Dimension estimation-based spectrum sensing for cognitive radio
- Bassem Zayen^{1}Email author and
- Aawatif Hayar^{2}
https://doi.org/10.1186/1687-1499-2012-64
© Zayen and Hayar; licensee Springer. 2012
Received: 1 May 2011
Accepted: 24 February 2012
Published: 24 February 2012
Abstract
In this article, we will derive closed-form expressions of false alarm probabilities for a given threshold for the dimension estimation-based detector (DED) using Akaike information criterion (AIC) and the minimum description length (MDL) criterion. Specifically, the DED algorithm will be formulated as a binary hypothesis test using AIC and MDL curves. Based on the proposed statistic test, we will express the probability of false alarm of the DED algorithm for a fixed threshold using the cumulative density function for the distribution of Tracy-Widom of order two. The derived analytical decision thresholds are verified with Monte-Carlo simulations and a comparison between simulation and analytical results to confirm the theoretical results are presented. These results confirm the very good match between simulation and theoretic results.
Keywords
1. Introduction
The discrepancy between current-day spectrum allocation and spectrum use suggests that radio spectrum shortage could be overcome by allowing a more flexible usage of the spectrum. Flexibility would mean that radios could find and adapt to any immediate local spectrum availability. A new class of radios that is able to reliably sense the spectral environment over a wide bandwidth detects the presence/absence of legacy users (primary users) and uses the spectrum only if the communication does not interfere with primary users (PUs). It is defined by the term cognitive radio [1–3]. Cognitive radio (CR) technology has attracted worldwide interest and is believed to be a promising candidate for future wireless communications in heterogeneous wideband environments.
CR has been proposed as the means to promote efficient utilization of the spectrum by exploiting the existence of spectrum holes. The spectrum use is concentrated on certain portions of the spectrum while a significant amount of the spectrum remains unused. It is thus key for the development of CR to invent fast and highly robust ways of determining whether a frequency band is available or occupied. This is the area of spectrum sensing for CR which will be considered in this article.
There are several spectrum sensing strategies that were proposed for CR. These strategies are categorized in two families: feature detection strategies and blind detection strategies. The feature detection approaches assume that a PU is transmitting information to a primary receiver when a secondary user (SU) is sensing the primary channel band. The elaboration of sensing techniques that use some prior information about the transmitted signal is interesting in terms of performance. In fact, feature detection algorithms employ knowledge of structural and statistical properties of PU signals when making the decision. The most known feature sensing technique is the cyclostationarity based detector (CD) [4]. Completely blind spectrum sensing techniques that do not consider any prior knowledge about the PU transmitted signal are more convenient to CR. A few methods that belong to this category have been proposed, but most of them suffer from the noise uncertainty and fading channels variations [5–8]. One of the most popular blind detectors is the energy detector (ED) [9]. This detector is the most common method for spectrum sensing because of its non-coherency and low complexity. [10] is an excellent reference on spectrum sensing methods. It gives a literature survey on feature detection and blind detection strategies. In this article, the CD and ED will serve as references when evaluating the performance of the dimension estimation-based detectors.
It is stated that current spectrum sensing techniques suffer from challenges in the low signal to noise range. The reasons for this have to be analyzed. It is suggested that information theoretic criteria is a possible area to look for a solution to overcome the problem. It is apparent that the problem at hand is wide and challenging. The initial attempt to apply information theoretic criteria for spectrum sensing was presented in [11, 12]. The study presented in [11] suggested to use model selection tools like Akaike information criterion (AIC) and the minimum description length (MDL) criterion to conclude on the nature of the sensed band. These tools were used as detection rules for the dimension estimation detector (DED) [12]. AIC criterion was first introduced by Akaike [13] and Schwartz [14] for model selection. It was shown in [13] that the classical maximum likelihood principle can be considered to be a method of asymptotic realization of an optimum estimate with respect to a very general information theoretic criterion [13]. In [11, 12], however, the AIC and MDL criterions were investigated in order to sense the signal presence. Specifically, the number of significant eigenvalues determined by the value which minimizes the AIC and/or MDL criterion was used as detection rule to decide on the presence/absence of data in the signal. The same idea was applied in [15, 16], published after [11], to develop two spectrum sensing algorithms exploiting the maximum or/and the minimum eigenvalue as detection rule. One is based on the ratio of the maximum eigenvalue to the minimum eigenvalue, the other is based on the ratio of the average eigenvalue to the minimum eigenvalue. However, in [15, 16], the model selection has not been considered. In [17], published in 2010, the authors used information theoretic criteria to detect PU presence. The proposed algorithm in [17] does not involve any threshold value when computing the false alarm probability. Otherwise, the false alarm probability derived in this work takes into account the threshold value. We will detail later, in Subsection 3.3, the difference between the proposed algorithm in this paper and the one derived in [17].
