On the detection probability of the standard condition number detector in finitedimensional cognitive radio context
 Hussein Kobeissi^{1, 2}Email author,
 Youssef Nasser^{3},
 Amor Nafkha^{1},
 Oussama Bazzi^{2} and
 Yves Louet^{1}
https://doi.org/10.1186/s1363801606340
© Kobeissi et al. 2016
Received: 29 September 2015
Accepted: 9 May 2016
Published: 23 May 2016
Abstract
Standard condition number (SCN) detector is an efficient detector in multidimensional cognitive radio systems since no a priori knowledge is needed. The earlier studies usually assume a large number of dimensions and a large number of samples per dimension and use random matrix theory (RMT) to derive asymptotic distributions of the SCN metric. In practice, the number of dimensions may not be large enough for the SCN distribution to be well approximated by the asymptotic ones. In this context, the false alarm probability is considered in literature and formulas for 2D, 3D, and infinitedimensional systems have been derived. However, the detection probability, which is of great importance in cognitive radio, has not been well discussed in literature. In this paper, we discuss, analytically, the detection probability of the SCN detector. Since the probability of detection is totally related to the SCN distribution, we derive new results on the joint ordered eigenvalues and SCN distributions for central semicorrelated Wishart matrices. These results are used to approximate the detection probability by the noncentral/central approximation. We consider systems with three or more dimensions, and we give an approximated form of the detection probability. The analytical results of this paper on probability of detection along with those on probability of false alarm present a complete performance analysis and are validated through simulations. We show that the proposed analytical expressions provide high accuracy and that the SCN detector outperforms the wellknown energy detector and the largest eigenvalue detector even with a small number of dimensions and low noise uncertainty environments.
Keywords
1 Introduction
Cognitive radio (CR) is considered as a promising solution for the scarcity and inefficient use of the spectrum [1, 2]. It allows the secondary (unlicensed) user (SU) to use the spectrum in an opportunistic way so that it does not cause harmful interference to the primary (licensed) user (PU). One possibility is to detect whether the PU is present or not to avoid any interference. Therefore, spectrum sensing has a key functionality in CR as it allows the CR to differentiate between the spectrum being used and the spectrum holes.
Several spectrum sensing techniques were proposed in the last decade [3]. Energy detector (ED) is among the most popular sensing techniques; it is simple, noncoherent, and needs no prior knowledge about the PU’s signal. However, ED is very sensitive to noise variations and requires perfect knowledge of the noise power. Thus, the performance of ED suffers from a high degradation under low signaltonoise ratio (SNR) and noise uncertainty conditions [4, 5].
Recently, eigenvaluebased detector (EBD) has been shown to overcome noise uncertainty challenges and performs adequately even in low SNR conditions. EBD is based on the eigenvalues of the received signal covariance matrix. In this context, the multidimensional aspect of a cognitive radio could be (i) multiantenna CR [6, 7], (ii) cooperative scenario between several CR nodes [8], (iii) oversampling within the CR node [9], or any combination of them. If the entries within these dimensions are considered Gaussian, then the receiver’s covariance matrix is known as a Wishart matrix in random matrix theory (RMT) [10]. Results from RMT have been used to study the EBD algorithms which include the largest eigenvalue (LE) detector proposed in [9, 11], the scaled largest eigenvalue (SLE) detector [12, 13], and the standard condition number (SCN) detector [9, 14–18]. The LE is a semiblind detector since it requires the knowledge of the noise power whereas the SLE and SCN are blind detectors that do not require this knowledge.
If we denote the maximum and the minimum eigenvalues of the receiver’s covariance matrix by λ _{max} and λ _{min}, respectively, then the standard condition number (SCN) is defined as the ratio between λ _{max} and λ _{min} (λ _{max}/λ _{min}). In multipleinput multipleoutput (MIMO) systems, the SCN has an effective connection with MIMO receiver performance in spatial multiplexing systems [19–21] and it indicates the multipath richness of the MIMO channel [22, 23]. In CR systems, the SCN detector compares this ratio to a threshold to decide whether the PU exists or not. The performance of a detection algorithm is, usually, characterized by two types of probabilities: the false alarm probability (P _{fa}) and the detection probability (P _{d}). The latter depends on the decision threshold and are related to the statistics of the detector metric. Thus, the characteristics of the SCN are of great importance in both MIMO and CR systems.
The SCN metric was well studied in the recent years; for example, this metric was studied asymptotically in [14] and the threshold was presented according to MarchenkoPastur (MP) law [24]. If the size of the receiver’s matrix is K×N, then MP law proves that the largest (λ _{max}) and the smallest (λ _{min}) eigenvalues of the receiver’s covariance matrix converge to constants as (K,N)→∞ with c=^{ K } /_{ N } is a constant ratio [25]. These constants are simply determined by c and the powers of the signal and the noise.
In [9], the authors improved the accuracy of the asymptotic statistical distribution of the SCN by using the TracyWidom (TW) distribution to model the largest eigenvalue (numerator of the SCN) [26] while maintaining the MP representation of the smallest one. TW distribution is a limiting distribution of the largest eigenvalue of a central Wishart matrix as (K,N)→∞ [27]. This approximation of the SCN, by TW distribution for the numerator and MP law for the denominator, results in an approximated relation between the decision threshold and the P _{fa}.
This work was further extended in [15, 16] by using the Curtiss formula for the distribution of the ratio of random variables (i.e., SCN) [28] where both the largest and the smallest eigenvalues converge to TracyWidom distributions when (K,N)→∞ as shown in [26, 29]. Moreover, by the exploitation of the normal and TW distributions and using the Curtiss formula, the authors in [17] provided a form for the probability of missed detection (P _{md}=1−P _{d}) for sufficiently large values of K and N. However, all these expressions include TW distribution and Curtiss formula that are hard to evaluate numerically online in addition to the considered asymptotic assumptions which are not necessarily practical.
By considering finite dimensions, the authors in [30] provided the exact generic form of the distribution of the SCN based on the joint distribution of the ordered eigenvalues of Wishart matrix. Consequently, [18] provides the exact forms of the P _{fa} of the SCN detector for CR with two dimensions and [31] provides P _{fa} form for three dimensions. In addition, the P _{d} for twodimensional systems was approximated using the noncentral/central approximation and provided in [18] following the results in [30]. When K≥3 (for Kdimensional system), the form of the P _{d} could be derived using the results in [30] as long as the noncentrality matrix Ω _{ K } is full rank. Moreover, the noncentral/central approximation is used to approximate the distribution of the noncentral uncorrelated Wishart matrix by the distribution of the central semicorrelated Wishart matrix. However, using this method to approximate the distribution of the SCN requires normalization and the behavior of this approximation could be affected especially at high SNR.

