DVB-T broadcast system on the one hand, IEEE802.11a/g networks on the other-the signal is modulated using Orthogonal Frequency Digital Multiplexing (OFDM); see, for example, [16]. The OFDM signal is a compound signal consisting of multiple frequency carriers, also called subcarriers or tones, that are each modulated in phase or in phase and amplitude. From a practical outlook, the modulated tones are multiplexed at the transmitter using an inverse FFT. Conversely, the subcarriers are demultiplexed at the receiver end by an FFT. The size of the FFT N, which defines that of the OFDM symbols, depends on the system. In the case of IEEE802.11a/g systems, 64 subcarriers are used whereas the DVB-T signal uses 1024, 2048, 4096, or 8192 tones. In order to avoid intersymbol interference, a Guard Interval (GI) is introduced. In the case of OFDM, this GI is designed as a copy of the last samples of the OFDM symbol. This approach provides the symbol with a cyclic nature which simplifies the receiver. For this reason, this D long GI is called the Cyclic Prefix (CP).
Let us now consider the autocorrelation of this signal, Under the condition that all subcarriers are used, the autocorrelation of an OFDM signal is written as [17] R y (u, m) = R y (u, 0)δ(m) + R y (u, N)δ(m − N) The first term corresponds to the energy of the signal. Energy detectors, which analyze this term only, provide poor performance at low SNR. Therefore, we focus on the two other terms, which stem from the repetition of the cyclic prefix present at the beginning and the end of each symbol. It can be shown that the term R y (u, N) is a periodic function of u [8] which characterizes the signal y. R y (u, N) has a period of α −1 = N + D. This cyclostationary nature of the signal is illustrated in Figure 1. Thus, R y (u, N) can be written as a Fourier series [5]: In (3), each Fourier coefficient R kα y (N) is the cyclic correlation at frequency kα at time lag N. This term is also written as which can be estimated as follows: The basic idea behind the cyclostationarity detector is to analyze this Fourier decomposition and assess the presence of the signal by setting a cost function related to one [18] or more [19] of these cyclic frequencies. This cost function is compared to some reference value. Several papers related to this algorithm have been proposed in the literature [17,[19][20][21]. They mainly differ in the way the harmonics are considered. In this paper, we consider the cost function suggested in [17]. By introducing the oversampling rate of T c /T e and by considering N b harmonics, this cost function can be derived from (5) as follows:   It can be observed that the cost function is only built upon R kα y (N) while R −kα y (−N) is omitted. Indeed, it is fairly easy to prove that for all k, R kα y (N) = R −kα y (−N) * (where * denotes the complex conjugation).

Cyclostationarity Detector Architecture for WiFi Signals
The cyclostationarity detector for IEEE802.11a/g signals is specified considering the scenario presented in [22]. In this scenario, the detector is used to check the presence of WiFi signals in order to trigger data transmission from a secondary system which is completely independent from the primary system (no messaging exchanged, no synchronization performed). Besides, in order to achieve the highest spectrum efficiency, the secondary system is expected to exploit time gaps (opportunities) in the time domain rather than to leave the channel to find a vacant one. Although this strategy may lead to some collisions, it is found acceptable due to the nature of the primary (unlicensed system) and in so far as the impact is not significant at application level [22]. Focusing on the design of the cyclostationarity detector, this scenario leads to the requirements of a low-latency detector. Detector latency directly impacts the duration of the time gaps that will be exploited by the secondary system. When the primary system is bursty, which is the typical nature of WiFi traffic, latency should be far shorter than the gaps between two consecutive bursts. The need for low latency calls for a parallel approach in which the Fourier coefficients are computed at the same time. Such a structure is described in Figure 2.
This architecture is directly derived from (6). The top left corner block computes one single observation of the  autocorrelation function. Each grey block then computes the Fourier coefficients in parallel. Each of these branches is accumulated over the observation time U before being aggregated by the sum blocks on the right of the figure.
The first point to consider when designing a parallel architecture is to analyze how many branches need to be instantiated. In other words, how the cyclostationary detector performs according to N b . For this purpose, the probability of detection is computed as a function of the SNR under AWGN conditions for various N b values. Other parameters are kept constant. For instance U is set to a large value (U = 1000). The results obtained are provided in Figure 3. For Figures 3-6, 1000 independent iterations have been carried out to build the curve.
Selecting N b = 0 corresponds to considering the fundamental frequency only, which is equivalent to performing energy detection. Detector performance is maximized for a small N b value, which implies that performance can be maximized for a limited hardware complexity. Aggregating harmonics still further causes performance to decrease since high harmonics, of low amplitude, are strongly impacted by noise. This shows that performance can be optimized with respect to N b while preserving a limited hardware complexity.
Another important parameter for the detector is the size of the integration window U (where U denotes the number of OFDM symbols considered for integration). Although this parameter has a more limited impact on hardware complexity (only the accumulators are slightly larger), U has a strong influence on latency, another major requirement in the scenario. As expected, increasing U does indeed improve performance significantly as shown in Figure 4.
Limiting detector latency while preserving performance of long observation time is possible by trading U against  T c /T e (F e = 1/T e = 20 MHz). Oversampling is expected to have a similar influence on performance in that it increases U, except for the fact that T c /T e cannot be reasonably increased to a similar extent to U. Therefore, whenever latency is not critical, increasing U should be considered. Besides, increasing T c /T e directly impacts the length of the delay line of the correlator, as well as the look-up tables used for storage of the sine waveforms, whereas U had only a slight impact on the complexity of the accumulators in each branch. Thus, priority should be given to increasing U in so far as detector latency fits into the latency specification. In the case of the WiFi detector, 5 OFDM symbols correspond to a latency of 20 μs. Additional performance can then be ensured by a reasonable increase in T c /T e to limit the additional complexity drawback. Figure 5 shows the influence of T c /T e.
Finally, the last parameter that needs to be determined is W, the width of the binary word representing the I/Q input data. Assuming that the full dynamic range is preserved throughout the architecture, it is obvious that this parameter will significantly impact hardware complexity. However, the impact on detector performance is less obvious, and some simulations must be quantified. These simulation results are provided in Figure 6. Figure 6 shows that near optimal performance can be obtained where W = 4. However, to preserve some additional margin, a value of 8 is preferred, with rescaling after each macro block to guarantee a good performance/complexity tradeoff. With these parameters, detector overall latency has been measured at 40.5 μs.
Finally, the complexity of detector hardware implementation is determined on a Xilinx Virtex 4 target technology using the ISE XST synthesis tool. Results are provided in Table 1 when the following parameter values are considered:

Cyclostationarity Detector Architecture for DVB-T Signals
In the same way as IEEE802.11a/g, the physical layer of DVB-T is based on an OFDM modulation. However, some key elements differ from WiFi systems. First, the DVB-T standard defines four FFT sizes: N = 1024, 2048, 4096, or 8192, and F e = 8 MHz. The cyclic prefix over FFT size ratio D/N can also vary: 1/32, 1/16, 1/8, and 1/4. However, in practice, implementation considers a smaller set of parameters depending on the country. For instance, in France, the set of parameters used is N = 8192, D/N = 1/32. Another key difference, which will be exploited in the architecture design, stems from the broadcast nature of the DVB-T signal. This means that detector sensitivity can be increased significantly by very long integration time which cannot be considered in the case of short signal bursts occurring in WiFi. This is, of course, a relevant feature since sensitivity requirements for primary user detection are very demanding (typically SNR = −10 dB, to which an additional margin for detector Noise Figure must be added [23]).
Another point derived from the broadcast nature of the signal is the way the reference signal used to define the decision threshold is computed. When undertaking this calibration phase, the secondary system needs to consider a reference value which is independent from signal presence. When considering long (ideally infinite) integration time, the autocorrelation function R y (u, N) defined in Section 2 tends to a rectangular signal as depicted in Figure 1, the cyclic ratio of which is D/(N + D). In this case, the Fourier coefficient is written as  Each coefficient power is given by It is obvious from (8) that R kα y (N) = 0 whenever kD/(N + D) is an integer value. This holds for instance when Figure 7 plots the Fourier coefficients of a rectangular signal when N/D = 32. It can therefore be concluded that Fourier harmonic 33 is not impacted by the presence of the signal and can thus be used for calibration purposes to define the reference noise level. As a comparison, calibration based on input power computation (i.e., (1/U) U−1 u=0 |y(u)| 2 ) is not relevant as this estimator is strongly impacted by the presence of the signal. When considering the first 4 harmonics [−3; +3], a decision variable V can be expressed as follows: Of course, this technique holds for infinite integration time to guarantee the rectangular shape of the autocorrelation  estimator ( Figure 1). Whenever a finite integration is performed, the convergence of the integrator needs to be considered. The integrator is a first order IIR filter, the z transform of which is given by where n can be tuned to adjust the raising time of the filter. Indeed, the indicial response of the filter is given by The raising time k r (in number of symbols) to reach 90% is then given by For large n values, the expression in (13) tends to 2.3n. Estimator performance is increased by increasing the integration ability of the filter. This is, however, at the cost of long integration time. Thus, this approach is to be considered for "always on" kind of systems, such as DVB-T broadcast signals to guarantee reliable detection under low SNR-conditions. Figure 8 shows the decision variable V as a function of the input SNR (under AWGN conditions) for several values of n. The area before the curve corresponds to a 0.5 detection probability and must be avoided. The aim of the curve is to show how increase in integration time impacts the performance of the detector for a given threshold value. For instance when an SNR of −7 dB is targeted and for a threshold set to 15, no detection is possible when considering n = 32. However, when n is set to 128, a reliable behavior is achieved. Setting n to 64 results in nonreliable decisions. From this graph, a trade-off between SNR detection condition and integration time can be set.
Detection and probability detection curves based on real signal measurements will be provided in a future paper. However, in order to evaluate a first implementation of the detector, parameter values used for the WiFi case were considered as an initial assumption. Finally, the cyclostationary detector hardware architecture for DVB-T is shown in Figure 9. First, the autocorrelation is computed on the I/Q complex samples. The IIR integrator then averages over a number of symbols tuned by setting the integration time parameter to achieve the required sensitivity. The supervisor, a Finite State Machine (FSM), then triggers the writing into a buffer that stores 8 k filter output samples (equivalent to the length of an OFDM symbol). Then, using a faster clock, the Fourier harmonics are computed sequentially. Unlike parallel computation over distinct instances in the architecture of Section 2, parallelism is achieved here using a faster clock and some control mechanisms provided by the FSM, even though latency constraints are not as critical as in the first case study. The sine generator computes sequentially the required sine function of the Fourier taps of interest. The Multiply ACcumulate (MAC) function enables the Fourier coefficient to be obtained for these taps. The sequence is as follows. First, the reference harmonics {−33; +33} are generated to compute the noise reference power. Then, the harmonics of interest for the DVB-T signal {0; −1; +1; −2; +2; −3; +3} are calculated. The power of each harmonic is summed up to obtain the cyclostationarity estimator value. Finally, the decision engine gives the final result by comparing the estimated value to the threshold value according to (10), which provides a hard decision output of the detector. Finally, the complexity of detector hardware implementation is determined on a Xilinx Virtex 5 target technology using the ISE XST synthesis tool. Results are provided in Table 2.

Conclusion
This paper presents 2 cyclostationarity detectors targeting different scenarios. It is shown in the paper that selection of the scenario has a strong influence on architecture and its performance tradeoffs. First, when aiming at secondary usage of ISM bands with time leftover reuse, latency is the key parameter. With this architecture, latency as low as 40.5 μs was measured. Besides, the cyclostationary detectors of this paper outperform classical energy detectors in terms of probability of detection (e.g., Pd is increased by 0.4 where SNR = −17 dB in the WiFi case). This has led to a parallel design in which sensitivity is traded against low latency as collisions with the primary system may be tolerated. On the other hand, when considering secondary spectrum usage of licensed bands, collisions are not permitted and much attention must be paid to sensitivity. This is achieved through long integration time which relies on the assumption that the signal is either "always on" or absent. This assumption makes the second architecture ideally suited to broadcast signal detection (e.g., DVB-T), but would be inapplicable to the first scenario.

Introduction
Many studies have shown that the static frequency allocation for wireless communication systems is responsible for the inefficient use of the spectrum [1]. This is so because the systems are not continuously transmitting. Cognitive Radios (CRs) networks try to make use of the gaps that can be found in the spectrum at a given time. This opportunistic behavior categorizes CR as secondary users of a given frequency band, by contrast with the systems that were permanently assigned this band (primary users) [2]. For the CR concept to be viable, it is required that it does not interfere with the primary user services. It means that the system must be able to detect primary user signals in low signal-to-noise ratio (SNR) environments fast enough. Efforts are being made to improve the performance of the detectors [3].
A radiometer (also called energy detector) can be used to detect completely unknown signals in a determined frequency band [4]. It is historically the oldest and simplest detector, and it achieves good performance when the SNR is strong enough. Unfortunately, since it is based on an estimation of the in-band noise power spectral density (PSD), it is affected by the noise level uncertainty (due to measurement errors or a changing environment), especially at low SNR [5], where it reaches an absolute performance limit called the SNR wall. Another type of detector is based on the spectral redundancy present in almost every manmade signal. It is called a cyclic feature detector and will be the kind of detector of interest in this paper.
Cyclic feature detectors make use of the cyclostationarity theory, which can be divided in two categories: the secondorder cyclostationarity (SOCS) introduced by Gardner in [6][7][8] and the higher-order cyclostationarity (HOCS) introduced by Gardner and Spooner in [9,10]. The SOCS uses quadratic nonlinearities to extract sine-waves from a signal, whereas the HOCS is based on nth-order nonlinearities. The idea behind this theory is that man-made signals possess hidden periodicities such as the carrier frequency, the symbol rate or the chip rate, that can be regenerated by a sine-wave extraction operation which produces features at frequencies that depend on these hidden periodicities (hence called cyclic features and cycle frequencies resp.). Since the SOCS is based on quadratic nonlinearities, two frequency parameters are used for the sine-wave extraction function. The result is called the spectral correlation density (SCD), and can be represented in a bifrequency plane. The SCD can be seen as a generalization of the PSD, as it is equal to the PSD when the cycle frequency is equal to zero. Therefore, the SOCS cyclic feature detectors act like energy detectors, but at cycle frequencies different from zero. The advantage of these detectors comes from the absence of features (at least asymptotically) when the input signal is stationary (such as white noise), since no hidden frequencies are present, or when the input signal exhibits cyclostationarity at cycle frequencies different than the one of interest. The HOCS cyclic-feature detectors are based on the same principles, but the equivalent of the SCD is a n-dimensional space (n > 2). Like SOCS detectors, HOCS detectors have originally been introduced in the literature to blindly estimate the signal frequency parameters.
It has been shown that the second-order cyclostationarity detectors perform better than the energy detectors in low SNR environments [7], and this has recently triggered a lot of research on the use of cyclostationarity detectors for spectrum sensing in the context of cognitive radios [11,12]. However the second-order detectors suffer from a higher computational complexity that has just become manageable. First field-programmable gate array (FPGA) implementations are presented in [13,14].
Higher-order detectors are generally even more complex, and since the variance of the features estimators increases when the order rises, most research results concern secondorder detectors. We will nevertheless demonstrate that it is possible to derive fourth-order detectors that bear comparable performances to second-order ones to detect linearly modulated baseband signals at SNR around 0 dB. The paper will include a mathematical analysis of the detection algorithm, the effects of each of its parameters and its computational complexity. Performance will be assessed through simulations and compared with the second-order detector.
After introducing the system model in Section 2, we will briefly review the basic notions of cyclostationarity theory in Section 3 in order to understand how second-order detectors work and identify their limitations. Afterwards, we will move on to HOCS theory, and present its most relevant aspects in Section 4, which will be used to characterize the linearly modulated signals in Section 5 and to derive an algorithm that may be used for signal detection of linearly modulated signals in Section 6. We will conclude by a comparison of the new detector performance with second-order detector and energy detector performances in Section 7.

