On the effect of false alarm rate on the performance of cognitive radio networks
 Isameldin M. Suliman^{1}Email author,
 Janne Lehtomäki^{2} and
 Kenta Umebayashi^{3}
https://doi.org/10.1186/s1363801504743
© Suliman et al. 2015
Received: 10 May 2015
Accepted: 30 October 2015
Published: 10 November 2015
Abstract
Spectrum sensing plays a significant role in enabling utilization of spectrum holes by unlicensed secondary users (SUs) in cognitive radio networks (CRNs). Most of the related work concerning spectrum sensing has focused on sensing carried out by incoming secondary users (SUs) aiming at locating spectrum opportunities. However, in order to appropriately protect returning licensed primary users (PUs), SUs should continuously perform spectrum sensing during their ongoing transmissions. An important issue associated with the continuous sensing is the false alarm rate (FAR), which is defined as the average number of false alarms per unit of time and can be modeled by a Poisson process with Poisson parameter λ _{FAR}. In this paper, we address this issue and develop a continuous time Markov chain (CTMC)based analytical model to evaluate the effect of the false alarm rate on the performance of CRNs. A major feature of the proposed analytical framework is that it takes into account the effects of sensing errors by both incoming SUs looking for free channels to transmit on and the already transmitting SUs expecting the presence of returning PUs. The analytical model also examines the interference tolerance among PUs and SUs as well as the impact of SUs residual selfinterference. The performance results show that high λ _{FAR} can severely degrade PUs performance and reduce the overall system resource utilization. However, with increasing PU interference tolerance, PUs performance improves as well. SU residual interference was found to decrease the detection probability resulting in a low PU performance. Extensive simulations validate the analytical model, demonstrating excellent agreement with the theoretical results.
Keywords
1 Introduction
With today’s inefficient utilization of the scarce radio spectrum, cognitive radio (CR) [1–3] is becoming an important tool for solving the problem of spectrum underutilization. As a result, there has been considerable research effort focusing on CR techniques that enable using radio spectrum efficiently. In CR networks (CRNs), unlicensed secondary users (SUs) employ spectrum sensing [4–6] to discover spectrum holes during the absence of licensed primary users (PUs) before attempting network access. Energy detection [7–9] is the simplest method for detecting the presence of PUs. It is based on calculating the energy of the received samples which is compared to a threshold. If the threshold is exceeded, it is decided that a signal or signals are present. If the sensed channel is free, SUs may be allowed to transmit on that channel.
No matter which detection scheme is used for protecting returning PUs, it will lead to the occurrence of false detection of returning PUs, i.e., we will erroneously assume that a PU has returned, when in fact the PU’s channel is free. A simple way to characterize the occurrence of false alarms for already transmitting SUs is to use the average number of false alarms per a time unit, similar to the rate parameter of a Poisson process. We call this parameter the false alarm rate parameter λ _{FAR} [10, 11]. It is to be mentioned that in multichannel systems with handoff capability (as studied here), an SU moving from its current operating channel (for example, due to false detection of a returning PU) will attempt to locate another free channel to continue its ongoing data transmission.

It motivates and develops a CTMCbased analytical framework that precisely evaluates the performance of CRNs. Unlike existing approaches, the proposed model thoroughly investigates the effect of FARs on the performance of CRNs. The model also takes into consideration the SUs residual selfinterference as well as interference tolerance among PUs and SUs.

It models the occurrence of FAR events as a Poisson process with parameter λ _{FAR} with a theoretical justification based on a shrinking Bernoulli process [13].

It proposes a new performance evaluation measure, the SU selftermination probability. The proposed metric can precisely measure the percentage of SU calls that are terminated because of the FAR occurrence. The new metric also allows for measuring the SUs’ ability of utilizing spectrum opportunities.

