Asymptotic Approximation of the Standard Condition Number Detector for Large Multi-Antenna Cognitive Radio Systems

Standard condition number (SCN) detector is a promising detector that can work e ﬃ ciently in uncertain environments. In this paper, we consider a Cognitive Radio (CR) system with large number of antennas (eg. Massive MIMO) and we provide an accurate and simple closed form approximation for the SCN distribution using the generalized extreme value (GEV) distribution. The approximation framework is based on the moment-matching method where the expressions of the moments are approximated using bi-variate Taylor expansion and results from random matrix theory. In addition, the performance probabilities and the decision threshold are considered. Since the number of antennas and/or the number of samples used in the sensing process may frequently change, this paper provides simple form decision threshold and performance probabilities o ﬀ ering dynamic and real-time computations. Simulation results show that the provided approximations are tightly matched to relative empirical ones.


Introduction
Cognitive Radio (CR), firstly proposed by Mitola [1], is the technology that provides solution for the scarcity and inefficiency in using the spectrum resource [2].For the CR to operate efficiently and to provide the required improvement in spectrum efficiency, it must be able to effectively detect the presence/absence of the Primary User (PU).Thus, Spectrum Sensing (SS) is the key element for the presence/absence detection process in any CR guarantee.
Several SS techniques were proposed in the last decade [3], however, Eigenvalue Based Detector (EBD) has been shown to overcome noise uncertainty challenges and performs adequately even in low SNR conditions as it does not need any prior knowledge about the noise power or signal to noise ratio.EBD is based on the eigenvalues of the received signal covariance matrix and it utilises results from Random Matrix Theory (RMT) [4].It detects the presence/absence of the PU by exploiting receiver diversity and includes the Largest Eigenvalue (LE) detector [5][6][7], the Scaled Largest Eigenvalue (SLE) detector [7][8][9] , and the Standard Condition Number (SCN) detector [5,[10][11][12][13][14][15][16][17].
Let W be the sample covariance matrix and denote by λ 1 ≥ λ 2 ≥ • • • ≥ λ K > 0 its ordered eigenvalues; then the SCN of W , defined as the ratio of the maximum to minimum eigenvalues, is given by: The SCN detector compares X with a certain threshold to decide about the existence of the PU.This threshold was set according to Marchenco-Pastur law (MP) in [5] which states that the largest eigenvalue (λ 1 ) and the smallest eigenvalue (λ K ) converge asymptotically to constants; however, this threshold is not related to any error constraints.In [10], the authors have provided an approximated relation between the threshold and the False-Alarm Probability (P f a ) by exploiting the Tracy-Widom distribution (TW) for the largest eigenvalue while maintaining the MP law for the smallest one.TW distribution is an asymptotic distribution for λ 1 [18].This work was further improved in [11,12] by considering the TW distribution and by using the Curtiss formula for the distribution of the ratio of random variables [19].In these two cases, the threshold could not be computed online and Lookup Tables (LUT) should be used instead.The exact distribution of the SCN was, also, derived in [13] for 2 antennas and in [14] for 3 antennas, however, it is very complicated to extend this work for CR with higher number of antennas.
On the other hand, Massive MIMO (Multiple-Input-Multiple-Output) is viewed as one of the foundational 5G technologies [20].Thus, CR with massive MIMO technology (equipped with tens to hundreds of antennas) could use any multi-antenna SS technique to identify the unused channels while achieving a significant increase of the performance of the SS detector.EBD presents an efficient way for multiantenna SS and could be used efficiently in CR with Massive MIMOs.However, in such practical scenario, the number of antennas and/or the number of samples used in the sensing process may frequently change.Accordingly, the implementation of the decision threshold must be dynamic and rely on real-time computations and thus simple form of the performance probabilities and the decision threshold are required.
In this paper, we are interested in finding a simple approximation for the SCN detector's performance probabilities and decision threshold.This allows the system, equipped with tens to hundreds of antennas, to dynamically compute its threshold online according to the instantaneous scenario.For this purpose, we propose to asymptotically approximate the SCN distribution with the Generalized Extreme Value (GEV) distribution by matching the first three central moments.This approximation yields a simple and yet accurate closed form expression for the SCN detector.Accordingly, the threshold could be simply computed.The main contributions of this paper are summarized as follows: • Derivation of the asymptotic central moments of the extreme eigenvalues.
• Derivation of an asymptotic approximated form of the central moments of the SCN from that of the extreme eigenvalues.
• Proposition of a simple and asymptotic closed form approximation for the SCN distribution using the central moments.
• Derivation of a simple form for the performance probabilities and the decision threshold for the SCN detector.
The rest of this paper is organized as follows.Section 2 provides the system model including the hypotheses analysis.Section 3 provides the asymptotic mean, variance and skewness of the extreme eigenvalues under H 0 and H 1 hypotheses.The asymptotic mean, variance and skewness of the SCN are derived in section 4.Then, we propose a new asymptotic approximation for the SCN distribution.In section 5, we derive the form of the performance probabilities as well as the decision threshold for the SCN detector.Theoretical findings are validated by simulations in section 6 while the conclusion is drawn in section 7.
Notations.Vectors and Matrices are represented, respectively, by lower and upper case boldface.The symbols |.| and tr(.) indicate, respectively, the determinant and trace of a matrix while (.) T , and (.) † are the transpose, and Hermitian symbols respectively.I n is the n × n identity matrix.Symbols ∼ stands for "distributed as", E[.] for the expected value and .2 for the norm.