System Model
This paper focuses on the detection of linearly modulated signals, like pulse amplitude modulation (PAM) or quadrature amplitude modulation (QAM) signals. The baseband transmitted signal is usually expressed as where I m is the sequence of information symbols transmitted at the rate F s = 1/T s and p(t) is the pulse shaping filter (typically a square-root Nyquist filter). After baseband-toradio frequency (RF) conversion, the RF transmitted signal is given by: where ω c = 2π f c and f c is the carrier frequency. In the PAM case, the symbols I m are real and only the cosine is modulated. In the QAM case, the symbols I m are complex and both the cosine and sine are modulated. A QAM signal can be seen as two uncorrelated PAM signals modulated in quadrature.
For the sake of clarity, we assume that the signal propagates through an ideal channel. Our results can nevertheless be extended to the case of multipath channels, if we consider a new pulse shape that is equal to the convolution of square-root Nyquist filter with the channel impulse response. However, this would make the new pulse random. Simulations have shown that both second-order and fourth-order detectors are affected in the same way by a multipath channel (equivalent degradation of performances). Therefore it does not seem critical to introduce multipath channels in order to compare the two, and it allows us to work with a constant pulse shape. Additive white Gaussian noise (AWGN) of onesided PSD equal to N 0 corrupts the signal at the receiver. Some amount of noise uncertainty can be added to N 0 . The detection of the signal at the receiver can be either done directly in the RF domain or in the baseband domain after RF-to-baseband conversion.

Second-Order Cyclostationarity
Two approaches are used to introduce the notion of cyclostationarity [8]. While the first approach introduces the temporal features of cyclostationary signals, the second approach is more intuitive and is based on a graphical representation of spectral redundancy. Both approaches lead to the same conclusion. This section reviews the main results of the second-order cyclostationarity theory, which will be generalized to higher-order cyclostationarity in the next sections.

Temporal Redundancy.
A wide-sense cyclostationary signal x(t) exhibits a periodic autocorrelation function [6,7] where E[·] denotes the statistical expectation operator. Since R x (t, τ) is periodic, it can be decomposed in a Fourier series where the sum is over integer multiples of the fundamental frequencies. The coefficient R α x (τ) is called the cyclic autocorrelation function, and represents the Fourier coefficient of the series given by EURASIP Journal on Wireless Communications and Networking 3 When the signal is cyclo-ergodic, the expectation in the definition of the autocorrelation can be replaced by a time average so that The cyclic autocorrelation is therefore intuitively obtained by extracting the frequency α sine-wave from the time-delay product We notice that the only cyclic frequencies α for which the SCD will not be null are the ones corresponding to the Fourier coefficients.

Spectral Redundancy.
Let X( f ) be the Fourier transform of x(t). The SCD measures the degree of spectral redundancy between the frequencies f − α/2 and f + α/2 (α being called the cyclic frequency). It can be mathematically expressed as the correlation between two frequency bins centered on f − α/2 and f + α/2 when their width tends toward zero [6,7] S α where X T (t, f ) denotes the short-time Fourier transform of the signal Since the SCD depends on f and α, it can be graphed as a surface over the bifrequency plane ( f , α). When α = 0, the SCD reduces to the PSD.

Baseband and RF Second-Order Features.
The performance of the cyclic feature detectors will first depend on the strength of the features they are trying to estimate. The two most common features exploited to detect the linearly modulated signals are linked with the symbol rate and the carrier frequency.
(i) The symbol rate feature is usually exploited after RFto-baseband conversion at the receiver. As its name suggests it, it originates from the symbol rate at the transmitter. Since this is a discrete signal, its frequency spectrum is periodic, with a period equal to the inverse of the sample rate (which is equal to the symbol rate before RF conversion). If there is some excess bandwidth in the system, or in other words, if the pulse shaping filter p(t) does not totally cut off the frequency components larger than half the inverse of the symbol rate, some frequencies will be correlated, as shown in Figure 1.
(ii) The doubled-carrier frequency feature is directly exploited in the RF domain. It is based on the symmetry of the RF spectrum, and it is much Figure 1: Baseband signal frequency spectrum (top) and SCD at the symbol rate (bottom). The frequency spectrum results from the repetitive discrete signal spectrum and the filter shaping. The SCD is measured by the correlation between two frequency bins centered on f − α/2 and f + α/2 where α is the symbol rate. The symbol rate feature exists for baseband PAM/QAM signals if there is some excess bandwidth in the system. stronger than the symbol rate feature (it is as strong as the PSD). Since it depends on the symmetry of the spectrum of the baseband signal, it only exists if the modulation used has no quadrature components. If a real PAM scheme is used, the carrier feature exists, as illustrated in the left part of Figure 2. If a complex QAM scheme is used, the carrier feature vanishes, as illustrated in the right part of Figure 2.
Since complex modulations are quite common, it would not be possible to implement a cyclic feature detector for CRs based on the doubled-carrier frequency feature. On the other hand, the symbol rate feature solely depends on the pulse shaping filter. Provided that there is some excess bandwidth, the symbol rate feature will exist, whatever the modulation. Unfortunately, that feature is relatively small and depends on the amount of excess bandwidth. We can therefore ask ourselves if it would not be possible to find greater features using a fourth-order detector.

Higher-Order Cyclostationarity
The higher-order cyclostationarity (HOCS) theory is a generalization of the second-order cyclostationarity theory, which only deals with second-order moments, to nth-order moments [9,10]. It makes use of the fraction-of-time (FOT) probability framework (based on time averages of the signals) which is closely related to the theory of high-order statistics (based on statistical expectations of the signals), by ways of statistical moments and cumulants. This section reviews the fundamentals of the HOCS theory and highlights the metrics that can be used for spectrum sensing.

Lag-Product.
We must always keep in mind that the goal of the HOCS theory is to extract sine-waves components Figure 2: RF signal frequency spectrum (top) and SCD at twice the carrier frequency (bottom). The SCD is measured by the correlation between two frequency bins centered on f − α/2 and f + α/2 where α is the carrier frequency. The doubled-carrier frequency feature exists for RF PAM signals as the baseband frequency spectrum exhibits a correlation between negative and positive frequencies.
In the absence of any filtering, this correlation produces a symmetric frequency spectrum (left part). The doubled-carrier frequency feature vanishes for RF QAM signals as the baseband frequency spectrum is uncorrelated (right part).
from a signal, in which they are hidden by random phenomena. To extract, or regenerate, these frequencies, a nonlinear operation must be called upon. The second-order theory uses the time-delay product L(t, τ) = x(t) · x * (t − τ) which will be transformed in the autocorrelation after averaging. A natural and intuitive generalization of this operation to the nth-order is called the lag-product and can be expressed as [9]: where the vector τ is composed of the individual delays τ j ( j = 1, . . . , n). The notation x ( * ) (t) indicates an optional conjugation of the signal x(t).

Temporal Moment Function and Cyclic Temporal Moment
Function. If the signal possesses a nth-order sine-wave of frequency α, then the averaging of the lag-product, multiplied by a complex exponential of frequency α, must be different from zero [9]: Obviously, R α x (τ) n is a generalization of the cyclic autocorrelation function described in (5). It is called the nthorder cyclic temporal moment function (CTMF). The sum of the CTMF (multiplied by the corresponding complex exponentials) over frequency α is called the temporal moment function (TMF) and is a generalization of the autocorrelation function described in (3): Each term of the sum in (12) is called an impure nthorder sine-wave. This is so because the CTMF may contain products of lower-order sine-waves whose various orders sum to n. In order to extract the pure nth-order sine-wave from the lag-product, it is necessary to subtract the lowerorder products. The pure nth-order sine-wave counter-part of the CTMF, denoted by C α x (τ) n , is called the cyclic temporal cumulant function (CTCF). The pure nth-order sine wave counter-part of the TMF, denoted by C x (t, τ) n , is called the temporal cumulant function (TCF).

Temporal Cumulant Function and Cyclic Temporal Cumulant Function.
The CTMF and TMF have been computed by using the FOT probability framework. In order to compute the CTCF and TCF, it is interesting to make use of the equivalence between the FOT probability framework and the high-order statistics theory. More specifically, the paper [9] demonstrates that the TMF of a signal can be seen as the nth-order moment of the signal, and that the TCF of a signal can be seen as the nth-order cumulant of the signal (hence their names). By using the conventional relations between the moments and the cumulants found in the high-order statistics theory, the TCF takes therefore the form: (13) where {P} denotes the set of partitions of the index set 1, 2, . . . n (10), p is the number of elements in the partition P, and R x (t, τ j ) nj is the TMF of the jth-element of order n j of the partition P. The CTCFs are the Fourier coefficients of the TCF and can be expressed in terms of the CTMFs: (14) where {β} denotes the set of vectors of cycle frequencies for the partition P that sum to α ( p j=1 β j = α), and R βj x (τ j ) nj is the CTMF of the jth-element of order n j of the partition P at the cycle frequency β j .
The CTCF is periodic in τ: C α x (τ +1 n φ) n = C α x (τ) n e j2πφα (1 n is the dimension-n vector composed of ones, meaning that φ is added to all elements of τ). Therefore, it is not absolutely integrable in τ. To circumvent this problem, one dimension is fixed (e.g., τ n = 0), and the CTCF becomes: This function is called reduced dimension-CTCF (RD-CTCF). It is the key metric of the ensuing algorithms for HOCS detectors. It should be noted that the equivalent exists for the CTMF and is called the RD- However the RD-CTMF is generally not absolutely integrable.

Cyclic Polyspectrum.
The need for integrability comes from the desire to compute the Fourier transform of the RD-CTCF, which gives the cyclic polyspectrum (CP). The CP is a generalization of the SCD plane for cyclostationnary signals. However it is not necessary to compute the CP of a signal for sensing applications since detection statistics can be directly derived from a single slice of the RD-CTCF. For this reason, and the computational complexity gain, we will put the spectral parameters aside and devote our attention to the RD-CTCF.

Fourth-Order Features of Linearly Modulated Signals
We have previously talked about the second-order cyclic features for communication signals, and we saw that the carrier frequency features tend to vanish from the SCD plane if the modulation is complex. We also asked ourselves if a fourth-order transformation of the signal may suppress the destructive interferences of quadrature components of a signal. We now have to gauge the potential of these fourthorder features. In this section, we compute the RD-CTCF of the baseband and RF linearly modulated signals and identify the interesting features that can be used for signal detection.

Baseband Signals.
The TCF of the baseband signal (1) has been computed in paper [10]. The mathematical derivation results in: p t + mT s + τ j (15) in which C I,n is the nth-order cumulant of the symbol sequence I m : where {P n } is the set of partitions of the set {1, . . . , n}, p n is the number of elements in the partition P n , and n j is the order of the jth-element in the partition P n ( j = 1 · · · p n ). R I,n is the nth-order moment of the symbol sequence I m : The expression of the moment R I,n can be understood this way: given a particular type of modulation, do the symbol variables I k elevated to the power n (with optional conjugation specified by the operator ( * ) q ) gives a constant result? The answer to this question is helpful in assessing if a given signal may exhibit nth-order features and what kind of conjugation must be used in the lag-product (10). The appendix illustrates this result for the binary PAM and the quaternary QAM constellations (see also [10,15]).
Computing the Fourier transform of the TCF and canceling τ n reveals the RD-CTCF in the form of: where the cycle frequencies are integer multiples of the symbol rate (α = kF s with k integer). The RD-CTCF of the baseband signal is nonzero only for harmonics of the symbol rate. The amplitude of the features tend to zero as the harmonic number k increases.

RF Signals.
The RD-CTCF of the RF signal specified by (2) can be inferred from the RD-CTCF of the baseband signal s(t) by noting that the RF signal is obtained by modulating two independent PAM signals in quadrature. We need to calculate the CTCFs of PAM, sine and cosine signals, and to combine them using the following rules: (i) The cumulant of the sum is equal to the sum of the cumulants if the signals are independent. Therefore, if y(t) = x(t) + w(t) where x(t) and w(t) are two independent random signals, we have: and, after Fourier transform, we obtain: (ii) The moment of the product is equal to the product of the moments if the signals are independent. Therefore, if EURASIP Journal on Wireless Communications and Networking are two independent random signals, we have: and, after Fourier transform, we obtain: Equation (22) means that we have to multiply all CTMFs of x(t) and w(t) which sum to α. If one of the signals is nonrandom (w(t) in our case), the CTMF of the random signal can be replaced by its CTCF: The CTCFs of the baseband PAM signals can be computed using (18). The only difference with a QAM signal resides in the cumulant of the symbol sequence C I,n , which must be computed for PAM symbols through (16) and (17) (see the binary PAM case in the appendix).
The CTMF of the sine and cosine signals can easily be determined from the expression of their lag-products: where φ j = ω c τ j . The lag-product can be decomposed into a sum of cosine signals at various frequencies using Simpson formulas: for the second order, and: for the fourth order. It is clear that the CTMF of sine or cosine signals is made of Dirac's deltas at cycle frequencies 4 f c , 2 f c , and 0.
Since the real and imaginary parts of s(t) are two statistically independent PAM signals, the CTCF of s RF (t) is the sum of two CTCFs of modulated PAM signals in quadrature. The CTCFs of R[s(t)] and I[s(t)] are equal and denoted by C α PAM (τ) n in our next results. We can finally write: For n = 2, we observe the destructive interference between the components of R γ cos (u) 2 and R γ sin (u) 2 at twice the carrier frequency, as was introduced in Section 3.
For n = 4, we also observe that the components of R γ cos (u) 4 and R γ sin (u) 4 at twice the carrier frequency cancel out, just as they do for the second order. There only remain the features at zero and four times the carrier frequency: Since R 0 cos (u) 4 is a sum of cosines that depend on u and  4 (at least when u is null) and are therefore less suited for sensing scenarios.

Baseband and RF Fourth-Order Features.
We have to choose between baseband or RF signals and decide on the cycle frequency that will be used by the detector. We have seen that baseband QAM signals have features at the cycle frequencies that are multiples of the symbol rate (0, F s , 2F s . . .), whereas RF signals have additional features at cycle frequencies that depend on the carrier frequency It has been shown that these additional features are small and that the strongest feature for both baseband and RF signals is obtained when the cycle frequency α is equal to zero. Since noise signals do not have any fourth-order feature (the fourth-order cumulant of a Gaussian random variable is equal to zero), even when α = 0. Note that α = 0 is a degenerated cycle frequency, which is present even in stationary signals. However, since it gives the strongest 4th-order feature, it is the frequency that will be preferred for our sensing scenario, even if the denomination "cyclic-feature detector" becomes inappropriate in this case. Simulations made with baseband or RF signals for α = 0 have shown that the two detectors exhibit similar performances. From now on, we will focus on the fourthorder feature detection for baseband signals and let aside the fourth-order feature detection for RF signals, as it enables a significant reduction of the received signal sampling frequency. The feature obtained in this situation is illustrated in Figure 3.

Fourth-Order Feature Detectors
6.1. RD-CTCF Estimator. In order to estimate the RD-CTCF of the baseband QAM signal, we would have to use (14). Luckily, the signal is complex and the second order features disappear if we do not use any conjugation in the lag product (see the quaternary QAM example in the appendix). Therefore the RD-CTCF is equal to the RD-CTMF: In practice, the RD-CTCF is estimated based on a size-N finite observation window of the received sequence s[n] obtained after sampling the received signal. (30) with N > 2 max |l j | and l j are the elements of the discrete lag-vector l of size n − 1.