Using the noncentral/central approximation, the eigenvalues of the correlation matrix of the central semicorrelated Wishart matrix may coincide. For this purpose, we generalize the joint distribution of the ordered eigenvalues of the central semicorrelated Wishart matrix. In our derivations, the eigenvalues of the correlation matrix can be equal or distinct.

The authors in [30] consider the case where all the eigenvalues of the correlation matrix are distinct. In this paper, we show that the generic form of the cumulative distribution function (CDF) of the SCN given in [30] is still valid if the eigenvalues are not distinct. However, the form of the parameters differ where we give the new forms.

We derive the analytical forms of the CDF and the probability density function (PDF) of the SCN of threedimensional systems when Ω _{ K } is not a fullrank matrix.

For the sake of completeness, we also derive the CDF and the PDF of the SCN of threedimensional systems when Ω _{ K } is a fullrank matrix. Thus, the results of this paper combined with results in [31] form a complete closedform system for the threedimensional cognitive radio.

Since the previous studies consider the noncentral/central approximation in low SNR regions [18], we study the impact of the system parameters on the accuracy of the noncentral/central approximation. This impact is given in terms of P _{d} as it is an important target in the general CR context.
The rest of this paper is organized as follows. In Section 2, the system model is introduced and discussed. In Section 3, the noncentral/central approximation is recalled and analyzed. In Section 4, we give the joint ordered eigenvalue distribution with a generalization to equal eigenvalues of the correlation matrix. In Section 5, we emphasize on the general form of the CDF of the SCN for the central semicorrelated Wishart matrix when L eigenvalues of the correlation matrix are equal. The exact CDF and PDF of this SCN for threedimensional systems are derived as well. Simulation results and approximation accuracy are discussed in Section 6, and we conclude in Section 7.
Notations: Vectors and matrices are represented, respectively, by lowercase and uppercase boldface. The symbols . and tr(.) represent, respectively, the determinant and trace of a matrix while (.)^{1/2}, (.)^{T}, and (.)^{ † } are the square root, transpose, and Hermitian symbols, respectively. I _{ n } is the n×n identity matrix. Symbol ∥.∥^{2} stands for the norm, E[.] stands for the expected value, and ∼ stands for distributed as.
2 System model
where y(n)=[y _{1}(n),⋯,y _{ K }(n)]^{T} is the observed K×1 complex samples across all the dimensions at instant n. η(n) is a K×1 complex circular white Gaussian noise. h(n) is a K×1 complex vector that represents the channel coefficients between the PU and each of the CR dimensions, and s(n) stands for the primary signal sample at instant n.
Without loss of generality, we suppose that K≤N and we define the received sample covariance matrix as W=Y Y ^{ † }.
Under \(\mathcal {H}_{0}\), the input of the matrix Y is a complex circular white Gaussian noise with zero mean and unknown variance \(\sigma _{\eta }^{2}\), then W is well known as a central uncorrelated complex Wishart matrix and is denoted by \( \boldsymbol {W} \sim \mathcal {CW}_{K}(N, \sigma _{\eta }^{2}\mathbf {I}_{K})\) where K is the size of the matrix, N is the number of degrees of freedom (DoF), and \(\sigma _{\eta }^{2}\mathbf {I}_{K}\) is the correlation matrix.
Under \(\mathcal {H}_{1}\), we suppose that the channel remains constant during the sensing time and we have a single PU whose signal is drawn independently from Gaussian process for every sample. Consequently, W follows a noncentral uncorrelated complex Wishart distribution which is denoted as \(\boldsymbol {W} \sim \mathcal {CW}_{K}(N, \sigma _{\eta }^{2}\mathbf {I}_{K}, \mathbf {\Omega }_{K})\), where Ω _{ K } is a rank 1 noncentrality matrix [32].