The study presented in [11, 12] were a preliminary step for this idea. Indeed, no threshold expression was given and the decision was taken using the values minimizing the AIC and/or MDL criterion computed by simulation. Also, in [12] all AIC and MDL values are computed to find the minimum values and to decide then on the availability of the PU band. In this article, however, we will simply compute the first and the second values of AIC and MDL to make this decision. For this purpose, we will present the DED detector as a binary hypothesis test. We will then give the exact threshold expressions of the DED detector using the two selection tools AIC and MDL. Specifically, we will derive closed-form expressions of false alarm probabilities for a given threshold using both AIC and MDL criterion. We will use in this derivation the cumulative density function (CDF) for the distribution of Tracy-Widom of order two [18]. The analytical results will be compared with simulation results.
The rest of this article is organized as follows. In Section 2, we will formulate the two users selection tools used throughout the development of the proposed algorithm. The DED algorithm will be presented in Section 3 using AIC and MDL criterion. We will derive in Section 4 closed-form expressions of false alarm probabilities for a given thresholds using both AIC and MDL criterion. Performance evaluation and advantages will be described in Section 5 and a comparison of the proposed detector with reference detectors will be given. The performance will be assessed under different conditions, using three simulation scenarios. Finally, Section 6 presents the conclusions of this article.
2. Background of information theoretic criteria
In this section, we will provide the background of information theoretic criteria. The general problem for model selection using information theoretic criteria is: given a set of N observations {x_{1}, x_{2},..., x_{ N }} and a family of operating models which are represented by a parameterized family of probability density functions f, determine the best fit model. The operating models are usually unknown, since only a finite number of observations is available. Therefore, approximating probability model must be specified using the observed data, in order to estimate the operating model. The approximating model is denoted as g_{ θ }, where the subscript θ indicates the U-dimensional parameter vector, which in turn specifies the probability density function.
3. Spectrum sensing algorithms
where ϒ(x) denotes the test statistic for the given detector. In the following, we will describe two existing spectrum sensing algorithms. We select an example of feature spectrum sensing algorithms, the CD detector, and an example of blind sensing algorithms, the ED detector. These algorithms will serve as references when evaluating the novel approaches resulting from the research.
3.1. ED algorithm
where Q denotes the cumulative distribution function [21] of a χ^{2} distributed random variable with 2p degrees of freedom. γ_{ED} is the detection threshold of the ED and σ^{2} is the noise variance [9].
3.2. CD algorithm
CD has received a considerable amount of attention in the literature. Recent bibliography on cyclostationarity, including a large number of references on cyclostationarity based detection, is provided in [22]. The CD algorithm used in this article was presented in [23]. We will give in this subsection a brief description of this algorithm.
where G(.) is the (lower) incomplete gamma function [23]. The main advantage of the cyclic autocorrelation function is that it differentiates the noise energy from the modulated signal energy.
3.3. DED algorithm
We define here the two thresholds γ_{AIC} and γ_{MDL} in order to decide on the nature of the received signal. These thresholds depend only on P_{FA} and are calculated in the following section. In [17], does not involve any threshold value. A simple comparison between AIC(0) and AIC(1) makes the decision about the presence or absence of PU signal. The threshold value was neglected, i.e., at hypothesis H_{0} for example, ϒ_{ AIC }(x) = AIC(0) < AIC(1).