Noise Mean and Variance.
When there is only noise in the system, the mean of the RD-CTCF is equal to 0 since the fourth-order cumulant of a Gaussian random variable is null. On the other hand, the variance of the RD-CTCF is a function of the lag-vector given by:  (31) in which σ 2 n is the variance of AWGN noise samples at the input of the RD-CTCF estimator. Simulations illustrated in Figure 4 confirm the result (31). Every discrete lag-vector l for which two or more values l i , l j are identical should be avoided, since it increases the noise variance. However, to afford the luxury of choosing lag values that are different from zero, we would have to increase the sampling rate at the receiver, which in turn would increase the noise power. Simulations have shown that it is better to use the lowest sampling rate that still satisfies Shannon's theorem, and set all lag values equal to zero. The RD-CTCF variance also quite naturally decreases as the observation window N is increased.

Detector.
The detector has to decide between two hypotheses: hypothesis H 0 implying that no signal is present, hypothesis H 1 implying that the linearly modulated signal is present. The absolute value of the feature (here the RD-CTCF) is compared to a threshold γ to make a decision: The threshold is usually fixed to meet a target probability of false alarm (decide H 1 if H 0 ). In order to compute the threshold level as a function of the probability of false alarm, we must know the distribution of the RD-CTCF. We already know its mean and variance values and using the centrallimit theorem, we assume that the output distribution is Gaussian (see also [16]). As a consequence, the absolute value of the RD-CTCF takes the form of a Rayleigh distribution and the threshold level can be found using: where P f a is the probability of false alarm.

Detector Comparison
We will now briefly review the principles of all detectors previously mentioned in this paper, and compare their performance and computational complexity. We assume that second-order and fourth-order detectors work only at a single location of the feature they exploit (the second-order detector works at most favorable frequency, the fourth-order detector works at the most favorable value of the discrete lagvector l). Monte-Carlo simulations were used, each of which used 5000 iterations.

Energy
Detector. This is the most widely used detector in wireless communication systems. It averages the square modulus of the received sequence over time: Its advantages are its simplicity and its ability to perform blind detection (since it does not require any information about the signal it is trying to detect). Unfortunately, it has been demonstrated that it cannot be used in low-SNR environments due to its sensitivity to noise uncertainty [6].

Second-Order
Detector. This detector computes an estimation of the SCD by averaging, over time and frequency domains, the cyclic periodogram of the signal spectrum S k ( f ) computed for a finite time window at time k: where K is the number of time windows and F is the number of frequency bins. It is a much more complex and less efficient detector, which requires some characteristics of the signal in order to work (e.g., the symbol rate must be known in advance). Its advantage resides in the absence of features (at least asymptotically) when the input signal is a white noise, which results in the output mean of the detector always being equal to zero in presence of noise, therefore shielding the detector from noise uncertainty effects. Its computational complexity evolves as N · log 2 (1024) = N.10 if the FFTs used to evaluate the cyclic periodogram [6] have a length of 1024 samples, and the total number of samples is equal to N.

Fourth-Order
Detector. This detector averages the lagproduct of the received sequence over time: This detector is simpler to implement than the previous one (no Fourier transform of the signal is required since we work in the time domain), which results in a computational complexity evolving as N, the total number of samples. It benefits from the same immunity to noise uncertainty, and is therefore suited for operations at low SNR.

Performance Comparison.
We may now take a look at the performance of the different detectors. Figure 5 illustrates the probability of missed detection (decide H 0 if H 1 ) curves as a function of the SNR for the three detectors under consideration. The threshold has been set in the three cases to achieve a target probability of false alarm equal to 10 −1 . These curves have been obtained without adding any noise uncertainty to the signal. In such conditions, the energy detector is the optimal detector for blind detection, and can be considered as a reference. It appears that the secondorder detector and the fourth-order detector, have similar performances when the SNR is around zero dB: for the same complexity, (that leads to an observation time ten times longer for the fourth-order detector), both detectors exhibit the same probability of missed detection (roughly 1 percent) at an SNR of −0.8 dB. However, when we consider an SNR of −4 dB, the fourth-order detector requires much more samples, which makes it more complex than the secondorder. Besides, the detection-time constraints that are part of the cognitive radios reglementation would not be met if the observation time is too long. If we add some amount of noise uncertainty, the energy detector cannot perform reliable detections and must be discarded, whereas the cyclic feature detectors remain unaffected. In order to verify this assumption, we computed the receiver operating characteristics (ROC) curves of the fourth-order detector for two situations, one without any noise uncertainty, and one with 0 dB of noise uncertainty. The results are illustrated in Figure 6. We observe that the energy detector, which had the best ROC curve in the first case is a lot more affected by the noise uncertainty than the fourth-order detector. ROC curves for the second-order detector can be found in [7], and show the same immunity to noise uncertainty than the fourth-order.

Conclusion
This paper has started from the need for robust detectors able to work in low SNR environments. A brief review of the second-order cyclostationarity and second-order cyclic feature detectors has exposed the advantages and drawbacks of such detectors, and explained the intuition that lead to the study of higher-order cyclostationarity (HOCS). The main guideline is to identify features of sufficient strength and to design a detector able to extract it from the signal. The most relevant aspects of HOCS theory have then been analyzed and we have derived a new fourth-order detector that can be used for the detection of linearly modulated signals. Simulation results have shown that fourth-order cyclic feature detectors may be used as a substitute for second-order detectors at SNR around zero dB, which could be needed if the received signals do not exhibit second-order cyclostationarity.

A. Cumulants of the Binary PAM
This section computes the second-and fourth-order cumulants of a binary PAM sequence. The symbols take the values I m = {±1}. Since the binary PAM constellation is symmetric, only the first partition has a chance to give a product of moments different from 0. We will limit our investigations to the first partition. The partition {1, 2} gives R I,2 = 1 for its single element, so that C I,2 = 1. Since the 4-QAM constellation is symmetric, only the first partition has a chance to give a product of moments different from 0. We limit therefore our investigations to the first partition.
Different results are obtained according to the number of conjugations in the lag-product (10): (i) When no conjugation or two conjugations are used in the lag-product, the partition {1, 2} gives R I,2 = 0 for its single element, so that C I,2 = 0.
(ii) When one conjugation is used in the lag-product, the partition {1, 2} gives R I,2 = 1 for its single element, so that C I,2 = 1.

Introduction
Cognitive radio networks and, more generally, dynamic spectrum access networks are becoming a reality. These systems consist of primary nodes, which have guaranteed priority access to spectrum resources, and secondary (or cognitive) nodes, which can reuse the medium in an opportunistic manner [1][2][3][4]. Cognitive nodes are allowed to dynamically operate the secondary spectrum, provided that they do not degrade the primary users' transmissions [5]. From a practical viewpoint, this means that the secondary terminals must acquire a sufficient level of knowledge about the status of the primary network. This information can be gathered through the use of techniques such as energy detection [6], cyclostationary feature detection [7], and/or cooperative distributed detection [8]. Due to the complexity and drawbacks of the detection phase, the FCC recently issued the statement that all devices "must include a geolocation capability and provisions to access over the Internet a database of protected radio services and the locations and channels that may be used by the unlicensed devices at each location" [9]. Furthermore, the positions of the primary nodes and other meta-information can be shared in the same way. Though the locations of the nodes and their configurations can be obtained easily, the exploitation of such information remains an open problem. Considering that any diversity technique can be used by cognitive nodes, several approaches have been proposed to allow for the coexistence of primary and secondary networks [10]. These include, for example, the use of orthogonal codes (code division multiple access, CDMA) [11], frequency multiplexing (frequency division multiple access, FDMA), directional antennas (spatial division medium access, SDMA) [12], orthogonal frequencydivision multiple access (OFDMA) [13], and time division multiple access (TDMA) [14], among others.
In this paper, we investigate a simple, yet powerful, diversity scheme by exploiting the polarimetric dimension [15][16][17]. More specifically, a dual-polarized wireless channel enables the use of two distinct polarization modes, referred to as copolar (symbol: ) and cross-polar (symbol: ⊥), 2 EURASIP Journal on Wireless Communications and Networking respectively. Ideally, cross-polar transmissions (i.e., from a transmitting antenna on one channel to the receiving antenna on the corresponding orthogonal channel) should be impossible. In reality, this is not the case due to an imperfect antenna cross-polar isolation (XPI) and a depolarization mechanism that occurs as electromagnetic waves propagate (i.e., a signal sent on a given polarization "leaks" into the other). Both effects combine to yield a global phenomenon referred to as cross-polar discrimination (XPD) [18][19][20].
The scenario of interest for this work is shown in Figure 1. The primary system consists of a single transmitter located at a distance of d 0 from its intended receiver. Without any loss of generality, the primary receiver is considered to be located at the origin of the coordinates system, leading to a receiver-centric analysis. The secondary (cognitive) terminals are deployed along with the primary ones. However, limitations on interference prevent them from entering a protected region around the receiver. This region, referred to as the "primary exclusive region" [21], is assumed to be circular and therefore, is completely characterized by its radius, denoted as d excl .
Since polarimetric diversity does not allow a perfect orthogonality between primary and secondary nodes' transmissions, its use is possible under the application of a socalled underlay paradigm [10,22,23]. This means that both cognitive and primary terminals carry out communications, provided that the capacity loss caused by cognitive users does not degrade communication quality for primary users. For this purpose, we can further characterize the underlaid paradigm by requiring that the primary system must be guaranteed a minimum (transmission) capacity during a large fraction of time. As will be shown, this can, in turn, be formulated as a probabilistic coexistence problem under the constraint of a limited outage probability in the primary network.
We argue that using the polarimetric dimension allows dynamic spectrum sharing to be efficiently implemented in cognitive systems. To this end, we propose a theoretical model of interference in dual-polarized networks and derive a closed-form expression for the link probability of outage. We theoretically prove that polarimetric diversity can increase transmission rates for the secondary terminals while, at the same time, can significantly reduce the primary exclusive region.
First, we validated the expected (theoretical) performance gains analytically. To the best of our knowledge, none of the past studies in literature has investigated the behavior of the XPD under a complete range of propagation conditions, such as indoor-to-indoor and outdoor-to-indoor. In particular, we conducted a vast experimental campaign to provide relevant insights on the proper models and statistical distributions which would accurately represent the XPD. Based on these measures, the achievable performance of these dual-polarized cognitive networks, considering both half-duplex and full-duplex communications, will be determined.
The medium access control (MAC) protocol considered is a variant of the slotted ALOHA protocol [24] such that in each time slot, the nodes transmit independently with a certain fixed probability [25]. This approach is supported by the observations in [26, page 278] and [25,27], where it is shown that the traffic generated by nodes using a slotted random access MAC protocol can be modeled by means of a Bernoulli distribution. In fact, in more sophisticated MAC schemes, the probability of transmission of a terminal's transmission can be modeled as a function of general parameters, such as, queuing statistics, the queue-dropping rate, and the channel outage probability incurred by fading [28]. Since the impact of these parameters is not the focus of the this study, for more details we refer the interested reader to the existing studies in the literature [29][30][31].
The remainder of this paper is organized as follows. In Section 2, we demonstrate how the polarimetric dimension increases spectrum-utilization efficiency and supports the coexistence of primary and secondary users in a probabilistic sense, which requires guaranteed capacity for the primary network. After these theoretical developments, several insights are presented to move from the concept to practical implementation. First, Section 3 presents an experimental determination of the main parameters used to characterize cognitive dual-polarized networks in indoor-to-indoor and outdoor-to-indoor situations. These results are then used in Section 4 for analytical performance evaluation. Section 5 concludes the paper.

The Dual-Polarized Cognitive
Network Architecture

Probabilistic Coexistence and Interference.
Consider the cognitive network shown in Figure 1 with two types of users: primary and secondary (cognitive). The primary network is supposed to be copolar and the cognitive network is cross-polar. Without cognitive users, the primary network would operate with background noise and with the usual interference generated by the other primary users. Let C p (dimension: [bit/s/Hz]) be the desired capacity for a user in the primary network (In this manuscript, bold letters refer to random variables). We impose that the secondary network operates under the following outage constraint on a primary user: where 0 < ε < 1 and C (dimension: [bit/s/Hz]) is a minimum per-primary user capacity. Equivalently, this constraint guarantees a primary user a maximum transmission rate of at least C for at least a fraction (1 − ε) of the time. Under the simplifying assumption of Gaussian signaling (Note that this assumption is expedient for analytical purposes. However, in the following the analytical predictions will be confirmed by experimental results.), the rate of this primary user can be written as a function of the signal-to-noise and interference ratio (SINR) as follows: Using (2) into (1) yields and, by introducing θ 2 C − 1, one has where P{SINR > θ} can be interpreted as the primary link probability of successful transmission for an outage SINR value θ. This value depends on the receiver's characteristics, modulation, and coding scheme, among others [32]. The SINR at the end of a primary link with length d 0 can be written as where P 0 (d 0 ) is the instantaneous received power (dimension: [W]) at distance d 0 , N 0 /2 is the noise power spectral density of the noise (dimension: [W/Hz]), B is the channel bandwidth, and P int is the cumulated interference power (dimension: [W]) at the receiver, that is, the sum of the received powers from all the undesired transmitters. We now provide the reader with a series of theoretical results, which stem from the following theorem.
Theorem 1. In a narrowband Rayleigh block-faded dualpolarized network, where nodes transmit with probability q on the copolar and the cross-polar channels, the probability that the SINR exceeds a given value θ on a primary transmission, given a fixed transmitter-receiver distance d 0 , N int copolar where P 0 is the transmit power, N 0 B is the average power of the background noise, θ is the SINR threshold, α is the path loss exponent, XPD 0 is the reference cross-polar discrimination of the antenna at a reference distance d ref , and G(d, d ref ) is a function that characterizes the polarization loss over distance.
Proof. We assume a narrowband Rayleigh block fading propagation channel. The instantaneous received power P(d) from a node is exponentially distributed [33] with temporalaverage received power E t [P(d)] = P(d) = P · L(d), where P denotes the transmit power and L(d) ∝ d −α is the path loss at distance d (it accounts for the antenna gains and carrier frequency). The received power is then a random variable with the following probability density function: In a dual-polarized system, the cross-polar discrimination (XPD) is defined as the ratio of the temporal-average power emitted on the cross-polar channel and the temporal-average power received in the copolar channel [15], that is, where d is the transmission distance, is the temporal-average value of the instantaneous leaked power P (⊥ → ) (d), and P ⊥ (d) E t [P ⊥ (d)] is the temporal-average value of the instantaneous crosspolar power P ⊥ (d). In a generic situation, the XPD is subject to spatial variability [19] and, therefore, in the context of this network-level analysis, we define the XPD in a spatialaverage sense, that is, where the operator X denotes the average of the value X computed on multiple different locations at the same distance d. Note that, even though the XPD is considered here in a spatial-average sense, it is possible to accommodate its expected variability for the purpose of ensuring a required minimum cross-polar discrimination. This will be detailed 4 EURASIP Journal on Wireless Communications and Networking in Section 4. Finally, it is shown in [17][18][19], that XPD(d), defined according to (9), can be expressed as follows: where XPD 0 ≥ 1 is the XPD value at a reference distance d ref and the function G(d, d ref ) ≤ 1 characterizes the depolarization experienced over the distance. Let the traffic at the N int primary and N ⊥ int cognitive interfering nodes be modeled through the use of independent In other words, {Λ i } and {Λ j } are sequences of independent and identically distributed (iid) Bernoulli random variables: if, in a given time slot, one of these indicators is equal to 1, then the corresponding node is transmitting; if, on the other hand, the indicator is equal to 0, then the node is not transmitting. We also assume that the traffic distribution is the same at all interfering nodes of the network, that is, for all i, P{Λ i = 1} = q and for all j, P{Λ j = 1} = q, which is supported by the analyses presented in [25,27,34]. The overall interference power at the receiver is the sum of the interference powers due to copolarized and cross-polarized (leaked because of depolarization) interference powers, that is, } are the (instantaneous) interfering powers at the receiver. The probability that the SINR at the receiver exceeds θ can thus be expressed as follows: where in the second passage, we have exploited the fact that, in a Rayleigh faded transmission, the SINR is also exponentially-distributed [33]. Since all terminals have an independent transmission behavior and are subject to noncorrelated channel fading, that is, j }, and {Λ j } are independent sets of random variables, it then holds that The generic first expectation term at the right-hand side of (13) can be expressed as follows: The generic second expectation term in (13) can be expressed, by using (8), in a similar way: By plugging (14) and (15) into (13), one finally obtains expression (6) for the probability of successful transmission.
Theorem 1 gives interesting insights on the expected performance in a dual-polarized transmission subject to background and internode interference. First, the leftmost term of the expression at the right-hand side of (6) is relevant in a situation where the throughput is limited by the EURASIP Journal on Wireless Communications and Networking 5 background (typically thermal) noise. In large and/or dense networks, the transmission is only limited by the interference and one can focus on the interference and polarization terms (i.e., the two other term of the expression, assuming N 0 B is negligible). The first exponential term can be easily evaluated if N 0 B / = 0. The second and the third terms of expression (6) relate to the interference generated by the surrounding nodes transmitting in co-and cross-polarized channels. These terms depend on (i) the polarization characteristics of the interfering nodes, (ii) the traffic statistics, and (iii) the topology of the network. Note that the impact of the topology has been largely investigated in [35] and we will limit our study to the effect of polarization.
Finally, channel correlation is neglected here, as often in the literature, for the purpose of analytical tractability and because these correlations do not change the scaling behavior of link-level performance. For the sake of completeness, we note that in [36] an analysis of the impact of channel correlation is carried out. The authors conclude that, when the traffic is limited (q < 0.3), the assumption of uncorrelation holds. On the other hand, when the traffic is intense (q ≥ 0.3), the link probability of success is higher in the correlated channel scenario than in the uncorrelated channel scenario.