Extensive simulations to validate the theoretical results.
We believe that the work presented in this paper contributes towards a better understanding and provides a new insight into the operation of CRNs and can be used to develop more accurate and realistic CRNs performance analysis models. The rest of the paper is organized as follows. Section 2 presents the related work. The system model is presented in Section 3. Section 4 contains a description of the continuous spectrum sensing and data transmission. Section 5 presents the CTMCbased analytical framework. In Section 6, we discuss performance evaluation metrics for the CRN. Section 7 summarizes results and provides comparison of simulation and theoretical results. Finally, Section 8 provides the conclusions and remarks on future work.
2 Related work
In recent years, several studies has been proposed to detect returning PUs in CRNs. In [14], the authors investigate the issues of how to maximize the overall discovery of opportunities in the licensed channels and how to minimize the delay in locating an idle channel in order to minimize interference on returning PUs. Similarly, the authors of [15] presented a dynamic spectrum access mechanism in a network where SUs do not have perfect knowledge of PUs’ communication behavior. The interference issue has also been studied. However, this study only considered perfect spectrum sensing and a network with only one PU. In [16], the authors consider a preemptive priority approach for the channel access where SUs must vacate their channels whenever the corresponding PUs appear. The work presented in [17] formulates a joint spectrum sensing and access problem as an evolutionary game by considering the mutual influence between spectrum sensing and access. Although the interference problem has been addressed in these works, the problem of the FAR has not been investigated.
In [18], continuous time Markovian process (CTMP) is used to model PU traffic in opportunistic spectrum access (OSA) systems. However, for analyzing SU’s behavior, discrete time queuing was used. In contrary to our work, the underlying assumption made therein is that sensing and data transmission cannot be carried out simultaneously and therefore the SU has to periodically suspend its data transmission in order to perform spectrum sensing. The problems with this technique are the overheads associated with the scheduling and synchronization of the suspension periods among SUs as well as the frequent interruption in the SU’s data transmission. Additionally, the SU can only detect a reappearing PU during the suspension period, even if the PU reappeared before the suspension period. This work also differs from our study because it only supports CRNs with one channel and the assumption that the spectrum sensing is perfect.
Simultaneous spectrum sensing and data transmission approach have been studied in [19–22]. The issue of selfinterference due to transmitting and receiving in the same band has been studied in [23, 24]. In spite of considering the problem of unnecessary false alarms, the authors of [12, 25] did not investigate their effect on performance metrics such as blocking and termination probabilities. Furthermore, the authors of [26] analyzed different types of unreliable sensing for both incoming and ongoing SUs and their impact on the performance of CRNs without addressing the FAR.
Most of the existing CTMC models [27–29] do not cover all the aspects of the spectrum sensing and CRN operation and some important factors were not fully addressed. In our previous work [30], we analyzed the performance of CRNs using a CTMC framework that supports multichannel, spectrum handoff, fullstate dependent transition rates, and the ability to handle spectrum sensing errors. In this paper, we extend the analysis in [30] to capture the effect of the FAR and to handle the residual selfinterference within the SU transceivers.
3 System model
We consider a CRN with N number of channels in which SUs are allowed to opportunistically utilize licensed spectrum bands with the constraint that the QoS of PUs remains at an acceptable level. There are two approaches for enabling PUs and SUs to coexist and share radio resources in CRNs: spectrum sharing (SS) and opportunistic spectrum access (OSA) [31]. In the SS model, SUs are allowed to transmit simultaneously with PUs on the same band. On the other hand, the OSA approach, which is more suitable for the model presented in this paper, allows SUs to access the licensed channels opportunistically when PUs are not present.
3.1 Primary user model
We assume that the primary channel occupancies are time varying alternating between idle and busy periods, and thus SUs must perform spectrum sensing continuously to detect the presence of returning PUs. PU connections arrive at the network according to a Poisson process at a rate of λ _{1}. The PU service rate which is assumed to be exponentially distributed is μ _{1}. We also assume that PUs can obtain primary channel occupancy information, for example, by accessing a core network that makes signaling or querying of the PUs’ base station [32], and thus it is further assumed that PUs do not collide with each other [28].
We assume that both PUs and SUs have some interference tolerance T _{TOL} of how many seconds of interference they will tolerate before withdrawing from the system. If the PU interference tolerance time T _{TOL} is 0, no SU transmission is allowed [33]. We assume equal interference tolerance for both systems leading to both colliding users withdrawing from the system simultaneously. A similar assumption has been considered previously in [28].
3.2 Secondary user model
We assume that SU connections arrive at the network according to a Poisson process with λ _{2}. The SU service rate is assumed to be exponentially distributed with μ _{2}. During the absence of PUs, SUs can opportunistically access the free channels if they are not occupied by other SUs. We also assume that SUs are capable of broadcasting control messages on a common control channel (CCC) [34] to show their existence to neighboring SUs in the proximity. Therefore, SUs do not attempt accessing channels occupied by other SUs. Upon detection of the presence of a returning PU, a SU leaves its current channel and starts the spectrum handover process in order to find a new free channel. If the channel search process ends without finding a free channel, the SU terminates its call and leaves the network.
As illustrated in Fig. 1 and similar to the distributed (coordination function) interframe space (DIFS) operation in IEEE 802.11, the SU has to keep sensing the PU channel from the beginning of its transmission, since the PU can arrive at any time instant of a slot. This process forms a continuous sequence of sensing slots with length equals T _{2}. For example, if the PU appears in the middle of a time slot, then the first slot will not get full PU energy, leading to a smaller detection probability than the later full slots. Since the first T _{2} may be wasted, we assume that the first partial slot sensing never leads to detection, i.e., the detection probability is close to zero. Hence, we should detect the PU arrival during T _{TOL}−T _{2} seconds which corresponds to \(\widehat {T_{\text {TOL}}}=\left \lfloor \frac {T_{\text {TOL}}T_{2}}{T_{2}} \right \rfloor \) slots. Although partial slot sensing can enable the SU to perform sensing immediately after the arrival of the PU and hence have a prompt reaction to protect PUs, for the sake of simplifying the analysis, we consider only the full slot sensing by assuming that the detection process will begin from the first full time slot following the SU arrival.
4 Continuous spectrum sensing model
One possibility for implementing continuous sensing is to leave the upper part of the PU channel empty (i.e., free from SU transmissions) [35]. As shown in Fig. 1, we split the PU channel into three subchannels: (A) SU communication channel, (B) a sufficient vacant guard band to reduce the effect of SU’s selfinterference, and (C) SU sensing channel. When the PU is active, it uses the whole bandwidth (A+B+C) for its communication. The secondary user uses subchannel (A) for its communication. A reappearing PU can be detected by sensing, during ongoing SU transmission from the subchannel (C). It is obvious that a problem here is the selfinterference due to the leakage of the SU’s transmitted signal back to its sensing device. However, the emergence of a large variety of selfinterference cancellation techniques [36–38] in the literature enabled efficient reduction in selfinterference and therefore allowing radios to operate in fullduplex mode. For example, the authors of [39] present a method for canceling a passband selfinterference signal using adaptive filtering in the digital domain. Therefore, in addition to the vacant guard band and bandpass filtering, selfinterference cancellation has also been assumed to remove most of the residual selfinterference. In this model, q∈{1,2} denotes an index with the interpretation that q=1 if the spectrum sensing is carried out by incoming SUs and q=2 if the spectrum sensing is performed by ongoing SUs.