4. False alarm probability computation
A theoretical probability of false alarm will be derived in this section using AIC and MDL criterion. The analytical results will be compared with simulation results to confirm the theoretical expression of thresholds and probabilities of false alarm.
4.1. DED-AIC false alarm probability
Infact, the sum of the eigenvalues of the estimated covariance matrix $\widehat{\mathbf{R}}$, given by (4.16), is equivalent to $\frac{1}{\mathsf{\text{pN}}}\mathsf{\text{Tr}}\left({\sum}_{n=1}^{N}{\mathbf{x}}_{n}{\mathbf{x}}_{n}^{H}\right)$. At hypothesis H_{0}, the received vector involves only the noise samples, thats why, $\frac{1}{\mathsf{\text{pN}}}\mathsf{\text{Tr}}\left({\sum}_{n=1}^{N}{\mathbf{x}}_{n}{\mathbf{x}}_{n}^{H}\right)$ is the unbiased estimation of the covariance of the white noise and it is equivalent to σ^{2}.
Generally, it is difficult to evaluate the function F_{2}. Fortunately, it can be computed using Matlab [18].
4.2. DED-MDL false alarm probability
4.3. Simulation and analytical results comparison
Simulation and analytical results comparison
P _{FA,AIC} | p= 100 0.0531 | p= 150 0.0518 | p= 200 0.0504 | |
---|---|---|---|---|
Simulation results | P _{FA,MDL} | 0.0549 | 0.0533 | 0.0520 |
γ _{AIC} | 3.857e04 | 2.590e04 | 2.152e04 | |
γ _{MDL} | 3.613e04 | 2.097e04 | 1.956e04 | |
P _{FA,AIC} | 0.0500 | 0.0500 | 0.0500 | |
Analytical results | P _{FA,MDL} | 0.0500 | 0.0500 | 0.0500 |
γ _{AIC} | 3.762e04 | 2.527e04 | 1.984e04 | |
γ _{MDL} | 3.484e04 | 1.825e04 | 1.754e04 |
5. Performances evaluation
The transmitted DVB-T primary user signal parameters
Bandwidth | 8 MHz |
---|---|
Mode | 2K |
Guard interval | 1/4 |
Channel models | Rayleigh/Rician (K = 1) |
Maximum Doppler shift | 100 Hz |
Frequency-flat | Single path |
Sensing time | 1.25 ms |
Location variability | 10 dB |
Three different scenarios with different properties have been chosen to evaluate the spectral detection performance, subject to provide different attributes so that the performance can be assessed under different conditions, aiming to provide fair conditions before making conclusions. OFDM is the modulation of choice for the three simulation scenarios to be used as evaluation tools in this report. In OFDM, a wideband channel is divided into a set of narrowband orthogonal subchannels. OFDM modulation is implemented through digital signal processing via to the FFT algorithm [25]. In Scenario 1, we use a DVB-T OFDM signal in an AWGN channel. It is assumed that the detection performance in AWGN will provide a good impression of the performance, but it is necessary to extend the simulations to include signal distortion due to multipath and shadow fading. Scenario 2 utilizes the same DVB-T OFDM signal as Scenario 1, but to make the simulations more realistic, the signal is subjected to Rayleigh multipath fading and shadowing following a log normal distribution in addition to the AWGN. The maximum Doppler shift of the channel is 100 Hz and the standard deviation for the log normal shadowing is 10 dB. Since the fading causes the channel to be time variant, it is necessary to apply longer averaging than in Scenario 1 to obtain good simulation results. Thus the number of iterations in the Monte Carlo simulation is increased from 500 to 1000. The third simulation scenario utilizes also a DVB-T OFDM signal in Rician multipath fading with shadowing. The K-factor for the Rician fading is 10, which represents a very strong line of sight component. The maximum Doppler shift of the channel and the standard deviation for the log normal shadowing are the same as in the second scenario.