The exact general form of the distribution of SCN under \(\mathcal {H}_{0}\) (F _{0}(t)) has been derived in [30]. Two particular cases have been considered in [18, 31] when K=2 and 3, respectively. Asymptotically, this distribution could be found in [9, 14, 15]. Under \(\mathcal {H}_{1}\), an asymptotic distribution of the SCN, F _{1}(t), could be found in [17]. In the finite case, the exact generic form of F _{1}(t) is derived in [30]; however, further numerical evaluation would either require NuttallQ function which could be replaced by Marcum Qfunction and a finite weighted sum of Bessel functions [33] or by hypergeometric functions that could be expanded to an infinite sum (see, for example, [34] for K=2 and N=2). Thus, and since both solutions are difficult to manipulate,^{1} a third solution is to use the noncentral/central approximation that approximates the distribution of the noncentral uncorrelated Wishart matrix by the distribution of the central semicorrelated Wishart matrix. This approximation is presented in the next section. The exact general form of the distribution of the SCN of central semicorrelated Wishart matrix is provided by [30] and is used to approximate F _{1}(t) for K=2 in [18, 30].
3 Noncentral/central approximation and parameter analysis
The noncentral/central approximation is recalled in Lemma 1.
Lemma 1 ([35]).
The complex noncentral Wishart matrix \(\boldsymbol {W} \sim \mathcal {CW}_{K}(N, \kappa ^{2}\mathbf {I}_{K}, \mathbf {\Omega }_{K})\) and the complex semicorrelated Wishart matrix \(\boldsymbol {W} \sim \mathcal {CW}_{K}(N, \widehat {\mathbf {\Sigma }}_{K})\) with effective correlation matrix \(\widehat {\mathbf {\Sigma }}_{K} = \kappa ^{2} I_{K} + {}^{\mathbf {\Omega }_{K}}\!/_{N}\) have the same first and secondorder moments differing by \({}^{\mathbf {\Omega }_{K}}\!/_{N}\).
Using Lemma 1, the distribution of the SCN under \(\mathcal {H}_{1}\) hypothesis could hence be approximated by the distribution of the SCN of complex semicorrelated Wishart matrix with a proper correlation matrix. It is worth noting that this approximation for the SCN distribution also requires a normalization by κ ^{2} (W _{ n }=κ ^{−2} W) which gives the same SCN as W. Therefore, following (8), the detection probability is, accordingly, approximated.
where Σ _{ K } is the covariance matrix of Y, defined as \(\mathbf {\Sigma }_{K} = E[(\boldsymbol {Y}\boldsymbol {M})(\boldsymbol {Y}\boldsymbol {M})^{\dagger }]=\sigma _{\eta }^{2} I_{K}\), and M is the mean of Y defined as M=E[Y]=h s ^{T} with h=[h _{1} h _{2}⋯h _{ K }]^{T} and s=[s(1)s(2)⋯s(N)]^{T}. It is worth mentioning that (11) will not be changed after normalization.
Thus, Ω _{ K } is not full rank and the eigenvalues of \(\widehat {\mathbf {\Sigma }}_{K}\) are not distinct. It is worth noting that for the exact form of P _{d}, it would be useful to refer to the joint distribution of the ordered eigenvalues of noncentral uncorrelated Wishart matrix with arbitrary noncentrality matrix rank derived in [36, App. I] and consider rank 1 case. However, as discussed before, it is hard to derive a tractable form for the SCN distribution since it requires NuttallQ functions or hypergiometric functions. Therefore, by considering the noncentral/central approximation, results based on assuming all correlation eigenvalues distinct could not be used and new results must be derived. Accordingly, an approximated and simpler form for the P _{ d } will be derived.
4 Joint ordered eigenvalues distribution generalization
As discussed in the previous section, all but one of the eigenvalues of the correlation matrix are equal. In this section, and since the distribution of the SCN metric is derived using the joint distribution of the eigenvalues of the Wishart matrix, we generalize the joint distribution of the ordered eigenvalues of the central semicorrelated Wishart matrix. This new form of the joint distribution covers any number (L) of equal eigenvalues positioned anywhere in vector σ and is given in Theorem 1 below.
In addition, Theorem 1 is also useful in MIMO systems as it could be used to derive marginal and joint distributions in correlated Rayleigh channels when the correlation eigenvalues may coincide. Moreover, these derivations could, also, approximate that of the Rician channels when the mean channel matrix is of rank 1; this could be satisfied when all channel vectors have the same mean value or in some other practical scenarios where the lineofsight (LOS) component of the channel matrix approaches the scaled allones matrix [37].
Theorem 1 (Central semicorrelated Wishart when Σ _{ K } is full rank with L equal eigenvalues).