4.1 Energy detectorbased spectrum sensing
In Eqs. (1) and (2), \(s_{\!_{\text {PU}}}(t)\) is the PU transmitted signal, \(s_{\!_{\text {SU}}}(t)\) represents the leakage from the SU transmitted signal, n(t) is the additive white Gaussian noise (AWGN), \(h_{\!_{\text {PU}}}\) is the PU channel gain while \(h_{\!_{\text {SU}}}\) represents the SU leakage signal gain, and t is the time. In the above equations, H _{0} is the null hypothesis meaning that PU is not present in the sensed band, and H _{1} represents the alternative hypothesis referring to the presence of the PU signal.
where Q _{ m(.,.)} is the generalized mth Marcum Qfunction [41].
In continuous spectrum sensing with fullduplex communication, the consideration of selfinterference is particulary important since the selfinterference can affect the sensing outcome and degrade SUs performance. Although the SU selfinterference signal can have nonzero mean, it has been been assumed in the majority of related works to have a zero mean. For instance in [43], the authors mentioned that in practical fullduplex systems, the selfinterference cannot be completely canceled, such that the signals received at each node is a combination of the signal transmitted by the other source, the residual selfinterference (RSI), and the noise. They also assume that the RSI can be typically modeled as zeromean additive white Gaussian noise (AWGN). The work reported in [42] assumed that the Gaussian distortion and noise follows central chisquare distribution in the absence of PU signals but potentially including RSI and noncentral chisquare distribution when PU signal is present.
Selfinterference mitigation in fullduplex MIMO relays has been investigated in [23] where the authors focused on minimizing the residual loop interference so that it can be regarded as additional relay input noise. They assumed that all signal from the relay output to the relay input (including loop interference (LI) signal) and noise vectors have zero mean. Furthermore, the authors of [44–46] assumed that the SU selfinterfering signal before carrying out selfinterference suppression (SIS) to be a zeromean random signal with selfinterference channel coefficient equal one. In [47], the residual selftransmitted signal is modelled with circular symmetric complex Gaussian variables. Following the common practice in existing models, the use of the assumption that SU’s leakage signal can be zero mean and follows central chisquare distributions is justified and can be hold in order to take into account the RSI signal and perform the analysis.
It should be noted that when we use a dedicated part of the bandwidth (subchannel C) for continuous sensing, the effect of the residual interference becomes much lower than when we use the full bandwidth for simultaneous sensing and transmission. Each incoming SU correctly detects channel occupancy with probability P _{ D1}, and falsely classifies a free channel as occupied with P _{FA1}. Similarly, each SU with ongoing calls detects the arrival of a PU with probability P _{ D2} and falsely classifies a free channel as occupied with P _{FA2}. The corresponding misdetection probabilities for incoming and outgoing SUs are P _{ M1}=1−P _{ D1} and P _{ M2}=1−P _{ D2}, respectively. The detection probability P _{ D2} refers to the probability of detecting incoming PU during the first \(\widehat {T_{\text {TOL}}}\) full slots of its arrival, instead of the perslot detection probability. If the perslot detection probability is denoted as z then \(P_{D2}=1(1z)^{\widehat {T_{\text {TOL}}}}\). This does not affect the FAR process since one perslot false alarm event is enough to initiate the spectrum handoff and channel searching process. Modeling of partial slot sensing is left for future work.
4.2 Poisson process approximation
We model the occurrence of the false alarm at each sensing decision with the Bernoulli process. The energy detector makes only one sensing decision in each slot which results into a binary variable (0 or 1). Since the sensing decisions with only white Gaussian noise present are independent, the resulting binary output of the sensing clearly follows the Bernoulli process (i.e., independent and identically distributed process generating 1 and 0 s), and the Bernoulli parameter corresponds to the probability of FAR occurrence (binary output 1) in each spectrum sensing decision.
At each spectrum sensing decision epoch T _{2}, a false alarm occurs with probability P _{FA2} and does not occur with probability 1−P _{FA2}, independently of the decision outcome of the last sensing period. The λ _{FAR} parameter is the product of the decision rate and the false alarm probability [10, 11, 48]. Therefore, λ _{FAR} is given by P _{FA2}/T _{2}. Let us assume that the sensing interval T _{2} is short and therefore we assume that the decision rate given by 1/T _{2} is large, and that the false alarm P _{FA2} is small as otherwise there would be too many false alarms for successful SU operation. Then, the arrival process of false alarms can be approximated by a Poisson process as a limit of a shrinking Bernoulli process [13] with parameter λ _{FAR}.
5 Continuous time Markov chain model (CTMC)
As an example, consider a CRN with three channels denoted by C1, C2, and C3. Let us assume that channel C1 is occupied by an PU, channel C2 is occupied by a SU, and the last channel C3 is free. The Markov chain is in state (1,1). We now explain a series of events that trigger the system to move from state (1,1) to state (0,0). State (0,0) indicates that all channels are free. On the occurrence of λ _{FAR}, the SU leaves C2 and starts the channel searching process. There are two channel selection possibilities for the SU for continuing its data transmission. The SU can first select C1 with probability 1/2 and then misdetect the presence of the PU on C1 with probability P _{ M1}. The second possibility is to select C3 with probability 1/2, then falsely classify the free channel as occupied by the PU with a false alarm probability P _{FA1}, and finally misdetect the presence of the PU on C1 with probability P _{ M1}. Both selections lead the SU to collide with the PU, and eventually both of them leave the network. Combining all these events, the transition rate from state (1,1) to state (0,0) can be obtained by \(\frac {{{\lambda _{\text {FAR}}}{P_{M1}}}}{2}(1 + {P_{\text {FA1}}})\).
Another example is the transition from state (1,1) to state (2,0). This happens with the arrival of a PU with rate λ _{1} and with probability 1/2 to channel C2 which is occupied by the SU. The SU correctly detects the presence of the PU with detection probability P _{ D2} and vacates the channel. After leaving the channel, the SU has two possibilities with probability 1/2 for each. The SU first falsely classifies the free channel C3 as being occupied by a PU with false alarm probability P _{FA1} and then detects the presence of the PU in channel C1 with probability P _{ D1}, ending up leaving the network. The other possibility is that the SU correctly detects the existence of the PU in channel C1 with probability P _{ D1} and then erroneously classifies the free channel C3 as occupied by a PU with false alarm probability P _{FA1}. The resulting transition rate is \(\frac {\lambda _{1}{P_{D2}}{P_{D1}}{P_{FA1}}}{2}\). Proceeding in a similar manner, the transition rates to and from the remaining states can be obtained.
5.1 Generalization of the CTMC
The goal of this section is to extend the results presented previously to describe a CRN with an arbitrary number of channels N. When N is large, constructing a state transition diagram and finding a solution to the corresponding balance equations is complicated and time consuming. Similar to [30], we use a recursive method to calculate the state transition rates to and from all different states of the CTMC representing the CRN network. The state transition rates are used to get all the possible balance equations. Recalling that the number of PUs is denoted by i and the number of SUs is denoted by j. Let us also assume that the number of free channels is denoted by k which is given by k=N−i−j.