Probabilistic Link Throughput.
Referring back to our definition of the probabilistic coexistence of the primary and secondary terminals in (1), a transmission is said to be successful if and only if the primary terminal is not in an outage for a fraction of time longer than (1 − ε), that is, if the (instantaneous) SINR of the cognitive terminal is above the threshold θ. Therefore, we denote the probability of successful transmission in a primary link as P s , that is, The probabilistic link throughput [37] (adimensional) of a primary terminal is defined as follows: (i) in the full-duplex communication case, it corresponds to the product of (a) P s and (b) the probability that the transmitter actually transmits (i.e., q); (ii) in the half-duplex communication case, it corresponds to the product of (a) P s , (b) the probability that the transmitter actually transmits (i.e., q), and (c) the probability that the receiver actually receives (i.e., 1 − q).
The probabilistic link throughput can be interpreted as the unconditioned reception probability which can be achieved with a simple automatic-repeat-request (ARQ) scheme with error-free feedback [38]. For the slotted ALOHA transmission scheme under consideration, the probabilistic throughput in the half-duplex mode is then τ (half) q(1 − q)P s and in full-duplex case τ (full) qP s .

Properties and Opportunities of Polarization Diversity.
Theorem 1 expresses a network-wide condition to support the codeployment of primary and cognitive terminals. In order to implement polarization diversity and make it work, proper considerations have to be carried out. In this section, we propose several lemmas, all derived from Theorem 1, that allow to design and operate dual-polarized systems.

Lemma 2.
In a dual-polarized system subject to probabilistic coexistence of primary and secondary networks, relocating a cognitive terminal from the copolar channel to the cross-polar channel increases its probability of transmission while keeping intact the transmission capacity of the primary network.
Proof. Let us consider a scenario with a single interferer located at distance d and transmitting with power P. For the ease of understanding, let us assume that if the terminal uses a polarized antenna, its probability of transmission will be denoted as q = q ⊥ , whereas if a classical (not dual-polarized) scenario is considered, then q = q .
If the cognitive terminal is using the copolar mode, the probabilistic coexistence condition (1) can be written as whereas if the cognitive terminal is using the cross-polar mode, it holds that Therefore, the maximum acceptable probability of transmission in the copolar mode is Note that, on average, XPD(d) ≥ 1 according to definition (8) and for physical reasons-the power leaked on the copolar dimension is at most equal to the power transmitted on the cross-polar channel. Finally, all other quantities in (19) are strictly positive and, therefore, one obtains that where the right-hand side expression for q ⊥ max derives directly from (18). Therefore, the thesis of the lemma holds. Lemma 2 indicates that polarization can be exploited as a diversity technique. Indeed, the achievable transmission rate will always be increased if the secondary network uses a polarization state that is orthogonal to that of the primary network and, furthermore, this remains true regardless of the values taken by the other system parameters (e.g., transmission power, acceptable outage rate ε, SINR value, etc.).

Lemma 3.
There exists a region of space, referred to as the primary exclusive region, where the cognitive terminals are not allowed to transmit and can be reduced by means of polarimetric diversity. Proof. As previously anticipated in Section 1, the primary exclusive region is completely characterized by the primary exclusive distance d excl , that is, the minimum distance at which a cognitive terminal has to be, with respect to a primary receiver, so that it does not impact the capacity of the primary user (in a probabilistic sense) [21]. Starting from (6), in the presence of a single cross-polar interferer, one can write This relation is equivalent to where the definition at the right-hand side allows to express the minimum distance d excl as a function of the distance d 0 and the other main system parameters as follows: Therefore, since α ≥ 2, using polarization diversity, that is, In Figure 2, the normalized primary exclusive distance, defined as d excl /d 0 , is shown, as a function of the terminal probability of transmission q, with ε = 0.2. It can be observed that in the case without polarization, one always has d excl d 0 , that is, the cognitive terminals must be located outside the transmission zone defined by the primary emitter-receiver distance. On the opposite, it is possible to operate a cognitive terminal inside this region (i.e., with d excl < d 0 ) when the polarimetric dimension is used. Furthermore, in both cases the exclusive distance increases as a function of the terminal probability of transmission but its gradient is smaller in the dual-polarized case.
It is interesting to observe that relation (21) can also be used to parameterize practical realizations of the antennas. Indeed, it yields that Therefore, the quantity at the right-hand side of (25) represents the minimum amount of XPD that the antenna of the cognitive terminal must possess. This value depends on the network configuration but also on the propagation environment (through the depolarization function G (d, d ref )).

Lemma 4.
If q < ε, polarization diversity is not required to achieve a probabilistic coexistence.
Proof. As previously introduced, the coefficient XPD 0 is greater than or equal to 1. Therefore, the minimum value of XPD 0 to guarantee error-free transmissions on the crosspolar channel is In (25), all quantities are greater than zero. Therefore, if q < ε, the quantity q − ε is always negative and the solution of (26) is XPD 0 = 1.
Lemma 4 indicates that, if the desired throughput remains limited, then the outage is guaranteed on the primary system without summoning up the diversity of polarization on the secondary terminal. Therefore, the crosspolar channel can be kept available for other terminals that may require higher data rates. This can be observed in Figure 2.

Theorem 5. Besides being limited by probabilistic coexistence considerations, there exists an optimum probability of transmission by a terminal in the primary network, denoted as q opt , that maximizes the throughput.
Proof. Let us define the optimal user probability of transmission as where the probabilistic throughput τ has been defined in Section 2.2. We first focus on half-duplex systems, using polarization diversity: in this case, the link throughput is τ = q(1 − q)P s . Since ln(·) is a monotonically increasing function, finding the maximum of τ is equivalent to finding the maximum of ln(τ), that is, In order to find the maximum, we compute the partial derivative of ln(τ) with respect to q: where By using the approximation (This approximation is accurate for 0 < q < η j /3, which is always verified since d j and XPD 0 need to be kept high because of the probabilistic coexistence constraint.) ln(1 + x) ≈ x and setting ∂ ln(τ)/∂q = 0, one has The positive solution of this equation is given by which is the probability of transmission that maximizes the throughput. The same derivation can be applied in the case of a full duplex system and leads to the solution q opt ≈ η. If the approximation ln(1+x) ≈ x is not used, then the optimal probability of transmission cannot be given in a closed-form expression but has to be numerically evaluated.
Obviously, the maximum value of q will be the minimum between (i) the optimum probability of transmission in a slotted transmission system (in a general sense), given by (32), and (ii) the maximum rate that can be achieved under the constraint of a probabilistic coexistence in (20). Therefore, before selecting its transmission rate, a cognitive terminal must evaluate these two quantities, on the basis of the available information stored in the databases (positions of the nodes, acceptable outage, etc.), and use the smallest one.
In Figures 3(a) and 3(b), the accessible and optimal terminal probabilities of transmission are presented as functions of d/d 0 , in the cases with (a) half duplex and (b) full duplex communications, respectively. In each case, two polarization strategies are considered: (i) no polarization and (ii) XPD 0 = 10 dB. The accessible regions are defined by means of the inequality (22). In particular, the leftmost border of each exclusive region, denoted as line q excl , is defined as the probability of transmission for a terminal at the boundary of the primary exclusive region, that is, with d = d excl .
From these figures, it can be observed that the probability of transmission of dual-polarized cognitive systems is mainly limited by the interference bound imposed to protect the primary system. In fact, the transmission rate of the terminals will nearly always be lower than the optimal transmission rate, except when the cognitive terminal is distant. In that specific case, the optimum probability of transmission (20) in the accessible region (in a probabilistic sense) saturates, that is, it reaches q opt ≈ 1/2 in the half-duplex case and q opt ≈ 1 in the full-duplex case. Note that these values correspond to the maximum achievable throughput observed in any half-duplex or full-duplex system. Indeed, the definitions of the probabilistic link throughput are τ (half) q(1 − q)P s and τ (full) qP s and the corresponding optimum terminal probabilities of transmission cannot exceed q = 1/2 and q = 1, respectively.
In the scenarios where polarimetric diversity is exploited, this crossover distance is smaller (d excl /d 0 ≈ 1.5) than in the classical case (d excl /d 0 ≈ 3.3). Comparing the results in Figure 3(a) with those in Figure 3(b), another observation can be carried out. In the half-duplex case, for each distance d > d excl , the optimal transmission probability q opt lies inside the accessible region. In other words, q has to be properly selected to maximize the throughput. In the fullduplex case, q opt ≈ 1 everywhere in the exclusive region. These observations will be confirmed by the results presented in Section 4.
Finally, it is confirmed that, in the accessible regions, one either has (i) (d j /d 0 ) α 1 with XPD 0 > 1 (i.e., q/η j 1) or (ii) q opt 0.3. Therefore, the approximation used in proof of Theorem 5 (i.e., ln(1 + x) ≈ x) holds and the value of q opt derived in Theorem 5 can be considered as an accurate approximation of the true value.

Considerations for Practical System Implementation.
In the previous subsections, we have shown that the capacity of a primary user can be guaranteed, while, at the same time, allowing efficient spectrum access, if the polarimetric dimension is exploited. Moreover, dual-polarized terminals will benefit from an increase of capacity by means of a higher transmission rate and reduced terminal-to-terminal interference. The efficiency of polarization diversity depends on the cross-polar discrimination of the antennas in use. More precisely, the value of the initial cross-polar discrimination (i.e., XPD 0 ) has to be as high as possible; yet, the XPD of well-designed antennas is typically on the order of 10 ÷ 20 dB [15,39], which allows a significant discrimination between copolar and cross-polar channels. Depending on the achievable value of XPD 0 , the outage rate of a primary terminal, and the location of the terminals, the transmission rate of a cognitive terminal can be adapted taking into account the relations (20) and (32). Finally, the primary exclusive region can be determined by means of (22) and notified to the cognitive terminals which, in turn, can use it as a constraint.

Experimental Determination of the Indoor-to-Indoor and Outdoor-to-Indoor XPD
Several previous works have been undertaken in order to model the XPD for different kinds of environment. In [20], a theoretical analysis is conducted for the small-scale variation of XPD in an indoor-to-indoor scenario and it is concluded that it has a doubly, noncentral Fisher-Snedecor distribution.  A mean-fitting (i.e., the pathloss) model of XPD as a function of the distance in an outdoor-to-outdoor scenario was studied in [16,19]. The corresponding performance is analyzed in [11].
In this paper, we provide the reader with original measurements campaigns in both indoor-to-indoor and outdoor-to-indoor scenarios. Indeed, these correspond to real-life situations where various technologies, such as WiFi, sensor networks, personal area networks (indoor-to-indoor scenarios) or WiMax, public WiFi, and 3G systems (outdoorto-indoor scenarios) are in use.
We consider three generic models to describe the variation of the XPD with respect to the distance. For instance, when the transmission ranges are long (several hundreds of meters or a few kilometers), the best expression for the path loss function is where β is a decay factor (0 < β ≤ 1). On the other hand, when distances are small (tens of meters) or in indoorto-indoor scenarios, the XPD value, in decibels, decreases linearly with respect to the distance. In other words, one can write which corresponds, in linear scale, to the following path loss function: Finally, in some indoor scenarios where the transmission distances are small, it was observed that the XPD remains constant, that is, In the remainder of this section, we characterize the applicability of the three XPD models just introduced. In other words, we consider an experimental setup and, on the basis of an extensive measurement campaign, we determine which XPD model is to be preferred in each scenario of interest (indoor-to-indoor and outdoor-to-indoor).

Setup.
The measurements were performed using a Vector Signal Generator (Rohde & Schwarz SMATE200A VSG) at the transmitter (Tx) side and a Signal Analyzer (Rohde & Schwarz FSG SA) at the receiver (Rx) side. The Tx chain was composed of the VSG and a directional antenna. The Rx antenna was a tri-polarized antenna, made of three colocated perpendicular antennas. Two of these antennas were selected to create a Vertical-Horizontal dual-polarized antenna. The three receiver antennas were selected one after another by means of a switch and were connected to the Signal Analyzer through a 25 dB, low-noise amplifier. The Rx antennas were fixed on an automatic positioner to create a virtual planar array of antennas. A continuous wave (CW) signal at the frequency of 3.5 GHz was transmitted and the corresponding frequency response was recorded at the receiver side. The antenna input power was 19 dBm.
The measurement site was the third floor of a building located on the campus of Brussels University (ULB) and referred to as "Building U." In the outdoor-to-indoor case, shown in Figure 4(a), the transmitter was fixed on the rooftop of a neighboring building (referred to as "Building L"), at a height of 15 m and was directed toward the measurement site. A brick wall was separating the lineof-sight (LOS) direction between this measurement site and the transmitter. The measurements were performed in a total of 78 distinct locations and in seven successive rooms. The rooms were separated by brick walls and closed wooden doors. The distance between the transmitter and the measurement points was in the range between 30 m and 80 m. In the indoor-to-indoor case, shown in Figure 4(b), the Tx antenna was fixed in the first room and was directed toward the seven next rooms, in which 65 measurement points were considered. The distance between the transmitter and the measurement points was in the range between 8 m and 55 m. In order to characterize the small-scale statistics of XPD a total of 64 spatially separated measurements were taken at each Rx position and in an 8 × 8 grid. The spacing between grid points was λ/2 = 4 cm. At each grid point, 5 snapshots of the received signal were sampled and averaged to increase the signal-to-noise-ratio.