Φ(λ) is a K×K Vandermonde matrix of entries \(\lambda _{j}^{i1}\).

Ψ(λ) is a K×K matrix of the form E(λ,σ) given, for all 1≤j≤K, by$$ \{\mathbf{E}(\boldsymbol\lambda,\boldsymbol\sigma)\}_{i,j} = \left\{\begin{array}{cc} \frac{\partial^{qi}(e^{\frac{\lambda_{j}}{\sigma_{p}}})}{\partial\sigma_{p}^{qi}} & p\leq i \leq q \\ e^{\frac{\lambda_{j}}{\sigma_{i}}} & otherwise\end{array}\right. $$(14)

ξ(λ _{ i }) is defined as \(\xi (\lambda _{i}) = \lambda _{i}^{NK}\).

The normalization constant is given by
$$ C = \frac{\prod_{i<j}^{K} \sigma_{i} \sigma_{j}}{\prod\limits_{i=1}^{K}(Ni)! {\sigma_{i}^{N}} \prod\limits_{{i<j}\atop{\sigma_{i} \neq \sigma_{j}}}^{K}(\sigma_{j}  \sigma_{i}) \, \Gamma_{L}(L)}, $$(15)
Proof.
See Appendix 1.
It can be easily shown that by taking L=1, the parameters in Theorem 1 will be equivalent to the case of all distinct eigenvalues in the literature [30, Table I].
5 SCN distribution
In this section, we will derive the general form of the CDF of the SCN of central semicorrelated Wishart matrix for arbitrary K and N. Then, we emphasize by considering the threedimensional case (K=3). Given (8) and the results of this section, P _{d} can then be directly computed.
5.1 The general form
where ϕ _{ i }(λ _{ j })= [ Φ(λ)]_{ i,j } and ψ _{ i }(λ _{ j })= [ Ψ(λ)]_{ i,j } with Φ(λ), Ψ(λ), ξ(λ _{ i }), and C are defined in [30, Table I].
If we have L equal correlation eigenvalues (say from p to q with q=p+L−1), then (18) is still valid by considering the parameters Φ(λ), Ψ(λ), ξ(λ _{ i }), and C that are defined in Theorem 1. This could be easily shown by comparing the joint distribution parameters in Theorem 1 and Table I of [30] and noting that they have the same general form used in deriving the SCN CDF in (18).
where γ(.,.) is the lower incomplete gamma function [38, Eq. (8.350.1)], the constant l=N−K+i, and 1≤i≤K.
5.2 SCN distribution for threedimensional systems
In the previous sections, we have considered an arbitrary number (L) of equal eigenvalues positioned anywhere in vector σ. Considering our assumptions that Ω _{ K } is a rank 1 matrix reveals, from Lemma 1, that the eigenvalues of the effective correlation matrix of the central semicorrelated Wishart matrix has (K−1)equal eigenvalues such that σ _{1}>σ _{2}=⋯=σ _{ K }>0.
Now, if we consider the case K=3, then \(\widehat {\mathbf {\Sigma }}_{K}\) has two equal eigenvalues (σ _{1}>σ _{2}=σ _{3}>0). In this case, the CDF and PDF of the SCN are given in Theorem 2 and Corollary 1, respectively.
Theorem 2 (CDF of SCN of central semicorrelated Wishart matrix when Σ _{3} has two equal eigenvalues).
with Δ _{3} defined in (35) in Appendix 2, Theorem 3.
Proof.
Then, the result comes after simplification.
Corollary 1.
with Δ _{4} defined in (38) in Appendix 2, Corollary 2.
Proof.
Obtained by differentiating Eq. (20).
For the sake of completeness, the reader could find, in Appendix 2, the CDF and PDF of the SCN of the central semicorrelated Wishart matrix when all the correlation eigenvalues are distinct.
6 Simulation and discussion
In this section, we discuss the analytical results through Monte Carlo simulations. We, firstly, validate the theoretical analysis presented in Sections 4 and 5. Given the noncentral/central approximation in Lemma 1, we study the approximation accuracy as well as the impact of the SNR on this approximation. Since the probability of detection is the main target, we study the effect of the approximation accuracy and the impact of system parameters on P _{d}. Finally, we show that the SCN detector outperforms the ED and the LE detector if noise power is not perfectly known.