Transition type 1: (i,j)→(i,j+1). This transition defines the increase in the number of SU by one and can be obtained by$$ \begin{aligned} f(\textit{i,k}) &= \frac{k}{{i + k}}(1  {P_{\text{FA1}}}) \\&\quad+ \frac{k}{{i + k}}{P_{\text{F}\text{A1}}}f(\textit{i,k}  1)\\ &\quad+ \frac{i}{{i + k}}{P_{D1}}f(i  1,k) \end{aligned} $$(9)
where function f(.) is used to define the increase in the number of SUs by 1 [30]. P _{ D1} and P _{FA1} have been defined earlier to denote the initial sensing’s detection and false alarm probabilities. They have been used to obtain more accurate state transition rates and state probabilities in comparison to results obtained in [30]. The overall state transition rate for this case is given by λ _{2} f(i,k)

Transition type 2: (i,j)→(i−1,j). This transition defines the decrease in the number of PUs by one. We use the recursive function g(.) [30] to define this transition$$ \begin{aligned} g(\textit{i,k}) &= \frac{i}{{i + k}}{P_{M1}} + \frac{i}{{i + k}}{P_{D1}}g(i  1,k)\\ &\quad+ \frac{k}{{i + k}}{P_{\text{F}\text{A1}}}g(\textit{i,k}  1), \end{aligned} $$(10)
where P _{ M1}, P _{ D1}, and P _{ F A1} denote the initial sensing’s misdetection, detection, and false alarm probabilities, respectively. The overall state transition rate for this case can be obtained by i μ _{1}+λ _{2} g(i,k) [30].