Experimental Results and Their
Interpretation. The analysis of the collected experimental results has shown that the values of the XPD, for a given distance, present a locationdependent variability. Therefore, in the following figures, where the XPD is shown as a function of the distance d, the average value is shown along with the 1σ and 2σ being confidence intervals. Since the spatial variations were found to be Gaussian, these intervals account for 68% and 95% of the observed sets, respectively.
The horizontal polarization was first used in an indoorto-indoor scenario and is reported in Figure 5. It was observed that the XDP can be accurately modeled by means of the propagation model G 2 (d, d ref ) where one has XPD 0 = 11.3 dB, d ref = 1 m, and γ = 0.16 dB/m. The variation around the average value was also analyzed and the corresponding cumulative distribution function (CDF) is shown in Figure 6. This variation was found to fit with a zeromean Gaussian random variable with standard deviation equal to 0.295 dB. It is interesting to note that, unlike the case of the outdoor-to-outdoor scenarios presented in [19], the behavior of the XDP depends on the initial polarization of the antenna. More precisely, the results in Figure 6 correspond to a horizontal polarization while the results in Figure 7 correspond to an initial vertical polarization. It can be seen that, in the latter scenario, the XPD is almost constant and equal to XPD 0 = 4 dB. In this case, the XPD variability can be modeled as a zero-mean Gaussian random variable with standard deviation equal to 2.75 dB.
Finally, the results collected in an outdoor-to-indoor scenario are presented in Figure 8. As expected, the XPD is a decreasing function of the distance and is suitably modeled by using the propagation model G 2 (d, d ref ), with XPD 0 = 12.87 dB, d ref = 20 m, and γ = 0.13 dB/m. The spatial variability can be modeled as a zero-mean Gaussian random variable with standard deviation equal to 2.95 dB. Note that full de-polarization occurs after a hundred of meters and the two initial polarizations (i.e., horizontal and vertical) lead to the same behaviour.

Numerical Performance Evaluation
In this section, a numerical analysis of the performance of the proposed dual-polarized cognitive systems is presented. In Section 3, it has been shown that the XPD experiences spatial shadowing: more precisely, at a fixed distance different values of the XPD can be observed at different locations. The system parameters for performance analysis are selected by taking into account this normal fluctuation. Therefore, instead of using the average value for XPD 0 , it is preferable to use a value (denoted as XPD min 0 ) that can be observed with a confidence equal to a predefined value δ ∈ (0, 1). Taking into account that XPD 0 has a Gaussian distribution, it follows that (37) where μ and σ are the average value and the standard deviation of the observed XPD 0 , respectively. Therefore, XPD min 0 can be expressed as For instance, if a confidence level of 80% is required (i.e., δ = 0.8), one has to select XPD min 81σ. This approach will be used to set the initial parameters in the following performance analysis.

Full Duplex Systems in an Outdoor-to-Indoor Scenario.
Cellular system typically corresponds to an outdoor-toindoor scenario. Examples include WiMax base stations or cellular mobile phone systems. A typical scenario is presented in Figure 9. Referring to the experimental results presented in Section 3, we used in our simulations the model Two different polarization strategies are investigated: (i) the primary and the cognitive networks do not use polarimetric diversity (this scenario is referred to as no polarization) and (ii) the systems reduce their interference by using two orthogonal polarization states (this scenario is referred to as full polarization).
In Figure 11, the performance of full duplex systems is presented. More specifically, in Figure 11(a), the throughput of the system is shown as a function of the terminal probability of transmission. It can be seen that the throughput of the dual-polarized system is significantly higher, particularly when the probability of transmission is high. In Figure 11(b), the corresponding link probability of success in the primary network is investigated. It can be seen that it confirms the conclusions of Lemma 2: for a given minimum value of the link probability of success, the achievable transmission rate is significantly higher in the dual-polarized mode with respect to the value observed with the classical approach. For instance, with ε = 0.8, one has q max = 0.15 while, by using the dual-polarized approach, the maximum probability of transmission can be increased up to q max = 1.0. In other words, virtually any transmission rate is achievable with a limited impact on the primary system.   works (BANs). A typical scenario is presented in Figure 10. In our simulations, we considered a primary transmission at distance d 0 = 15 m and subject to interference from 5 terminals located at d = 25 m from the central base station.  In Figure 12, the performance of these half-duplex systems is presented. More particularly, in Figure 12(a), the throughput is shown as a function of transmission rate of the terminals, in a scenario with copolar interferers (i.e., without diversity of polarization) and under the dualpolarized scheme under study. It can be observed that the diversity of polarization drastically increases the throughput, even when the terminal probability of transmission is small. Regarding the probabilistic coexistence, in Figure 12(b) the link probability of success at the primary terminal is shown as a function of the transmission rate of the cognitive terminals. It can be observed that the use of polarization diversity gives a clear advantage in terms of interference limitation and available throughput for the cognitive terminals. For instance, with ε = 0.8 and a horizontal initial polarization, one has q max = 0.07 while, by using the dual-polarized approach, this quantity can be increased up to q max = 0.25 at each terminal. Finally, it can be seen that the optimum probability of transmission with XPD = 10 dB is approximately q opt ≈ 0.5, which matches with the value of q opt found in Figure 3(a).

Conclusions
In this paper, we have presented a novel theoretical framework to demonstrate the network-level performance increase that can be achieved in a polarimetric diversityoriented system subject to Rayleigh fading and probabilistic coexistence of primary and secondary (cognitive) networks. The theoretical approach was supported by an extensive measurement campaign. It has been shown that different mathematical expressions must be used in order to suitably model the dependence of the XPD on the distance between transmitter and receiver. These models depend not only on the scenario of interest, but also on the initial antenna polarization. For instance, in an indoor-to-indoor scenario,    we have observed that the horizontal polarization provides a significant diversity (XPD 0 around 10 dB) while the vertical polarization leads to a more limited gain (XPD 0 around 4 dB).
Our results suggest that dual-polarized networks are of interest, even if orthogonality (indicated by the XPD value) is limited. Indeed, with respect to the classical implementation of probabilistic coexistence of primary and secondary networks on the same (single polarization) channel, the use of polarization diversity allows to remarkably increase the perlink throughput and reduce the primary exclusive region. In some cases (i.e., at low transmission rates), it could even be possible to deploy a cognitive terminal closer to a primary receiver than the primary transmitter itself, that is, inside the primary exclusive region.

Appendix
The performance analysis carried out throughout the paper applies to networking scenarios with narrowband fading. In this appendix, we present a preliminary, yet insightful, extension of our approach to encompass the presence of wideband fading.
In the presence of a transmission channel experiencing wideband fading, the transmitted symbols of the considered packet suffer from interference of the other symbols that have been delayed by multipath [33]. This phenomenon is referred to as Inter-Symbol-Interference (ISI) and it depends on the channel model, modulation format, and symbol sequence characteristics, among others [40][41][42]. Therefore, the expression of the ISI is hard to obtain and typically is not in closed form. In the network-level approach, we follow in this paper, an approximation to SINR in presence of wideband fading can be obtained by treating the ISI as an additive, uncorrelated, Rayleigh-faded noise power proportional to the received power [41]. The expression of the link-level SINR introduced in (5) becomes where P ISI is noise power associated with the ISI. Its average value (noted P ISI = E[P ISI ]) is supposed to be proportional to the received power [41] and can be defined as Note that ν = 0 refers to the narrowband scenario. Theorem 1 can now be extended to incorporate the case of wideband Rayleigh fading as follows. The probability that the SINR at the receiver exceeds a given value θ is where the expectation of the term containing the ISI power becomes The definition of P ISI gives and, finally, Following the derivation outlined in the proof of Theorem 1, the link probability of successful transmission (A.3) in the wideband fading case finally becomes By comparing (A.7) with (6), it can be observed that the presence of wideband fading reduces the probability of successful link transmission by the factor 1/(1 + νθ). Since this factor is lower than 1 for ν ∈ (0; 1], it can be concluded that the presence of ISI has a negative impact on the link probability of outage. Moreover, for a given value of ν, that is, for a given level of ISI, the stronger this negative impact is, the higher is the considered SINR threshold θ. This, in turn, results in (i) an increase of the primary exclusive region (i.e., a reduction of the accessible region) and (ii) a degradation of system throughput. More precisely, in Figure 13(a) we clearly show the reduction of the comparison between the accessible transmission regions in the presence of narrowband fading (shown in Figure 3(a)) and in the presence of wideband fading (with P 0 /P ISI = 20 dB). As one can see, the presence of a limited ISI has a detrimental impact, significantly increasing the primary exclusive region. In Figure 13(b), the throughput in the presence of ISI is shown in a scenario with half-duplex communications. In this case as well, the negative impact of wideband fading is evident.
Although the impact of frequency selective fading is detrimental, from these figures it can be concluded that, even in presence of wideband fading channel, the use of polarimetric diversity significantly increases the overall performance of the whole system and is thus of interest in the context of cognitive radio networks.

Introduction
Cognitive radio networks are envisaged as the key technology to realize dynamic spectrum access (DSA). Such paradigm shift in wireless communications aims at solving the scarcity of radio spectrum [1][2][3][4]. The DSA concept proposes to boost spectrum utilization by allowing DSA users (SUs) to access the licensed wireless channel in an opportunistic manner so that interference to licensed users (PUs) is kept to a minimum. The idea of DSA is undoubtedly compelling and its realization will induce a huge advance in wireless communications. However, there are many challenges and open questions that have to be addressed before DSA networks become practically realizable [5,6].
To fulfill the requirement of minimum interference to PUs, a SU with an ongoing communication must vacate the channel when a licensed user is detected. The SU may switch to a different unused spectrum band which is referred to as spectrum mobility or spectrum handover (SH). If no available bands can be found or the SH procedure is not implemented, one or more SUs will be forced to terminate their sessions. From the user's perspective, it is generally assumed that the interruption of an ongoing session is more annoying than denying initial access [7]. Therefore, blocking the request of a new SU session, even if there are enough free resources, can be employed as a strategy to reduce the number of SU sessions forcedly terminated and the interference caused to PUs.
A variety of studies that focus on priority mechanisms to handle conventional handovers in cellular networks have appeared in the literature, see [8] and references therein. However, SH and conventional handover are different in nature and also from a modeling perspective.
In this paper, we focus on the study of the Quality of Service (QoS) perceived by PUs and SUs at the session level. We employ the same rather simple model than [9], which is enhanced to include an extension of the reservation scheme so that a noninteger number of channels can be reserved for SH. Such extension borrows the idea from the Fractional Guard Channel scheme that was introduced in cellular networks [10]. Furthermore, our numerical results for the system throughput are qualitatively different from those obtained in [9] leading to completely different conclusions, especially in what concerns the optimum system configuration.
Interference management has been identified as one of the critical challenges to make DSA networks work in practice [6]. Common DSA proposals take a reactive approach, in which SUs perform SH only after detecting PU interference. To detect PU activity in the same band, a SU must perform spectrum sensing, which requires to pause any ongoing transmission and causes a considerable performance penalty [6]. Additionally, SUs must execute spectrum sensing frequently to react quickly when a PU occupies the same band [11]. To handle both requirements, transmission and spectrum sensing episodes are typically interleaved in a cyclic manner [12,13].
We study the interference management problem from the traffic perspective. Our perception is that the mechanisms we propose might have a complementary role with respect to those defined at the physical layer. Our work is motivated by the fact that although simple spectrum access and channel repacking algorithms have been proposed in the classical communications literature their application to DSA systems has not been explored yet. In this paper, we assume that the primary network follows a predefined deterministic pattern when searching for free channels to set up a new session. The secondary network is aware of the rule followed by the primary network and uses this information in its own benefit but also in that of the primary network. The secondary network senses and assigns free channels to SUs in the reversed order that they will be occupied by PUs, hence reducing the probability of SUs having to vacate the assigned channel and causing interference to PUs. The probability of causing interference may be further reduced by performing a channel rearrangement to SUs after the release of channel. The mechanisms described above entail a minimal cooperation of the primary network, which in turn redound in a reduced interference for PUs. The idea of the primary network cooperating with the secondary one has also been proposed in [14].
We will show that both the forced termination probability and the interference created by the operation of SUs upon PUs can be controlled by limiting the access of SUs. This finding motivated us to design an admission control scheme for SUs that is able to limit simultaneously both the forced termination probability of SUs and what we define as the probability of interference. We show that both the forced termination probability and the interference caused to PUs are highly dependent on system parameters and on the arrival processes and service distributions. However, the proposed scheme is self-adaptive and does not require any configuration parameters beyond the targeted QoS objectives. Besides, it does not rely on any particular assumptions on the traffic characteristics; that is, it can operate with any arrival process and distribution of the session duration.
The rest of the paper is structured as follows. The different models of the systems studied are described in Section 2. In Section 3, we evaluate numerically the impact of incorporating admission control on the forced termination of SUs and also the impact of deploying channel allocation with preference and repacking on the interference. In Section 4, we propose and evaluate a novel adaptive admission control scheme that is able to limit simultaneously both the forced termination probability and the interference. Finally, Section 5 concludes the paper.

Model Description
We consider an infrastructure-based DSA network where PUs and SUs cooperate. Infrastructure-based DSA networks have been proposed in [2,6,15]. We assume that channels available for system operation are numbered according to the order in which they are assigned by the primary network; that is, we consider that to setup a PU session, the system searches from left (low-channel numbers) to right (high-channel numbers) until enough free channels can be allocated to the new session. Conversely, to setup a new SU communication the system searches from right (high-channel numbers) to left (low-channel numbers). We call this mechanism channel allocation with preference (CAP). Additionally, once a PU or a SU session has finished, a channel repacking of ongoing SU sessions can be performed to avoid interfering with future PU arrivals. Channel repacking can be triggered when, after a session completion, there exist ongoing SU sessions that can be moved to higher channel numbers; that is, there exist ongoing SU sessions that can perform a preventive SH to avoid creating future interference.
The system has a total of C resource units, being the physical meaning of a unit of resource dependent on the specific technological implementation of the radio interface. For the sake of mathematical tractability, we make the common assumptions of Poisson arrival processes and exponentially distributed service times. However, we also study the impact that distributions different than the exponential for the session lifetime have on system performance. The arrival rate for PU (SU) sessions to the system is λ 1 (λ 2 ), and a request consumes b 1 (b 2 ) resource units when accepted, b i ∈ N, i = 1, 2. For a packet-based air interface, b i represents the effective bandwidth of the session [16,17]. We assume that b 1 = N, b 2 = 1 and C = M × N, therefore the system resources can be viewed as composed by M = C/N bands for PUs or M × N subbands or channels for SUs. In other words, the maximum number of ongoing PU sessions is M and of SU sessions is M×N. The service rates for primary and secondary sessions are denoted by μ 1 and μ 2 , respectively.
We study seven different systems that can be aggregated into three groups. The characteristics of each of the seven systems are defined in Table 1. The second (SH), third (AC-FT) and sixth (AC-FT&I) columns refer, respectively, to spectrum handoff mechanism, the admission control (AC) scheme to limit the forced termination (FT) of SUs, and the adaptive AC scheme that limits simultaneously the forced termination probability perceived by SUs (P ft 2 ) and the interference caused to PUs. On these columns, a "Y" means that the systems implements the corresponding mechanism and a "N" that it is not implemented. The fourth (CA) and fifth (RP) columns refer, respectively, to the channel allocation, which can be either random ("R") or with preference ("P"); and the the repacking mechanism, which is either implemented ("Y") or not ("N").
In the following subsections, we introduce analytical and simulation models to study the systems described in Table 1. In Section 2.1 we present two continuous-time Markov chain (CTMC) models that define the operation of systems 1 (S1) and 2 (S2). The aim is to use these models to evaluate the effectiveness of AC to limit P ft 2 . Numerical results of this evaluation are shown in Section 3. In Section 2.2, we briefly outline two CTMC models that define the operation of systems 3 (S3) and 4 (S4). The aim is to use these models to compare the interference in a system deploying the proposed CAP and repacking schemes with the interference in the conventional random channel allocation scheme. Numerical results of this evaluation are also shown in Section 3. Finally, the model of the adaptive AC scheme deployed in system 5 (S5) and its evaluation is described in Section 4.