5.1. Non-cooperative sensing evaluation
It is obvious from Figure 2b,c how the absolute detection performance deteriorates when the signal is subjected to channel fading. The P_{ D }slope for all the detectors starts dropping at higher SNR values than for the AWGN case. While the P_{ D }curves started dropping off in the range from approximately -3 dB to about -5 dB for the four detectors in the AWGN channel of scenario 1, all curves start dropping off before 8 dB under the fading applied in Scenarios 2 and 3.
In general, some detectors like CD assume that some information about PU signal are explicitly known and their performance is optimal or close to optimal when this assumption is valid. The performance of such detectors deteriorates rapidly even for small departures from the underlying assumptions. Nonparametric or robust detectors on the other hand make no such assumptions and maintain their reliable performance in all conditions. There is a trade-off between robustness and optimality and robust techniques exchange the optimality to reliability. We developed in this work a realistic detector since it does not require any a priori knowledge about PU signal.
5.2. Cooperative sensing evaluation
The challenges of cooperative sensing include the development of efficient information sharing algorithms and increased complexity. Cooperative sensing decreases also the probability of false alarms considerably. In addition, cooperation can solve the hidden PU problem and can decrease sensing time. It can also mitigate the multi-path fading and shadowing effects, which improve the detection probability. However, the cooperation causes adverse effects on resource-constrained networks due to the additional operations and overhead traffic.
In this article the cooperative spectrum sensing is performed as follows:
Step 1. Every SU performs local spectrum measurements independently and then makes a binary decision.
Step 2. All the SUs forward their binary decisions to a FC.
Step 3. The FC combines those binary decisions and makes a final decision to infer the absence or presence of the PU in the observed band.
In the above mentioned cooperative spectrum sensing algorithms, each cooperative partner makes a binary decision based on its local observation and then forwards one bit of the decision to the FC. At the FC, all one-bit decisions are fused together according to an "OR" logic. This cooperative sensing algorithm is referred to as decision fusion.
5.3. Complexity study
Complexity comparison of the different sensing techniques
Sensing method | Complexity |
---|---|
CD | ${p}^{2}+\mathcal{O}\left(p\text{log}\left(p\right)\right)$ |
ED | p |
DED | $N{p}^{2}+Np+\mathcal{O}\left({p}^{3}\right)$ |
6. Conclusion
In this article, we derived the exact threshold expressions of the dimension estimation based spectrum sensing using AIC and MDL criterion. This is based on the distribution of Tracy-Widom of order two. Simulations using three different scenarios with different properties DVB-T PU systems were presented in order to verify the derived threshold values based on the probability of detection performance. It has been shown that analytical and empirical results are coincide with each other.
This paper provided also a number of simulations aimed at assessing the performance of the proposed detectors in comparison with two reference detectors, ED and CD detectors. From the presented results, we show that the ED has lost its detecting ability when decreasing the SNR. For sufficiently low SNR, robust detection becomes impossible. These results come from the fact that the theoretical analysis for the ED algorithm assumes the noise variance to be known, and the underlying noise to have a perfect stationary Gaussian distribution. This assumption does not hold. In reality, the noise variance will usually not be completely stationary. The assumption about the distribution of the noise is also known to be weak. On the other hand, we find that if knowledge of signal parameters is provided, the CD detector can still perform a high probability of detection. Since this type of detection requires a priori knowledge about the received signal, they are not blind. Therefore, a CD can perform better than other detectors in discriminating against noise due to its robustness to the uncertainty in noise power. However, it is computationally complex and requires a significantly long observation time. The proposed detectors however do not require any information about PU transmitted signal and can detect the presence or the absence of PU blindly. Blind detection of telecommunication signals in a radio band is very helpful in some CR environments; especially, when SU does not have enough information about the PU. Finally, to adopt a spectrum sensing algorithm for a given situation depends on the CR environments and the spectrum sensor characteristics.
Declarations
Acknowledgements
The research leading to these results had received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement SACRA n°249060, and the European projects ACROPOLIS and CROWN.
Authors’ Affiliations
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