Transition type 3: (i,j)→(i+1,j). This transition is given by$$\begin{array}{*{20}l} T_{(i + 1,j)}^{(i,j)} = {\lambda_{1}}\left({\frac{{N  i  j}}{{N  i}}} \right)+ \qquad\frac{{j{\lambda_{1}}{P_{{D2}}}}}{{N  i}}f(i,N  i  j) \end{array} $$(11)
to reflect the increase in the number of PUs by one.

Transition type 4: (i,j)→(i,j−1). The state transition rate for decreasing the number of SUs by one is given by$${} \begin{aligned} T_{(i,j  1)}^{(i,j)} &= j{\mu_{2}} + {\lambda_{1}}{P_{{M2}}}\frac{j}{{N  i}} \\ &\quad+ j{\lambda_{\text{FAR}}}\left({1  f(i,N  i  j)  g(i,N  i  j)} \right)\\&\quad+ {\lambda_{1}}{P_{{D2}}}\frac{j}{{N  i}}g(i,N  i  j) \end{aligned} $$(12)

Transition type 5: (i,j)→(i+1,j−1). The state transition rate for this case is given by$$ \begin{aligned} T_{(i + 1,j  1)}^{(i,j)} &= \left({{\lambda}_{1}}{{P}_{{D2}}}\frac{j}{N  i} \right)\\ & \quad{\times} \left({1  f(i,N  i  j)  g(i,N  i  j)} \right) \end{aligned} $$(13)

Transition type 6: (i,j)→(i−1,j−1). The number of PUs is decreased by one and the number of SUs is decreased by one. This transition occurs if after the occurrence FAR, the SU ends up colliding with a PU. We get the transition rate as$$ T_{(i  1,j  1)}^{(\textit{i,j})} = j{\lambda_{\textit{F}{AR}}}\left[ {g(i,N  i  j)} \right] $$(14)
5.2 Construction of state transition rate matrix and computation of the steady state probability vector

Step 1: Solve the recursive Eqs. (10–15) to obtain the state transition rates.

Step 2: Drive the balance equations using the rule that incoming transition rates to each state must equal outgoing transition rates from that state [50].

Step 3: Use the balance equations to build the infinitesimal generator matrix Q. All elements not on the main diagonal of Q represents state transition from one state to another. The elements on the main diagonal of Q make the sum of the elements in the respective row equal zero [51].

Step 4: Apply the normalization condition \(\sum \limits _{d} {{\boldsymbol {\pi }_{d}}} = 1\)