AC Scheme to Limit the Forced Termination of SUs.
We denote by x = (x 1 , x 2 ) the system state vector, when there are x 1 ongoing PU sessions and x 2 SU sessions. Let b(x) represent the amount of occupied resources at state x, b(x) = x 1 N + x 2 . The system evolution along time can be modeled as a multidimensional Markov process whose set of feasible states is We develop two analytical models to evaluate the performance of DSA systems measured by the forced termination probability of SUs.

System 1.
This first system is characterized by not supporting SH, deploying the Complete Sharing admission policy, that is, all SU requests are accepted while free resources are available, deploying a random channel allocation scheme with no repacking. A PU arrival in state x will force the termination of k SUs, k = 0, . . . , min(x 2 , N), with probability when k SUs are in the channels occupied by the newly arrived PU session, while the other (x 2 − k) are distributed in the other (M − x 1 − 1)N channels. Clearly, Let r xy be the transition rate from x to y, x, y ∈ S, and be e i a vector whose entries are all 0 except the ith one, which is 1, then min(x 2 , N), (4) Figure 1 shows the state diagram and transition rates of the CTMC that models the system dynamics. The global balance equations are expressed as where π(x) is the stationary probabilityof state x. The stationary distribution {π(x)} is obtained from (5) and the normalization equation. The blocking probability for SU requests, P 2 , and the SUs forced termination probability, P ft 2 , can be determined from the stationary distribution. Let us define Clearly, k(x) is the mean number of SUs that are forced to terminate upon the arrival of a PU in state x. Then, Note that P ft 2 is the ratio of the forced termination rate to the acceptance rate.
Finally, the SUs throughput, that is, the successful completion rate of SUs is determined by 2.1.2. System 2. This system is characterized by supporting SH, deploying the Fractional Guard Channel admission policy and deploying the random channel allocation scheme with no repacking. When a SU new setup request arrives and finds the system in state x, an admission decision is taken according to the number of free resource units available: where we denote by t ∈ [0, C], the admission control threshold; that is, the average number of resource units that must remain free after accepting the new SU requests must be t or higher. Clearly, these resources are reserved for SUs performing SH. Then, increasing t causes a reduction of the forced termination probability but, at the same time, increases the blocking probability perceived by new SU requests and vice versa. Note also that PUs are unaffected by the admission policy, as SUs are transparent to them. A PU arrival in state x will not force the termination of SUs when the system state complies with C − b(x) ≥ N, as the execution of SH will allow SUs to continue their ongoing session in a new unused channel, which are guaranteed to exist given the condition above. On the other hand, when C − b(x) < N, x 1 < M, a PU arrival will preempt b(x +e 1 )−C SUs. Let k(x) be the number of preemptions in state x, then Note that k(x) = 0 when C − b(x) > N, that is, it will be null for a high portion of the state space. As before, let r xy be the transition rate from x to y, x ∈ S, then min(x 2 , N), The coefficients a 1 (x) and a 2 (x) denote the probabilities of accepting a PU arrival and a SU arrival in state x, respectively. It is clear that a 1 (x) = 1, if x + e 1 − k(x), e 2 ∈ S, and 0 otherwise. Given a policy setting t, a 2 (x) is determined as follows: otherwise.
(13) Figure 1 shows the state transition rates of the CTMC that models the system dynamics. The stationary distribution, {π(x)}, is obtained by solving the global balance equations (5) together with the normalization equation. The blocking probability for SU requests, P 2 , the SUs forced termination probability, P ft 2 , and the SUs throughput, Th 2 , are then computed using (7), (8) and (9), respectively.
The analytical models described above have been validated through computer simulations. The simulation models we designed mimic the behavior of the physical system, in other words, the original system itself is simulated instead of simulating just the CTMC. Thus, the validation offers a guarantee on the correctness of the whole modeling process, and not only about the generation and solution of the global balance equations of the CTMC.

CAP Scheme to Limit the Interference Caused to PUs.
We assume that the SUs vacating rate induced by the arrival of new PU sessions is a measure of the interference caused by SUs to PUs, and we pursue to determine its value when deploying the spectrum access and channel repacking algorithms described in Section 1. Besides, we compare it to the one obtained when deploying the conventional random allocation scheme. A similar metric was used in [13] to measure the interference.
When the system supports SH the channel allocation and repacking algorithms have no impact on the performance perceived by the SUs; that is, their blocking and forced termination probabilities are not affected. Clearly, the finding of free channels by arriving or vacated SUs depends only on the number of ongoing PU and SU sessions and not on their physical disposition on the spectrum.
It should be noted that repacking for PUs is not considered. If the system deploys SH, CAP and repacking for SUs, doing repacking for PUs would only affect the algorithm followed to find a free channel upon the arrival of a SU, but not to the system performance (P ft 2 and interference). As described above, P ft 2 is not affected by the channel allocation and repacking algorithms used. In the same system, a PU arrival will experience interference when there are SUs occupying the PU band with the lowest order available. Clearly, this occurs when there are not enough free channels to accommodate the newly arrived PU without some SUs vacating the channel they are using (C − b(x) < N) then a previous repacking of PUs would have not helped.

System 3.
System 3b (3a) is characterized by supporting (not supporting) SH, deploying the Complete Sharing admission policy, deploying CAP and no repacking.
For the type of system under study, the state space of its CTMC model grows very quickly with the number of channels, as the state representation must describe not only the number of ongoing PU and SU sessions, but also the disposition of the allocated channels on the spectrum. More specifically, the number of states is (N + 2) M . This makes the solution of the CTMC intractable for any practical scenario.
Instead, we developed a simulation model and validated it with the analytical model of a simple scenario. This scenario has M = 2 bands for PUs and M × N subbands or channels for SUs. The set of feasible states is S := y = y 1 , y 2 : y 1 , y 2 ∈ {P, 0, . . . , N} , where y 1 (y 2 ) describes the state of the N leftmost (rightmost) channels. When y i = 0 the band is empty, when y i = P it is occupied by a PU, otherwise the number of SUs in the band can be y i = 1, . . . , N. The transition rates of the CTMC that models system 3b are displayed in Table 2. Note that, for example, at state (1, P), where there is one SU occupying one channel (out of N) in the first band of N channels and one PU occupying the second band, the actual channel allocated to the SU cannot be determined, but this information is irrelevant for the performance parameters of interest. When N = 2, the system has 16 states, independently of SH being supported or not.
As an example, for a system supporting SH and CAP, the vacating rate γ v and the forced termination rate γ ft can be determined from (15) and (16). The first term in (15) accounts for the contribution to the SUs vacation rate of states with no PUs in the system. In these states, a PU arrival will occupy the first band, vacating i SUs. The second and third terms account for the contribution of the states where there is a PU in the first or the second band, respectively. Then, the arrival of a new PU would vacate j or i SUs, respectively. The first term in (16) accounts for the contribution to the SUs forced termination rate of states with no PUs in the system. Note that if i SUs are found in the first band, the arrival of a PU will force the termination of one SU when there are N − i + 1 SUs in the second band, of two SUs when there are N − i + 2 SUs in the second band, and so on. The second and third terms clearly account for the contribution of states where there is a PU in the first and second band, respectively, To compare the results of the analytical and simulation models we selected three parameters: the blocking probabilities of PUs and SUs, and the forced termination probability of SUs. For both systems, with and without SH support, results clearly indicate a close agreement between the analytical and simulation models.

System 4.
This system is characterized by supporting SH, deploying the Complete Sharing admission policy, deploying CAP and repacking (CAP+RP).
Clearly, repacking can be triggered when either a PU or a SU leaves the system. Using the notation defined in the previous section for a system with M = N = 2, repacking would take place, for example, when a SU leaves from the upper band and the system state changes from (1,2) to (1,1). Note that as N = 2, a maximum of two SUs fit into the upper band. At this point, it is more convenient to move the SU in the lower band to the empty channel in the upper band, avoiding in this way future interference if a PU arrives. Then, repacking would make the system move from state (1, 1) to state (0, 2) instantaneously.
As in the previous section, we evaluate the system by simulation and validate the simulation model by a simple analytical model. For M = N = 2, the analytical model has 12 states, clearly less states than in a system without repacking, as now some states are not feasible, as shown in the previous example.
To compare the results of the analytical and simulation models we selected the same parameters of merit. Again, these results indicate an excellent agreement between the analytical and simulation models.

Effectiveness of the Proposed Mechanisms
In this section we evaluate the effectiveness of incorporating the Fractional Guard Channel admission policy to limit the P ft 2 , as well as the effectiveness of incorporating CAP and repacking to limit the interference caused to PUs.
Unless otherwise specified, the reference scenario for the numerical evaluation is defined by: M = 10, N = 8, C = M × N = 80, μ 1 = 1 and μ 2 = 1. In some scenarios, we consider that the load offered by PUs is such that their 6 EURASIP Journal on Wireless Communications and Networking blocking probability is P 1 = 0.01, which is achieved at λ 1 = 4.4612. Following common conventions, we do not specify the unit of the rates although typical values are expressed in s −1 . For the simulation result 95% confidence intervals are represented. The confidence intervals have been computed using 15 different simulation runs initialized with different seeds.

Effectiveness of AC to Limit the
The throughput achieved by SUs in systems 1 and 2 is shown in Figure 2, where we depict both the results of the analytical and the simulation models. Note the excellent agreement between the analytical and simulation results. Note also that the diameter of the confidence intervals are really small. This is the reason why confidence intervals will not be shown in the rest of the figures.
The authors of [9] suggest that a natural way of configuring a DSA system of similar characteristics to ours is to choose t for each SU arrival rate, such that the Th 2 is maximized. As observed in previous figures, it is not possible to determine an optimum operating point beyond the obvious one that is to deploy SH and t = 0. We believe that the role of reservation in DSA systems might be the same as its classical role in cellular systems; that is, to limit the forced termination probability of SUs. Note also that for the reservation values deployed, Th 2 is always higher when deploying SH and reservation than when not deploying SH. Deploying SH reduces the forced termination rate, which increases the successful completion rate.
One of the most interesting results of the study is the evolution of P ft 2 with the SUs arrival rate, which is shown in Figure 3. Observe that it seems to have a counterintuitive behavior. Intuitively, one would expect that P ft 2 should increase with the SUs arrival rate. However in a system without SH it has the opposite behavior. Note also that in a system with reservation, and particularly for some reservation values like t = 10 or higher, the forced termination first decreases, attaining a minimum, and then increases. The P ft 2 depends on the ratio of forced terminations to accepted sessions. By comparing the evolution of the forced termination rate with the SUs acceptance rate for the interval of arrival rates of interest (not shown here), these phenomena can be easily explained.
As expected, the P ft 2 can be controlled by adapting the threshold t according to the system traffic load.

Effectiveness of CAP and Repacking to Limit γ v .
To evaluate the effectiveness of CAP and repacking we obtained the evolution of the SUs vacating rate γ v with λ 1 in systems 2, 3a, and 4, when λ 2 = 20. We chose λ 2 = 20 as the P ft 2 is around 0.1 for a system with SH and λ 1 = 4.4612, which we consider a practical value. Recall that system 2 (S2) deploys the conventional random channel allocation algorithm, while systems 3a (S3a) and 4 (S4) deploy CAP and CAP and repacking (CAP+RP), respectively. To highlight the results of the study, we represent in Figure 4 what we define as the interference reduction factor; that is, the ratios Clearly, the proposed mechanisms are quite effective as they reduce the vacating rate induced by the arrival of PUs by approximately one order of magnitude or more for practical operating values. Note also that, as expected, the interference reduction factor is higher when repacking is used.

Adaptive Admission Control Scheme
In this section, we describe an adaptive admission control scheme that is able to limit simultaneously both the forced termination probability of SUs and the interference caused to PU communications by the operation of the SUs.
Our scheme generalizes a novel adaptive AC strategy introduced in [18] and developed further in [19], which operates in coordination with the well-known trunk reservation policy named Multiple Guard Channel (MGC). However, one of the novelties of the new proposal is that now the adaptive scheme is able to control simultaneously multiple objectives for the same arrival flow (SU arrivals), as opposed to only one objective per flow in previous proposals.
The definition of the MGC policy is as follows. One threshold parameter is associated with each objective. For example, in a system with two objectives, one for the P ft 2 and another for the interference. Let t ft , t if ∈ N be their associated thresholds. Then, a SU arrival in state x is accepted if b(x + e 2 ) ≤ t, t = min{t ft , t if }, and blocked otherwise. Therefore, t is the amount of resources that SUs have access to and decreasing (increasing) it reduces (augments) the acceptance rate of SU requests, which will in turn decrease (increase) both P ft 2 and the interference. Note that the definition of t in this section and in Section 2 are different.
For the sake of clarity, the operation of our scheme is described assuming that arrival processes are stationary and the system is in steady state. We denote by B ft 2 the objective for the forced termination probability perceived by SUs (P ft 2 ). In practice, we can assume without loss of generality that B ft 2 can be expressed as a fraction n ft /d ft , n ft , d ft ∈ N. When P ft 2 = B ft 2 , it is expected that, in average, n ft forced termination events and (d ft − n ft ) successfully completed SU session events, will occur out of d ft accepted SU session events. For example, if the objective is B ft 2 = 1/100, then n ft = 1 and d ft = 100. It seems intuitive to think that the adaptive scheme should not change t ft when the system is meeting its forced termination probability objective and, on the contrary, adjust it on the required direction when the perceived P ft 2 is different from its objective.
Given that the MGC policy uses integer values for the threshold parameters, to limit P ft 2 to its objective B ft 2 = n ft /d ft , we propose to perform a probabilistic adjustment in the following way.
(i) At the arrival of a PU, if it forces the termination of m SUs, do {t ft ← t ft − m} with probability 1/n ft .
(ii) When a SU session is accepted, do {t ft ← t ft + 1} with probability 1/d ft .
Intuitively, under stationary traffic conditions, if P ft 2 = B ft 2 then, on average, t ft will be increased by 1 and decreased by 1 every d ft accepted requests, that is, its mean value is kept constant.
We define a new measure for the interference by considering the fraction of PU arrivals that vacate exactly n SUs, n > 0, and denote it by P if (n). Let us denote its objective by B if (n) = n if n /d if n and the admission control threshold associated to it by t if n . Then, to limit P if (n) to its objective, we propose to perform the following probabilistic adjustment at the arrival of each PU. offered PU requests, that is, its mean value is kept constant. When the traffic is nonstationary, the adaptive scheme will continuously adjust the thresholds in order to meet the objectives if possible, adapting to any mix of traffic. Clearly, in the operation of this simple scheme no assumptions have been made concerning the arrival processes or the distributions of the session duration.
An important consequence of the definition of the interference probabilities {P if (n)} is that now we have the possibility to limit what we call the interference distribution. That is, we can define one objective for each of the elements of {P if (n)}, n = 1, . . . , N, or combinations of them, in order to give less importance (allow higher probabilities) to events that create lower interference (small values of n) and more importance (allow smaller probabilities) to events that create higher interference (high values of n). Figure 5 describes the procedure followed at a SU arrival to decide upon the acceptance or rejection of the new request. If the system defines multiple objectives for the interference and therefore manages multiple thresholds, then t if would be the minimum of all these thresholds.