System Model
Consider a CR system equipped with K receiving antennas aiming to detect the presence/absence of a single PU during a sensing period corresponds to N samples.After collecting N samples from each antenna, the received signal matrix, Y , is given by: where For this detection problem, two hypotheses exist: (i) H 0 : there is no PU and the received sample is only noise; and (ii) H 1 : the PU exists.The received vector, at instant n, under both hypotheses is given by: with η k (n) is a complex circular white Gaussian noise with zero mean and unknown variance σ 2 η , h k (n) is a the channel coefficient between the PU and antenna k at instant n, and s(n) stands for the primary signal sample modeled as a zero mean Gaussian random variable with variance σ 2 s .Without loss of generality, we suppose that K ≤ N and the channel is considered constant during the sensing time for simplicity.
H 0 hypothesis.By considering H 0 hypothesis, the received samples are complex circular white Gaussian noise with zero mean and unknown variance σ 2 η .Consequently, the sample covariance matrix is a central uncorrelated complex Wishart matrix denoted as W ∼ CW K (N , σ 2 η I K ) where K is the size of the matrix, N is the number of Degrees of Freedom (DoF), and σ 2 η I K is the correlation matrix.
H 1 hypothesis.By considering H 1 hypothesis and following our assumptions, the sample covariance matrix is a non-central uncorrelated complex Wishart matrix denoted as where Ω K is a rank-1 non-centrality matrix [21].
Let Σ K be the correlation matrix defined as: and denote by σ Then W , under H 1 , could be modeled as a central semi-correlated complex Wishart matrix denoted as W ∼ CW K (N , Σ K ) [22].Since Ω K is a rank-1 matrix, then Σ K belongs to the class of spiked population model with all but one eigenvalue of Σ K are still equal to σ 2 η while σ 1 is given by: with ω 1 is the only non-zero eigenvalue of Ω K .Denote the channel power by σ 2 h , then the average signal to noise ratio (SNR), under H 1 , is defined by: where σ 2 s could be estimated by ( s 2 /N ), and the channel power σ 2 h = ( h 2 /K).The non-centrality matrix is given by: where Σ K is the covariance matrix of Y , defined as As a result, and by using the property that the trace of a matrix equals the sum of its eigenvalues, then ω 1 could be written as:

Assymptotic Moments of Extreme Eigenvalues
This section considers the statistical analysis of the extreme eigenvalues (λ 1 and λ K ) of the sample covariance matrix (W ) by considering both hypotheses.Since SCN is not affected by the noise power, let σ 2 η = 1 and define the Asymptotic Condition (AC) and the Critical Condition (CC) as follows: These conditions are important as will be stated through this paper.