Step 5: Solve the system of linear equations π Q=0 to obtain the CTMC’s steady state probabilities.
Each element in the steady state probability vector π represents the percentage of time that the system spends in that state.
Since the number of states of the CTMC grows exponentially with the number of the channels in the network, it would be impossible to derive the CTMC transition rates by hand for large number of states. In this sense, the utilized recursive approach solves one part of this problem. However, the number of states is still exponential, which lead to higher memory and processing time requirements when the number of channels increases since the fullstate transition matrix is used to obtain exact results. With very large number of channels, approximation solutions with reduced number of channel states would be beneficial. In the literature, some approximation methods have been presented for CTMCs with a large number of states [51, 52]. The results presented in this paper have been obtained using the exact full CTMC. However, when the number of channels is very large which brings some inefficiency, we can apply approximate solutions of large CTMCs to overcome this problem [51, 52].
6 Performance evaluation measures
In order to measure the performance of the CRN, we define several performance evaluation measures: secondary forced termination probability (S U _{ FTP }), primary forced termination probability (P U _{ FTP }), and secondary self termination probability (S U _{ STP }). Those performance metrics are calculated by using the state transition rates and the steady state probabilities π _{(i,j)} and state transition rates derived in the previous section. The reader is referred to [30] for more details on the definition and derivation of other performance metrics such as secondary successful probability (S U _{ SP }), primary blocking probability (P U _{ BP }), secondary blocking probability (S U _{ BP }), as well as system resource utilization.
6.1 Secondary forced termination probability (S U _{ FTP })
The secondary forced termination probability, denoted by S U _{ FTP }, is the probability of terminating SU calls because of SU’s failure to find a new free channel after moving from its current channel. The S U _{ FTP } is calculated and defined by Eq. (15). It reflects the ratio of terminated SUs’ call to total SU call arrivals λ _{2}.
6.2 Primary forced termination probability (P U _{ FTP })
The primary forced termination probability, denoted by P U _{ FTP } and given by Eq. (16), is calculated as the ratio of terminated PU calls because of collisions with SUs to the total primary call arrivals λ _{1}.
6.3 Secondary self termination probability (S U _{ STP })
where \({{\overline {T}}_{(i,j  1)}^{(\textit {i,j})}}\) represents the portion of the transition rate from state (i,j) to state (i,j−1) that occur because of the λ _{FAR}.
7 Simulation and numerical results
In this section, we report results obtained both through theoretical analysis and simulations. We conduct simulations with MATLAB using an eventbased approach and Poisson arrival processes. The parameters in simulations and theory are chosen as follows: We set the primary licensed bandwidth as 20 MHz, the initial sensing bandwidth is 20 MHz meaning that a SU carries out spectrum detection over the whole spectrum. However, the continuous sensing bandwidth is chosen to be 2 MHz. The initial sensing time =20 μs. We set the primary and secondary service rates as μ _{1}=μ _{2}=4. PU signal power is –91 dBm. The PU power has been set to a low level since SUs should be able to detect even weak PUs signals. The noise level is –160 dBm/Hz. To include the effect of the residual interference signal, we set the interference distortion factor to 0.1. We present plots for different performance metrics. It can be observed from all plots that the analytical results are in excellent agreement with the simulation results, which demonstrates the accuracy and validity of the CTMC analytical model.
7.1 ROC curves
The impact of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the detection and false alarm probabilities is demonstrated by the ROC curves shown in Fig. 3. The SNR value for the initial sensing is 19 dB. However, the SNR values for the continuous sensing vary depending on the spectrum sensing time duration T _{2} that affects the time bandwidth product. We assume that the PU signal power is uniformly distributed over the PU channel. It can be seen that the residual interference affects the ROC curves, as the ROC performance drops significantly with increases in \(\alpha _{_{\text {SU}}}\) values. We can also see from the figure that the curve for the initial sensing with spectrum sensing time duration T _{1}=10 μs is identical with the curve for continuous sensing with spectrum sensing time duration T _{2}=100 μs and \(\alpha _{_{\text {SU}}}=0\). This is due to the fact that their time bandwidth products are the same and equal to 200. Figure 3 also shows the effect of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the false alarm and detection probabilities.
7.2 Effect of the interference tolerance \(\widehat {T_{\text {TOL}}}\)
7.3 Effect of the false alarm rate
7.4 Performance under perfect spectrum sensing
8 Conclusions
We studied the effect of the false alarm rates λ _{FARs} on the operation of CRNs. We developed a CTMCbased analytical model to evaluate the performance of CRNs under realistic network operating conditions. The proposed model not only includes sensing errors by incoming SUs but also takes into account the misdetection and false alarm probabilities by ongoing SUs. The modeling approach described here is capable of examining other performance evaluation parameters such as the effect of interference tolerance \(\widehat {T_{\text {TOL}}}\) among PUs and SUs as well as the effect of SU residual selfinterference. We derived formulas for different performance metrics, including primary and secondary forced termination probabilities as well as secondary selftermination probability. Furthermore, we performed extensive simulations to validate the accuracy of the analytical model. Simulation results are in excellent agreement with the analytical results.