Numerical Results.
The adaptive scheme has been evaluated in systems 5a and 5b by simulation. We used the parameter values defined in Section 3.
As an example, let us consider P if (n ≤ N) = N n=1 P if (n); that is, the fraction of PU arrivals that are interfered by SUs. Figure 6 shows the variation of P ft 2 and the interference with the SUs arrival rate when the objectives are B ft 2 ≤ 0.05 and B if (n ≤ N = 8) ≤ 0.1. As observed, the scheme is able to limit P ft 2 and P if (n ≤ N) to their objectives or below, and (2) Execute at every SU arrival: the interference is lower when repacking is used. Note that the limiting objective in both systems is B ft 2 , as P if (n ≤ N) remains below its objective. In other words, t ft is lower than t if (n ≤ N) in both systems for the load range considered. Note also that we have chosen a wide arrival rate range to show the effectiveness of the adaptive scheme. However, if the system does not reserve resources to accommodate SHs then P ft 2 > 0.05 even for small values of λ 2 . Figure 7 shows the variation of the SUs throughput with the SUs arrival rate. As a reference, we also plot the results obtained for systems 3a and 4. Recall that systems 3a and 5a do not support SH, deploy CAP but no repacking, while systems 4 and 5b do support SH, deploy CAP and repacking. However, S5a and S5b deploy the adaptive AC scheme, while S3a and S4 do not. We consider that system loads that make P ft 2 > 0.1 are of no practical interest. Although not shown, in systems 3a and 4, P ft 2 > 0.1 for λ 2 > 20. Then, restricting to the load range of interest for S3a and S4, Th 2 is higher in S5a and S5b than in S3a and S4. The improvement comes from the fact that limiting P ft 2 increases the rate of SUs that complete service successfully. As λ 2 keeps on growing, the blocking of SU setup  requests increases as the AC scheme must keep on limiting P ft 2 . This higher SUs blocking limits the SUs acceptance rate and therefore the growth of Th 2 .
As another example, let us consider P if (n ≤ 3) and P if (n > 3); that is, the fraction of PU arrivals that perceive low interference (n ≤ 3) and the fraction that perceive high interference (n > 3). objectives. However, for λ 2 > 20 the limiting objective in S5a

4.2.
Adaptivity of the AC Scheme. As discussed above, the adaptive scheme can operate with any arrival process and distribution of the session duration. As an example, we study in system 5a the adaptivity of the scheme to different distributions of the SUs session duration random variable (s 2 ). We consider three distributions: exponential (CV[s 2 ] = 1), Erlang (CV[s 2 ] < 1) and hyperexponential (CV[s 2 ] > 1). Please refer to any textbook, for example [20], for the definition of the probability density functions of these distributions. For an Erlang-k distribution with E[s 2 ] = 1/μ 2 , the standard deviation and the coefficient of variation are: σ 2 = 1/(μ 2 √ k) and CV[s 2 ] = 1/ √ k. We set k = 4 to obtain CV[s 2 ] = 1/2. We use a special type of a two stage hyperexponetial distribution that requires only two parameters (mean and standard deviation) for characterization [21]. The standard deviation is selected to obtain CV[s 2 ] = 2. Note that in our results we also vary the mean (E[s 2 ] = 1/μ 2 ), then the offered load (λ 2 /μ 2 ) is maintained constant to make results comparable.
To motivate the interest of deploying adaptive schemes, Figure 9 shows the variation of P ft 2 in system 3a. Note that both the CV and the mean of s 2 have a great impact on P ft 2 . In fact, in Figure 9 we get one order of magnitude variation in the values of P ft 2 for a constant offered load. The effectiveness of the adaptive scheme to cope with traffic having different characteristics is clearly shown in Fig that the proposed scheme is able to adapt and limit the forced termination and the interference under all conditions. In Figure 10, we observe that for μ 2 < 0.75 the limiting objective is B ft 2 , as the interference probabilities are below their objectives. However, for μ 2 > 0.75 this behavior is reversed. This is due to the fact that to meet one of the interference objectives the rate of admitted SUs into the system is reduced (the threshold is reduced), as observed in Figures 11 and 12. Note that a similar phenomenon was described in Figure 8 objective is B if (n ≤ 3), that is, the fraction of PU arrivals experiencing low interference (P if (n ≤ 3)) is at its objective or close, while the fraction experiencing high interference (P if (n > 3)) is considerably below its objective. Finally, if we compare Figures 9 and 10 we conclude that the operation of the adaptive scheme makes P ft 2 insensitive to the distribution of the SUs service time, which is an additional robustness advantage. A similar conclusion can be obtained for P if (n ≤ 3) and partially for P if (n > 3).

Conclusions
We studied the effectiveness of the Fractional Guard Channel admission policy to guarantee the QoS perceived by SUs, defined in terms of their forced termination probability. We modeled the system as a CTMC which was validated by computer simulation. Results showed that, contrary to what has been proposed, the throughput of SUs cannot be maximized by configuring the reservation parameter. We also showed that the probability of forced termination can be limited by setting appropriately the reservation threshold.
We also studied the QoS perceived by PUs, defined in terms of the interference caused to PU communications by the operation of SUs. We proposed and evaluated different mechanisms to reduce the interference based on simple spectrum access and channel repacking algorithms. In this case, to cope with the state explosion as the number of system channels grows, we resorted to simulation models that were validated by developing analytical models for systems of manageable size. We compared the interference in a system that uses the proposed mechanisms with the interference in a system that uses the common random access scheme. Numerical results showed that the interference reduction can be of one order of magnitude or higher when using the new mechanisms with respect to the random access case.
Finally, we proposed and evaluated a novel adaptive admission control scheme for SUs that is able to limit simultaneously the probability of forced termination of SUs and the interference. The operation of our scheme is based on simple balance equations which hold for any arrival process and holding time distribution. Our proposal has two relevant features, its ability to guarantee a certain degree of QoS for PUs and SUs under any traffic characteristics, and its implementation simplicity.

Introduction
Recently, there has been an increasing interest for small-cell networks. In fact, they have been recognized as an effective and low-cost architecture to provide wireless data rate access to Internet users [1,2]. These networks consist of numerous and densely deployed APs, known as outdoor femto cells or small-cells, connected to an existing backbone network with heterogeneous links, for example, fibers, ADSLs, and power lines. The general idea is to provide signal coverage and high data rates in dense environments, that is, areas with high user concentrations, by installing low-cost wireless access nodes and exploiting the existing heterogeneous wired infrastructures without a new high-cost cabling. In reality, the femto nodes may belong to different service providers eventually organized in coalitions to maximize their own revenues. In such a context, there is a critical trade-off between cooperation and competition among different providers who may share information and resources to maximize their own revenues. In order to enable both cooperation among providers and network scalability, the femto nodes need selforganizing mechanisms to perform communications and network control functions. Thus, distributed algorithms accounting for the revenues of different providers play a key role in this context.
In contrast to the legacy cell networks, in a small-cell network a user may be served by more than one femto node. This feature is strategic to cope with the heterogeneity of the core network. In fact, if a user were only connected to a single out-door femto-cell, it would suffer from low throughput from time to time due to the limited-backhaul capacity, despite the presence of a high-speed wireless link. As a result, users would access simultaneously to different femto-cells in order to aggregate the sum capacity of the backhaul links.
In this paper, we describe a small-cell network with N MTs served simultaneously by M femto nodes over N orthogonal channels, for example, FDMA, TDMA, and OFDM. For such a system, and we study the power allocation problem under the constraint of maximum transmit power at each femto node (The issue of load balancing [3] in the wired network, and how the different packets are split with respect to the backhaul capacity from a main decentralized scheduler, although important, is not investigated in this paper. We assume that perfect load balancing holds). This system is substantially different from the ones typically analyzed in literature. In fact, it does not reduce to a classical downlink of a cellular network modeled as a broadcast channel since there are several APs transmitting information simultaneously to the same MT. Nor does it reduce to N independent multiple access channels when considering each mobile as a receiver because of the power constraints at the APs. Finally, the considered system does not reduce to a multicellular or an adhoc network modeled as an interference channel since all the signals received at each MT carry useful information to be decoded. In this paper we assume that each signal of interest is decoded considering the remaining signals as interference. This scheme is susceptible to improvement by joint decoding of all the received signals. However, this decoding approach exceeds the scope of this paper.
In traditional wireless cellular networks, the power allocation is often implemented with centralized algorithms aiming at maximizing the sum of the Shannon transmission rate [4]. The maximization problem is solved by waterfilling algorithms [5][6][7][8] extended to multiuser contexts. The optimization is in general nonconvex but algorithms that reach local maximum are available [9][10][11]. Such a centralized power control scheme usually requires a unique shared resource allocation controller and complete channel state information (CSI) with consequent feedback and overhead. It is worth noting that this overload scales exponentially with the number of transmitters and receivers. Thus, such a fully centralized approach is not suitable for small-cell networks without centralized devices and with multiple service providers interested in their own revenues. Additionally, it is not scalable in dense networks.
Game theory [12] provides a possible analytical framework to develop decentralized and/or distributed algorithms for resource allocation in the context of interacting entities having eventually conflicting interests. Recently, noncooperative game theory and its analytical methodologies have been widely applied in wireless systems to solve communication control problems [13]. Distributed power allocation algorithms based on noncooperative games have been proposed for uplink single cell systems, that is, multiple access channels, and downlink multicellular networks or ad hoc networks, that is, interference channels. In [14], general results on potential games are provided and specialized to an uplink single-cell system with multiple access channel based on code division multiple access (CDMA). In [15], a digital subscriber line (DSL) is modeled as a multiple access system based on an OFDM scheme and an iterative waterfilling algorithm is proposed along the lines of the results in [16]. The classical uplink single-cell scenario is relaxed in [17] to include a jammer in the system and an iterative waterfilling algorithm is proposed.
In [16], power allocation on the interference channel is modeled as a noncooperative game, and the conditions for the existence and uniqueness of Nash equilibrium (NE) are established for a two-player version of the game. Similar conditions for the existence and uniqueness have been extended to the multiuser case in [18], where the authors focus on the practical design of distributed algorithms to compute the NE and propose an asynchronous iterative waterfilling algorithm for an interference channel. In [9], the so-called symmetric waterfilling game was studied. The authors assume that for a set of subchannels and receivers the channel gains from all transmitters are the same. The game is shown to have an infinite number of equilibria. The framework of the interference channel has been relaxed in [19] to include cognitive radio systems with transmitters and receivers equipped with multiple antennas, that is, multiple input multiple output (MIMO) systems. A distributive algorithm for the design of the beamformers at each secondary transmitter based on a noncooperative game is developped. Uniqueness and global stability of the Nash equilibrium are studied. Finally, it is worth to note that the DSL power allocation game in [15] is similar to our game from the mathematical point of view. However, it can be shown that with DSL crosstalk link channel coefficients the game in [15] is not a potential game. Therefore, in general, all the nice properties from potential games do not necessarily hold in their case.
In this paper, we adopt game-theoretical methodologies for power allocation problem in the downlink of small-cell networks (Note that a similar power allocation game can be considered for the uplink where MTs are the players taking decisions. However, it is impractical for MTs to have complete uplink CSI. Then, realistic models should take into account the assumption of knowledge reduction at the transmitters. The interested readers are referred to [20] for the framework of Bayesian games). We model femto cells of different operators as players who adaptively and rationally choose their transmission strategies, that is, their transmit power levels, with the aim of maximizing their own transmission sum-rates under maximum power constraints. We first consider the case where each femto cell decides its own power allocation based on the assumption of complete CSI. Later we remove this assumption, and we show that the same equilibrium can still be reached. In such a context it is important to characterize the NE set, for example, the existence and uniqueness of NE. This aspect plays a key role for the application of a distributed game-theoretical-based algorithm. In fact, the existence and uniqueness of an NE guarantees a predictable power allocation and the behavior of a self-organizing network. An answer to this relevant issue depends strongly on the channel fading statistics and the number of players of the investigated channel setting, as is apparent from the comparison of the results in [9][10][11]. We show that, for a quasi-static fading channel (a fading channel is quasi-static if it is constant during the transmission of a codeword but it may change from a codeword to the following one) with continuous probability density functions of the channel power attenuations, an NE exists and is unique with unit probability. Additionally, we point out that the considered game is a potential game and a simple decentralized algorithm based on the bestresponse algorithm can be readily proposed. However, a straightforward decentralized algorithm based on complete CSI would not be scalable since the required overhead would scale exponentially with the number of transmitters and receivers. Then, we propose a distributed iterative algorithm which requires the transmission of the total received power at each MT at each iteration step. With this distributed algorithm, the overhead scales only linearly with the number of receivers. The convergence rate of the proposed algorithm is analyzed. The price of anarchy is also investigated by numerical analysis.
The paper is organized as follows. In Section 2, we introduce the system model and formulate the problem. In Section 3, we study the existence and uniqueness of NE and characterize the NE set. In Section 4, we show that the game at hand is a potential game. Based on the property of potential games and observations on the required information, we propose a distributed algorithm converging to the NE. We investigate the convergence issue. Numerical analysis of the price of anarchy and the convergence rate are provided in Section 5. Section 6 concludes the paper by summarizing the main results and insights on the system behaviour acquired in this work.

MultiSource MultiDestination System
Model. We consider a wireless system in downlink with M noncooperative APs simultaneously sending information to N MTs over N orthogonal channels, for example, different time slots, frequency bands, or groups of subcarriers in time division multiple access (TDMA), frequency division multiple access (FDMA), or OFDM systems, respectively, as shown in Figure 2. Each channel is preassigned to a different MT by a scheduler and each MT receives signals only on the assigned channel. Without loss of generality, throughout this paper we assign channel n to MT n, for n = 1, . . . , N. This implies that both the MT set and the channel set share the same index in our model. Note that the system model at hand does Figure 2: The multiuser OFDM model. not reduce to a classical multiple access channel, a broadcast channel, or an interference channel [6]. We assume that the channels are block fading (in different scientific communities these channels are also referred to as quasi-static fading or delay constrained channels), that is, the fading coefficients are constant during the transmission of a codeword or block. Within a given transmission block, let G ∈ R M×N ++ be the channel gain matrix whose (m, n) entry is g m,n , the channel gain of the link from AP m to MT n on the preassigned channel n. The matrix G is random with independent entries. We further assume that the distribution function of each positive entry g m,n is a continuous function.
By assuming that the MTs use low-complexity singleuser decoders [6], the signal-to-interference-plus-noise-ratio (SINR) of the signal from AP m received at MT n is given by γ m,n = g m,n p m,n σ 2 + M j=1, j / = m g j,n p j,n , where p m,n is the power transmitted from AP m on subchannel n, and σ 2 is the variance of the white Gaussian noise. For AP m, write the maximum achievable sum-rate as [6] R m = N n=1 log 1 + γ m,n , ∀m, (2) and the power constraint as where P max m is maximum transmit power of AP m and P max m > 0, for all m.

Power Allocation as a NonCooperative Game.
Here, we introduce the power allocation problem as a noncooperative strategic game. Because of the competitive nature of the APs, belonging in general to different service providers, AP m aims to maximize its own transmission rate R m (2) by choosing its transmit power vector p m [p m,1 , . . . , p m,N ] T , subject to its power constraint (3). Denote by vector p = [p T 1 , . . . , p T M ] T the outcome of the game in terms of transmit power levels of all M APs on the N channels. We can completely describe this noncooperative power allocation game as where the elements of the game are where p −m denotes the power vector of length (M − 1)N consisting of elements of p other than the mth element, that is, In such a noncooperative game setting, each player m acts selfishly, aiming to maximize its own payoff, given other players' strategies and regardless of the impact of its strategy may have on other players and thus on the overall performance. The process of such selfish behaviors usually results in Nash equilibrium, a common solution concept for noncooperative games [21]. Definition 1. A power strategy profile p is a Nash equilibrium If, for every m ∈ M, for all p m ∈ P m .
From the previous definition, it is clear that an NE simply represents a particular "steady" state of a system, in the sense that, once reached, no player has any motivation to unilaterally deviate from it. The powers allocated in our system correspond to an NE.