H 0 hypothesis
Let λ and λ be the maximum and minimum eigenvalue of W under H 0 respectively, then:

Distribution of λ
with a 1 (K, N ) and b 1 (K, N ), the centering and scaling coefficients respectively, are defined by: then, as AC is satisfied, λ 1 follows a TW distribution of order 2 (TW2) [23].

Distribution of λ
with a 2 (K, N ) and b 2 (K, N ), the centering and scaling coefficients respectively, are defined by: then, as AC is satisfied, λ K follows a TW2 [24].

Central Moments of λ
and λ The mean, variance and skewness of λ 1 and λ K are that of the TW2.They are given by µ T W 2 = −1.7710868074,σ 2 T W 2 = 0.8131947928 and S T W 2 = 0.2240842036 respectively [25].Accordingly, using (12), the mean, variance and skewness of λ H 0 1 are, respectively, given by: and using (15), the mean, variance and skewness of λ H 0 K are, respectively, given by:

H 1 hypothesis
Let λ and λ be the maximum and minimum eigenvalue of W under H 1 respectively, then: with a 3 (K, N ) and b 3 (K, N ), the centering and scaling coefficients respectively, are defined by: then, as AC and CC are satisfied, λ 1 follows a standard normal distribution (λ 1 ∼ N (0, 1)) [26].

Distribution of λ
As mentioned in [27], when Σ K has only one non-unit eigenvalue such that CC is satisfied, then only one eigenvalue of W will be pulled up.In other words, and as it could be deduced from [28, Proof of Lemma 2], the remaining K − 1 eigenvalues of W (λ Denote the centered and scaled version of λ with a 2 (K, N ) and b 2 (K, N ) are, respectively, given by ( 16) and (17).Then, as the AC and CC are satisfied, λ K follows a TW2.
It is worth mentioning that as CC is not satisfied and AC is satisfied, then λ [26].Thus, non of the eigenvalue of W will be pulled up and λ Accordingly, the PU signal has no effect on the eigenvalues and could not be detected.It follows that the same analysis under H 0 hypothesis is applied for this case. 1 are, due to (24), given respectively by:

Central Moments of λ
and using (27), the mean, variance and skewness of λ are respectively given by: As a result, this section provides a simple form for the central moments of the extreme eigenvalues.These moments are used, in the next section, to derive an approximation for the mean, the variance and the skewness of the SCN under both hypotheses.

SCN Distribution Approximation
This section approximates the asymptotic distribution of the SCN by the GEV distribution using moment matching.First, we consider both detection hypotheses and we derive an approximation of the mean, the variance and the skewness of the SCN to be used in the next subsection for the approximation.

Asymptotic Central Moments of the SCN
The bi-variate first order Taylor expansion of the function X = g(λ 1 , λ K ) = λ 1 /λ K about any point θ = (θ λ 1 , θ λ K ) is written as: with g λ i is the partial derivative of g over λ i .Let θ = (µ λ 1 , µ λ K ) with µ λ 1 and µ λ K are the means of λ 1 and λ K respectively, then it could be proved that: Accordingly, we give the following theorems that formulate a simple approximation for the central moments of the SCN.Theorem 1.Let X be the SCN of W ∼ CW K (N , σ 2 η I K ).The mean, the variance and the skewness of X, as AC is satisfied, can be tightly approximated using the mean, the variance and the skewness of the λ H 0 1 and λ H 0 K as follows: Proof.The result follows (35), ( 36) and (37) while considering λ H 0 1 and λ H 0 K asymptotically independent [29].The mean, the variance and the skewness of λ Theorem 2. Let X be the SCN of W ∼ CW K (N , Σ K ) where Σ K has only one non-unit eigenvalue.The mean, the variance and the skewness of X, as the AC and CC are satisfied, can be tightly approximated using the mean, the variance and the skewness of λ as follows: Proof.The result follows (35), ( 36) and (37) while considering λ 1 and λ H 1 K asymptotically independent [30].The mean, the variance and the skewness of λ