Results have shown that λ _{FAR} greatly influences the performance of CRNs by degrading SU performance and reducing network resource utilization. Results have also shown that decreasing the interference tolerance \(\widehat {T_{\text {TOL}}}\) has negative effect on the performance of PUs as it reduces primary successful probability and increases their forced termination probability. A similar effect was also observed with the increase in the SU residual interference distortion factor. Large amount of residual interference deteriorates the detection probability and leads to a reduced PU performance. The incorporation of λ _{FAR} into the CTMC model allows for obtaining exact and accurate state transition probabilities that improves calculation of the performance evaluation measures. The results of the proposed analytical model provide a new insight into the operation of CRNs and can be used to develop practical and more accurate CRN performance evaluation models. In future work, cooperative spectrum sensing can be considered for improving detection performance and/or for mitigating the effects of fading. Further study is also needed to investigate the case where on collisions, only SU calls will be terminated. Additionally, adaptive sensing parameters based on PU channel utilization can also be studied.
Declarations
Acknowledgements
The work of Janne Lehtomäki was supported by the Research Council of the University of Oulu. The work of Kenta Umebayashi was supported by the Strategic Information and Communications R&D Promotion Programme (SCOPE).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 J Mitola III. Cognitive radio: an integrated agent architecture for software defined radio. PhD thesis, Royal Institute of Technology, Sweden (2000).Google Scholar
 S Haykin, Cognitive radio: brainempowered wireless communications. IEEE J. Sel. Areas. Commun.23(2), 201–220 (2005).View ArticleGoogle Scholar
 IF Akyildiz, WY Lee, MC Vuran, S Mohanty, A survey on spectrum management in cognitive radio networks. IEEE Commun. Lett.46(4), 40–48 (2008).View ArticleGoogle Scholar
 A Ghasemi, ES Sousa, Spectrum sensing in cognitive radio networks: requirements, challenges and design tradeoffs. IEEE Commun. Mag.46(4), 32–39 (2008).View ArticleGoogle Scholar
 L Luo, S Roy, Efficient spectrum sensing for cognitive radio networks via joint optimization of sensing threshold and durations. IEEE Trans. Commun.60(10), 2851–2860 (2012).View ArticleGoogle Scholar
 T Yucek, H Arslan, A survey of spectrum sensing algorithms for cognitive radio applications. IEEE Commun. Surv. Tutorials. 11(1), 116–130 (2009).View ArticleGoogle Scholar
 J Lehtomäki. Analysis of energy based signal detection. PhD thesis, University of Oulu (2005). http://herkules.oulu.fi/isbn9514279255/isbn9514279255.pdf.
 JJ Lehtomäki, R Vuohtoniemi, K Umebayashi, On the measurement of duty cycle and channel occupancy rate. IEEE J. Sel. Areas Commun.31(11), 2555–2565 (2013).View ArticleGoogle Scholar
 H Urkowitz, Energy detection of unknown deterministic signals. Proc. IEEE. 55(4), 523–531 (1967).View ArticleGoogle Scholar
 RA Dillard, GM Dillard, Detectability of SpreadSpectrum Signals (Artech House, Norwood, 1989).Google Scholar
 GM Dillard, M Reuter, J Zeiddler, B Zeidler, Cyclic code shift keying: a low probability of intercept communication technique. IEEE Trans. Aerosp. Electron. Syst.39(3), 786–798 (2003).View ArticleGoogle Scholar
 DJ Lee, MS Jang, Optimal spectrum sensing time considering spectrum handoff due to false alarm in cognitive radio networks. IEEE Commun. Lett.13(2), 899–901 (2009).MathSciNetView ArticleGoogle Scholar
 RG Gallager, Discrete Stochastic Processes (Kluwer Academic Publishers, Norwell, Massachusetts, 1996).View ArticleGoogle Scholar
 H Kim, KG Shin, Efficient discovery of spectrum opportunities with MAClayer sensing in cognitive radio network. IEEE Trans. Mob. Comput.7(5), 533–545 (2008).MathSciNetView ArticleGoogle Scholar
 C Jiang, Y Chen, KJR Liu, Y Ren, Renewaltheoretical dynamic spectrum access in cognitive radio network with unknown primary behavior. IEEE J. Sel. Areas Commun.31(3), 406–416 (2013).View ArticleGoogle Scholar
 C Zhang, KG Shin, What should secondary users do upon incumbents’ return?IEEE J. Sel. Areas Commun.31(3), 417–428 (2013).View ArticleGoogle Scholar
 C Jiang, Y Chen, Y Gao, KJR Liu, Joint spectrum sensing and access evolutionary game in cognitive radio networks. IEEE Trans. Wirel. Commun.12(5), 2470–2483 (2013).View ArticleGoogle Scholar
 Jl Wang, Yh Xu, Z Gao, Qh Wu, Discretetime queuing analysis of opportunistic spectrum access: single user case. Frequenz. 65(11–12), 335–341 (2011).Google Scholar
 J Ko, H Ko, C Kim, Fast primary user detection during ongoing opportunistic transmission in OFDMbased cognitive radio. Wirel. Pers. Commun. 70(4), 1463–1471 (2013).View ArticleGoogle Scholar
 E Ahmed, A Eltawil, A Sabharwal, in proc. IEEE International Symposium on Antennas and Propagation Society (APSURSI). Simultaneous transmit and sense for cognitive radios using fullduplex: A first study (Chicago, USA, 2012), pp. 8–14.Google Scholar
 T Ihalainen, A Viholainen, TH Stitz, M Renfors, in proc. IEEE International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM). Reappearing primary user detection in cognitive radios (Cannes, France, 2010), pp. 1–5.Google Scholar