Characterization of Nash Equilibrium Set
In many cases, an NE results from learning and evolution processes of all the game participants. Therefore, it is fundamental to predict and characterize the set of such points from the system design perspective of wireless networks. In the rest of the paper, we focus on characterizing the set of NEs. The following questions are addressed one by one.
(i) Does an NE exist in our game?
(ii) Is the NE unique or there exist multiple NE points?
(iii) How to reach an NE if it exists?
(iv) How does the system perform at NE? Throughout this section we investigate the existence and uniqueness of a Nash equilibrium.
It is known that in general an NE point does not necessarily exist. In the following theorem we establish the existence of a Nash equilibrium in our game.

Theorem 1. A Nash equilibrium exists in game G.
Proof. Since P m is convex, closed, and bounded for each m; u m (p m , p −m ) is continuous in both p m and p −m ; and u m (p m , p −m ) is concave in p m for any set p −m , at least one Nash equilibrium point exists for G [12,22].
Once existence is established, it is natural to consider the characterization of the equilibrium set. The uniqueness of an equilibrium is a rare but desirable property, if we wish to predict the network behavior. In fact, many game problems have more than one NE [12]. As an example of games with infinite NEs, we could consider a special case of our game G, namely, the symmetric waterfilling game [9] where the channel coefficients are assumed to be symmetric. Then, in general, our game G does not have a unique NE. But with the assumption of independent and identically distributed (i.i.d.) continuous entries in G, we will show that the probability of having a unique NE is equal to 1.
For any player m, given all other players' strategy profile p −m , the best-response power strategy p m can be found by solving the following maximization problem: which is a convex optimization problem, since the objective function u m is concave in p m and the constraint set is convex. Therefore, the Karush-Kuhn-Tucker (KKT) conditions for optimization are sufficient and necessary for the optimality [5]. The KKT conditions are derived from the Lagrangian for each player m, ν m,n p m,n = 0, ∀n, where λ m ≥ 0, ν m,n ≥ 0, for all m and for all n are dual variables associated with the power constraint and transmit power positivity, respectively. The solution to (11)- (13) is known as waterfilling [6]: where (x) + max{0, x} and λ m satisfies In order to analyze the equilibrium set, we establish necessary and sufficient conditions for a point being an NE in the game G.
The proof can be found in Appendix A. From (16), it is easy to verify that necessarily λ m > 0, since ν m,n ≥ 0 and g m,n > 0, for all m and for all n. Also, from (17) This equation implies that, at the NE, all APs transmit at their maximum power by conveniently distributing the power over all the orthogonal channels. However, it is still difficult to find an analytical solution from (16)-(18), since the system consisting of (14) and (15) is nonlinear. To simplify this problem, we could consider linear equations instead of nonlinear ones. The following lemma provides a key step in this direction.
The proof can be found in Appendix B. Now, let Z be the following (M + N) × MN matrix: where g n is the nth column of G, I M is the M × M identity matrix, and 0 M is the zero vector of length M. Let c be the following vector of length M + N: Then, (19) and (20) can be written in the form of linear matrix equation Define the following sets: and denote by |X| and |N | their cardinalities. From (18), if an index (m, n) / ∈ X we must have p m,n = 0. Without loss of generality, we assume that N = {1, . . . , N} for N ≤ N. Let Z be the (M+ N)×M N matrix formed from the first M+ N rows and first M N columns of Z, p is formed from the first M N elements of p, and c is formed from the first M + N elements of c. Then, any NE solution must satisfy Let Z be the (M + N) × |X| matrix formed from the columns of Z that correspond to the elements of X. Similarly, let p be the vector of length |X| with entries p m,n such that (m, n) ∈ X (same order as they were in p). Then, any NE solution satisfies  The proofs of Lemmas 2 and 3 can be found in Appendices C and D, respectively.
Based on Lemmas 1, 2, and 3, we derive the following theorem.
Theorem 3. For any realization of a random M × N channel gain matrix G with statistically independent continuous entries, the probability that a unique Nash equilibrium exists in the game G is equal to 1.
The proof can be found in Appendix E. Thus, from Theorems 1 and 3, we have established the existence and uniqueness of NE in our game G.

Distributed Power Allocation and Its
Convergence to the Nash Equilibrium An equilibrium has practical interests only if it is reachable from nonequilibria states. In fact, there is no reason to expect a system to operate initially at equilibrium. The convergence of an algorithm to an equilibrium is in general a very hard problem usually related to the specific algorithm and requiring the analysis of synchronous or asynchronous update mechanisms (for power allocation algorithms in interference channels see [18,23]).

Potential Game Approach.
Fortunately, our game G can be studied as a potential game (The notation of potential games was firstly used for games in strategic form by Rosenthal (1973) [24], and later generalized and summarized by Monderer (1996) [25]). Potential games are known to have appealing properties for the convergence of the bestresponse or greedy algorithms to the equilibrium. All the potential games admit a potential function. This potential function is a unique global function that all the players optimize when they optimize their own utility functions. Thus, the set of pure Nash equilibria can be found by simply locating the local optima of the potential function. Such games have received increasing attention recently in wireless networks [14,26,27], since the existence of potential function enables the design of fully distributed algorithms for resource allocation problems. In fact, there are various notions of potential games such as exact potential, weighted potential, ordinal potential, generalized ordinal potential, pseudo potential, and so forth. These potential games could possess slightly different properties for the existence and convergence of NE. Here, we consider only the exact potential games, since they are closely related to our game. Exact potential games are defined in the following statement.
for all (p m , p −m ), (q m , p −m ) ∈ P . The function v is referred to as exact potential of the game.
Equation (27) implies that the NE of the original game G must coincide with the NE of the potential game, which is defined as a new game with v as an identical utility function for all the players. Therefore, we can transform the noncooperative strategic game G into a potential game, if we can find a potential function that quantifies the variation in terms of utility due to unilateral perturbation of each player's strategy, as indicated in (27).
Taking inspiration from the result derived in the single channel case [14], we have the following lemma.
Proof. From (28) and (6), we observe that the first derivatives of v and u m are equal, that is, which implies that the property of exact potential (28) is satisfied. This completes the proof.
We denote by ζ m,n the term (σ 2 + j / = m g j,n p j,n ) which stands for the aggregate interference plus noise of user m on subchannel n. In order to find user m's single-user bestresponse in the potential game, one needs to solve the following maximization problem: Note that the problem (30) can be solved as a convex optimization, when the private channel gain g m = {g m,1 , . . . , g m,N } and the aggregate interference plus noise ζ m = {ζ m,1 , . . . , ζ m,N } are both known to player m. It is easy to verify that this single-user best-response is the same waterfilling solution expressed in (14), due to the property of potential function.

Distributed Algorithm and Convergence Property.
Note that if each AP has complete CSI, that is, knowledge of the channel gain matrix G, defined as in Section 2, the uniqueness of the NE guaranties that each AP can determine independently the power allocation at the NE in a decentralized manner. In order to acquire information about the whole matrix G at each AP, a feedback channel is usually needed to transmit the channel estimations from MTs to APs. With this information, each AP can solve locally the system of equations (16)- (18) or perform locally a bestresponse algorithm based on the repeated maximization of problem (30) by starting from a random point p −m ∈ j / = m P j . However, the structure of problem (30) suggests an alternative distributed approach to reduce eventually the signalling on the feedback channel. In fact, the repeated optimization of problem (30) can be performed in a distributed way by feeding back at each AP m only the private channel gain g m and the aggregate interference plus noise ζ m . Nevertheless, note that such a distributed implementation of the algorithm would lead to a transition phase where the APs are not transmitting at an equilibrium point. In our numerical results, we ignore the cost of feedback, and we focus on analyzing the theoretical upper-bound.
The above discussion yields a simple algorithm based on the iterative waterfilling [28] detailed in the following.
In this algorithm, we assume that the same game could be myopically played repeatedly: in each round, every myopic player (a myopic player has no memory of past gamerounds) chooses its best-response according to the singleplayer waterfilling that depends on the current state of the game. The following theorem shows the convergence and optimality of the algorithm. The proof can be found in Appendix F. A more general discussion about the convergence and stability properties of potential games can be found in [25,29]. In [25], it shows that every bounded potential game (a game is called a bounded game if the payoff functions are bounded) has the approximate finite improvement property (AFIP), that is, for every > 0, every -improvement path is finite. Then, it is obvious that every such finite improvement path of the exact potential games terminates in anequilibrium point (an -equilibrium is a strategy profile that approximately satisfies the condition of Nash equilibrium). In other words, the sequential best-response (players move in turn and always choose a best-response) converges to theequilibrium independent of the initial point. Note that this is a very flexible condition for the convergence, since order of playing can be deterministic or random and need not to be synchronized. It is one of the most interesting properties of the potential games, especially in order to distributively find the equilibrium in self-organizing systems. In [29], it shows that potential games are characterized by strong stability properties (Lyapunov stable, see its definition in Theorem 5.34 of [29]). Also note that if the game has a unique NE, then it is globally stable.
In the simultaneous best-response algorithm all the players choose their best-responses simultaneously at each iteration. It is not difficult to verify that, in the general case, it does not necessarily converge, due to the "ping-pong" effect generated by myopic players. However, [30] has shown that for infinite pseudopotential games, a general class of games including also exact potential games, with convex strategy space and single-valued best-response (games with strictly multiconcave potential, concave in each players' unilateral deviation, have single-valued best-response), the sequence of simultaneous best-responses, reminiscent of fictitious play, also converges to the equilibrium.
It is interesting to note that for many practical systems with finite transmit power states, similar results still hold for the convergence of the sequential best-response. The only difference is that, in the finite case, the existence of exact potential function implies the finite improvement property (FIP), and therefore, the sequential best-response converges to the exact NE instead of an -equilibrium.
Although the final convergence of the DPIWF algorithm is proved, one may wonder whether the optimum of the potential function (28) coincides with the optimum social welfare, that is, the optimal total information rate transmitted in the network. We discuss the price of anarchy in the following section.

Numerical Evaluation
In this part, numerical results are provided to validate our theoretical claims and assess the price of anarchy, that is, the performance loss in terms of the transmit sum-rate of all APs in the network due to a noncooperative game compared to the maximum social welfare. We denote this transmit sumrate in the network as the actual total network rate, and defined it as We consider frequency-selective fading channels with channel matrix G of size M × N, where M is the total number of transmitters (players) and N is the total number of receivers. We assume that the Rayleigh fading channel gain g m,n are i.i.d. among players and channels. The maximum power constraint for each player m is assumed to be identical and normalized to P m = 1.
In Figure 3, we show the convergence behaviors of potential function and the actual total network rate, shortly referred to as "actual rate", by using the proposed DPIWF algorithm for a random channel realization. We set the number of transmitters to M = 10 and the number of receivers to N = 10. As expected, in both Figures 3(a) and 3(b) the potential function converges rapidly (at the 4th iteration). In Figure 3(a), the actual rate converges slightly slower (at the 6th iteration) and maintains a monotonically increasing slope. However, in Figure 3(b), the actual rate finally converges, but unfortunately it does not increase monotonically and it converges only at the 34th iteration with a convergence rate much slower than the potential function. Note that we use this example to show that a "defective" convergence may happen during the iteration steps.  In order to measure the performance efficiency of distributed networks operating at the unique NE, we provide here the optimal centralized approach as a target upperbound for the total network rate. We ignore the performance loss caused by the necessary uplink and downlink signalling transmission. The total network rate maximization problem can be formulated as The optimization problem (32) is difficult to solve since the objective function is nonconvex in p. However, a relaxation of this optimization problem [11] can be considered as a geometric programming problem [31]. As well known, a geometric programming can be transformed into a convex optimization problem and then solved in an efficient way. A low-complexity algorithm was proposed in [11] to solve the dual problem by updating dual variables through a gradient descent. Note that the algorithm always converges, but may converges to a local maximum point in a few cases. We use this algorithm in our simulations.
In the following part, we address two main practical questions through numerical results.
(1) How does the network performance behave in average at the unique NE in comparison to the global optimal solution or global welfare? More precisely, we are interested in comparing the average total network rate instead of the instantaneous total network rate. We denote by u(M, N) the average total network rate for a M transmitters and N receivers system, that is,  (2) What about the convergence behavior for the actual total network rate when using DPIWF algorithm? Does it converge as rapidly as in Figure 3(a) for the most of the cases?
Let us consider the first question. In Figure 4, we compare the average total network rate of both decentralized and centralized networks for two different channel noise levels σ 2 = 0.1 and 1, respectively. The plots are obtained through Monte-Carlo simulations over 10 4 realizations for the channel gain matrix G. Figures 4(a) and 4(b) show the total network rate as a function of the number of transmitters M for different number of receivers N. More specifically, N = 5, 10, 15. We note that in both Figures 4(a) and 4(b), the centralized optimal approach always outperforms the decentralized noncooperative algorithm. Additionally, for a fixed number of transmitters N, when we increase the number of receivers M, the performance loss of decentralized systems compared to the centralized social welfare becomes greater and greater. This phenomenon can be intuitively understood as follows: when there is a great number of selfish players, the hostile competition turns the multiuser communication system into an interference-limited environment, where interference significantly degrade the performance efficiency.
In Figure 4, we also note that for a fixed N the average performance of centralized systems is an increasing function of M, and the average performance of decentralized systems corresponding to NE reaches a maximum and then decreases flatting out. For the typical values of N, that is, N = 5, 10, 15, in Figure 4(a), when σ 2 = 0.1 the average performance of decentralized systems are maximized at M = 4, 9, 14, respectively; in Figure 4(b), when σ 2 = 1 the average performance of decentralized systems are maximized at M = 6, 11, 16, respectively. This comparison simply shows that different noise variance (in general channel condition) have a different impact on the decentralized system performance. This observation is fundamental for improving the spectral efficiency of a distributed multiuser small cell networks: For a given area, that is, a given number of receivers N and given channel conditions, there exists an optimal number of access points, denoted as M , to be installed in the network. Roughly speaking, when M > M , the system is saturated due to the increasing competition for the shared limited resources; when M < M , the system operates in a unsaturated state, since system resources are not fully exploited.
Let us now consider the second question. In Figure 5, we show the probability of convergence to the NE within 5 iterations for σ 2 = 0.1 and 1, respectively. To be more precise, we say that the algorithm converges at the fifth iteration if the total network rate exceeds 99% of the rate at the NE. We find that the probability of convergence is satisfactory. It is greater than 0.982 in all cases and tends to 1 when M N and M N. Another interesting observation is that the minimal convergence probability always occurs when M = N, regardless of the noise value σ 2 .

Conclusions and Future Works
In this paper, we study the power allocation problem in the wireless small-cell networks as a strategic noncooperative game. Each transmitter (AP) is modeled as a player in the game who decides, in a distributed way, how to allocate its total power through several independent fading channels. We studied the existence and uniqueness of NE. Under the condition of independent continuous fading channels, we showed that the probability of having a unique equilibrium is equal to 1. The game at hand is shown to be a potential game. A distributed algorithm requiring very limited feedback has been proposed based on the potential game analysis. The convergence and stability issues have been addressed. Numerical studies have shown that the DPIWF algorithm can converge rapidly within 5 iterations with very high probability.