Approximating the SCN using GEV
GEV is a flexible 3-parameter distribution used to model the extreme events of a sequence of i.i.d random variables [31].These parameters are the location (δ), the scale (β) and the shape (ξ).In the following two propositions, we approximate the distribution of the SCN under H 0 and H 1 hypotheses respectively.Proposition 1.Let X be the SCN of W ∼ CW K (N , σ 2 η I K ) with defined skewness −0.63 ≤ S X < 1.14 1 .If AC is satisfied, then the CDF and PDF of X can be asymptotically and tightly approximated respectively by: where ξ, β and δ are defined respectively by: where µ X , σ 2 X and S X are defined in Theorem 1 and with defined skewness −0.63 ≤ S X < 1.14 and Σ K has only one non-unit eigenvalue.If AC and CC are satisfied, then the CDF and PDF of X can be asymptotically and tightly approximated by ( 44) and (45) respectively.The parameters ξ, β and δ are defined respectively by ( 46), ( 47) and (48) with µ X , σ 2 X and S X are defined in Theorem 2.
It is worth mentioning that the shape, the scale and the location parameters are respectively derived using the skewness, the variance and the mean of the GEV distribution.The moments of the GEV distribution are provided by [32].

SCN Detector Analysis
Recall that the SCN is given by: Denoting by α the decision threshold, then the probability of false alarm (P f a ), defined as the probability of detecting the presence of PU while it does not exist, and the detection probability (P d ), defined as the probability of correctly detecting the presence of These probabilities depend on the decision threshold being used.However, if the expressions of the P f a and P d are previously known, then a threshold could be set according to a required error constraint.If we denote the CDF of X under H 0 and H 1 hypotheses respectively by F 0 (.) and F 1 (.), then we can write: Consequently, for a given decision threshold ( α) the SCN detector algorithm could be summarized as follows: Performance Probabilities.Based on Propositions 1 and 2, and using (52) and (53), the P f a and P d are respectively expressed as: )ξ 1 ) where δ 0 , β 0 and ξ 0 are the location, scale and shape parameters under H 0 and are computed according to proposition 1 while δ 1 , β 1 and ξ 1 are the location, scale and shape parameters under H 1 and are computed according to proposition 2.
Decision Threshold.The threshold could be computed using (54) and (55) according to a required error constraint.For example, for a target false alarm probability ( γ), the threshold is given by:

Numerical validation
In this section, we verify the analytical derivation results through Monte-Carlo simulations.We validate the theoretical analysis presented in sections 3, 4 and 5.The simulation results are obtained by generating 10 5 random realizations of Y .For H 0 case, the inputs of Y are complex circular white Gaussian noise with zero mean and unknown variance σ 2 η while for H 1 case the channel is considered flat.
Table 1 shows the accuracy of the analytical approximation of the mean, the variance and the skewness of the SCN provided by Theorems 1 and 2. It can be easily seen that these Theorems provide a good approximation for the statistics of the SCN, however, it could be noticed that the skewness is not perfectly approximated.In fact, the skewness is affected by the slow convergence of the skewness of λ K that must converge to −S T W 2 (i.e.−0.2241) as AC is satisfied.For example, for K = 50, the empirical skewness increases from S λ K = −0.1504 to S λ K = −0.1819as the number of samples increases from N = 500 to N = 1000.Comparing these results with SCN results in Table 1, one can notice that the empirical and approximated SCN skewness become closer as λ K skewness converges to that of TW2.Accordingly, Theorems 1 and 2 are good approximations for the mean, the variance and the skewness of the SCN under both hypotheses.It is worth noting that one could approximate the SCN moments using second order bi-variate Taylor series to get a slightly higher accuracy, however, this will cost higher complexity and it is not necessary as shown in Table 1 and the following figures.
Figure 1 shows the empirical CDF of the SCN and its corresponding GEV approximation given by Proposition 1.The results are shown for K = {10, 20, 50, 100} antennas and N = {500, 1000} samples per antenna.Results show a perfect match between the empirical results and our proposed approximation.Also, it could be noticed that the convergence of the skewness does not affect the approximation and thus the skewness in Theorem 1 holds for this approximation  even though the convergence of the skewness of λ K is slow.
Figure 2 shows the empirical CDF of the SCN and its corresponding GEV approximation given by Proposition 2. The results are shown for K = {20, 50} antennas and N = {500, 1000} samples per antenna and ρ = −10dB.Results show high accuracy in approximating the empirical CDF.Also, the difference in the skewness shown in Table 1 does not affect the approximation.
Figure 3 shows the empirical Receiver Operating Characteristic (ROC) of the SCN detector and its corresponding proposed approximation.The results are shown for K = {20, 25, 50} antennas and N = 500 samples per antenna for SN R = −18dB.Results show that the proposed approximation matches the empirical results with high accuracy.In addition, Fig. 3 shows the gain in the performance as the number of antenna increases.
Figure 4 plots the empirical P d versus SNR (ρ) and its corresponding proposed analytical approximation.It is worth mentioning that Figures 3 and 4 show high improvement in the system performance by a simple increase of the number of antennas or number of samples used in the sensing process.Accordingly, it is very important to have a simple form for the performance probabilities and thus for the decision threshold so a CR system with large number of antennas2 can dynamically adapt its threshold  according to pre-defined error constraints and channel conditions.

Conclusion
In this paper, we have considered the SCN detector for large number of antennas in cognitive radios.
We have derived the asymptotic mean, variance and skewness of the SCN using those of the extreme eigenvalues of the sample covariance matrix by means of bi-variate Taylor expansion.A simple closed form approximation for the distribution of the SCN under H 0 and H 1 hypotheses are proposed.This approximation is based on the extreme value theory distributions and uses results from random matrix theory.Consequently, simple forms for the false-alarm probability, detection probability and the decision threshold are derived for real-time computations such that a CR system with large number of antennas can dynamically adapt its threshold according to pre-defined error constraints and channel conditions.In addition to their simple forms, simulation results show high accuracy of the proposed approximation for different number of antennas and different number of samples on various SNR values.

H 0 1 .
Denote the centered and scaled version of λ H 0 1 of the central uncorrelated Wishart matrix W ∼ CW K (N , I K ) by:

H 0 K.
Denote the centered and scaled version of λ H 0 K of the central uncorrelated Wishart matrix W ∼ CW K (N , I K ) by:

1 K.
The mean, variance and skewness of λ H 1

K
are given in Section 3.1.

1 K
are given in Section 3.2

EAI
Endorsed Transactions on Cognitive Communications 02 2017 -05 2017 | Volume 3 | Issue 11 | e1 Asymptotic Approximation of the Standard Condition Number Detector for Large Multi-Antenna Cognitive Radio Systems PU, are, respectively, given by:

Figure 1 .
Figure 1.Empirical CDF of the SCN and its corresponding proposed GEV approximation for different values of K and N under H 0 hypothesis.

Figure 2 .
Figure 2. Empirical CDF of the SCN and its corresponding proposed GEV approximation for different values of K and N under H 1 hypothesis with ρ = −10dB.

FalseFigure 3 .
Figure 3. Empirical ROC of the SCN detector and its corresponding proposed approximation for different values of K with N = 500 and ρ = −18dB.

Figure 4 (
a) shows how the P d is improved as N increases while Figure 4(b) shows the P d improvement as K increases.Both Figures show a high P d when using large number of antennas and relatively large number of samples.

Figure 4 .
Figure 4. Empirical P d of the SCN detector as a function of SNR and its corresponding proposed approximation for different values of N and K with P f a = 0.001.

Table 1 .
Empirical mean, variance and skewness of the SCN under H 0 and H 1 hypotheses and it corresponding proposed analytical approximation using Theorems 1 and 2 respectively.