 W Lee, DH Cho, in proc. IEEE Wireless Communications and Networking Conference (WCNC). Concurrent spectrum sensing and data transmission scheme in a CR system (Las Vegas, USA, 2012), pp. 1326–1330.Google Scholar
 T Riihonen, S Werner, R Wichman, Mitigation of loopback selfinterference in fullduplex MIMO relays. IEEE Trans. Sig. Process. 59(12), 5983–5993 (2011).MathSciNetView ArticleGoogle Scholar
 K Umebayashi, JJ Lehtomäki, S Hatakeyama, Y Suzuki, in proc. IEEE Personal Indoor and Mobile Radio Communications (PIMRC), Wireless distributed network workshop. Cyclostationary spectrum sensing under fourlevel hypothesis for spectrum sharing (Toronto, Canada, 2011), pp. 2338–2343.Google Scholar
 W Han, J Li, Q Liu, L Zhao, Spatial false alarms in cognitive radio. IEEE Commun. Lett.5(5), 518–520 (2011).Google Scholar
 S Tang, Y Xie, Performance analysis of unreliable sensing for an opportunistic spectrum sharing system. Int. J. of Commun. Netw. Inf. Secur. (IJCNIS). 3(3), 240–246 (2011).MathSciNetGoogle Scholar
 B Wang, Z Ji, KJR Liu, TC Clancy, Primaryprioritized Markov approach for dynamic spectrum allocation. IEEE Trans. Wirel. Commun.8(4), 1854–1865 (2009).View ArticleGoogle Scholar
 S Tang, BL Mark, Modeling and analysis of opportunistic spectrum sharing with unreliable spectrum sensing. IEEE Trans. Wirel. Commun.8(4), 1934–1943 (2009).View ArticleGoogle Scholar
 Y Zhang, in proc. IEEE Int. Conf. Commun.Dynamic spectrum access in cognitive radio wireless networks (Beijing, China, 2008), pp. 4927–4932.Google Scholar
 IM Suliman, JJ Lehtomäki, K Umebayashi, M Katz, Analysis of cognitive radio networks with imperfect sensing. IEICE Trans. Commun.E96B(06), 1605–1615 (2013).View ArticleGoogle Scholar
 R Zhang, YC Liang, S Cui, Dynamic resource allocation in cognitive radio networks. IEEE Signal Process. Mag.27(3), 102–114 (2010).MathSciNetView ArticleGoogle Scholar
 R Urgaonkara, MJ Neely, Opportunistic scheduling with reliability guarantees in cognitive radio networks. IEEE Trans. Mob. Comput.8(6), 766–777 (2009).View ArticleGoogle Scholar
 M Höyhtyä. Adaptive power and frequency allocation strategies in cognitive radio systems. PhD thesis, University of Oulu 2014. http://www.vtt.fi/inf/pdf/science/2014/S61.pdf.
 BF Lo, A survey of common control channel design in cognitive radio networks. Phys. Commun.4(1), 26–39 (2011).View ArticleGoogle Scholar
 W Yin, P Ren, Q Du, Y Wang, Delay and throughput oriented continuous spectrum sensing schemes in cognitive radio networks. IEEE Trans. Wirel. Commun.11(6), 2148–2159 (2012).View ArticleGoogle Scholar
 S Hong, J Brand, JI Choi, M Jain, J Mehlman, S Katti, P Levis, Applications of selfinterference cancellation in 5G and beyond. IEEE Commun. Mag.52(2), 114–121 (2014).View ArticleGoogle Scholar
 S Li, RD Murch, An investigation into baseband techniques for singlechannel fullduplex wireless communication systems. IEEE Trans. Wirel. Commun.13(9), 4794–4806 (2014).View ArticleGoogle Scholar
 D Choi, D Park, Effective self interference cancellation in full duplex relay systems. Electron. Lett.48(2), 129–130 (2012).View ArticleGoogle Scholar
 JR Krier, IF Akyildiz, in in proc. IEEE Personal Indoor and Mobile Radio Communications (PIMRC). Active selfinterference cancellation of passband signals using gradient descent (London, UK, 2013), pp. 1212–1216.Google Scholar
 FF Digham, MS Alouini, MK Simon, in proc. IEEE International Conference on Communications (ICC). On the energy detection of unknown signals over fading channels (Alaska, USA, 2003), pp. 3575–3579.Google Scholar
 AH Nuttall, Some integrals involving the Qm function. IEEE Trans. Inf. Theory. 21(1), 95–96 (1975).MATHMathSciNetView ArticleGoogle Scholar
 T Riihonen, R Wichman, in IEEE International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM). Energy detection in fullduplex cognitive radios under residual selfinterference (Oulu, Finland, 2014), pp. 57–60.Google Scholar
 L Yun, L Song, Z Han, Y Li, Full duplex cognitive radio: a new design paradigm for enhancing spectrum usage. IEEE Commun. Mag.53(5), 138–145 (2015).View ArticleGoogle Scholar
 W Afifi, M Krunz, Incorporating selfinterference suppression for fullduplex operation in opportunistic spectrum access systems. IEEE Trans. Wirel. Commun.14(4), 2180–2191 (2015).View ArticleGoogle Scholar
 W Afifi, M Krunz, in proc. IEEE International Symposium on Dynamic Spectrum Access Networks (DYSPAN). Adaptive transmissionreceptionsensing strategy for cognitive radios with fullduplex capabilities (McLean, USA, 2014), pp. 149–160.Google Scholar
 W Afifi, M Krunz, in proc. IEEE International Conference on Computer Communications (INFOCOM). Exploiting selfinterference suppression for improved spectrum awareness/efficiency in cognitive radio systems (Turin, Italy, 2013), pp. 1258–1266.Google Scholar
 Y Xiao, I Ahmad, M Gao, J Yang, Y Zhang, Z Feng, Y Zhang, in proc. IEEE Vehicular Technology Conference (VTC Fall). Improved energy detector for full duplex sensing (Vancouver, Canada, 2014), pp. 1–5.Google Scholar
 E Brookner, False alarm rate and false alarm number for discrete and continuous time sampling. IEEE Trans. Aerosp. Electron. Syst.AES17(6), 809–814 (1981).View ArticleGoogle Scholar
 VG Kulkarni, Modeling and Analysis of Stochastic Systems (CRC Press, London, 1995).MATHGoogle Scholar
 L Kleinrock, Queueing Systems. Volume 1: Theory, 1st edn. (WileyInterscience, New York, 1975).Google Scholar
 G Ciardo, AS Miner, M Wan, AJ Yu, Approximating stationary measures of structured continuoustime Markov models using matrix diagrams. ACM SIGMETRICS Perform. Eval. Rev.35(3), 16–18 (2007).View ArticleGoogle Scholar
 P Buchholz, WH Sanders, in proc. IEEE First International Conference on the Quantitative Evaluation of Systems (QEST 2004). Approximate computation of transient results for large Markov chains (Enschede, The Netherlands, 2004), pp. 126–135.Google Scholar