where G is the Gaussian noise matrix and Σ _{ K } is the correlation matrix.
Figure 5 shows that the dissimilarity between the curves decreases as N increases and becomes very close to zero at N=100. This is related to the threshold selection criteria. In CR, the threshold is—usually—selected according to a target P _{fa} (i.e., the distribution of the SCN under \(\mathcal {H}_{0}\) as shown in Fig. 1). As N increases, the detection probability is improved as the part of the distribution under \(\mathcal {H}_{1}\) to the right of the threshold increases. This reveals that, in the sense of detection performance, the normalized noncentral/central approximation could be considered as a good fit for the empirical distribution of the SCN for sufficiently large values of N. Indeed, both distributions perfectly match at low SNR values and exceed the threshold for higher SNR values where the approximation accuracy degrades. Moreover, the same result could be deduced from Fig. 6 where the performance of the detector increases as K increases.
7 Conclusions
In this paper, we have considered the detection probability of the SCN detector. We have provided new statistics for the SCN metric, particularly for a Kdimensional cognitive radio system, by generalizing the form of the joint distribution of the ordered eigenvalues of central semicorrelated Wishart matrices. Then, we have considered a threedimensional system and have derived the approximated detection probability by the use of normalized noncentral/central approximation. The analytical results are validated through Monte Carlo simulations in different environments and with different system parameters. Results show that the approximation perfectly fits the empirical results at low SNR values; however, the approximation accuracy degrades as SNR increases. For a target false alarm probability, results prove that the probability of detection could be approximated using normalized noncentral/central approximation for all SNR values when the values of N and K are large enough. In addition, and by considering the noise uncertainty problem, results show that the SCN detector outperforms the LE detector and the ED even with very small noise uncertainty value.
8 Endnotes
^{1} The infinite sum of the hypergeometric function could be truncated to a finite sum with good accuracy if the K, N, and SNR are small. However, for a CR system, N is usually large.
^{2} The norm of the PU signal, ∥s∥^{2}, is still a random variable; however, this randomness decreases fast as N increases and (∥s∥^{2}/N) can be well approximated by \({\sigma _{s}^{2}}\) for sufficient value of N.
9 Appendices
9.1 Appendix 1: proof of Theorem 1.
Before we start the proof, it is required to give the following Lemma presenting the n ^{th} derivative of any function of the form \(e^{\frac {a}{x}}\) with respect to the variable x.
Lemma A.1.
with \(\mathcal {L}(n,k)\) the Lah number defined in (17).
Proof.
It should be noted that, for any value of n, the n ^{th} derivative of \(e^{\frac {a}{x}}\) will result into the function \(e^{\frac {a}{x}}\) multiplied by sum of (a/x) with different powers for a and x. Regardless of the sign, it is found that the number multiplied by each component of this sum is exactly equals to the Lah number.
Following the pattern seen by calculating the derivative of f(x) for small values of n, and matching the numbers at each derivative level, “n,” with the Lah number \(\mathcal {L}(n,k)\), it can be proved, by recurrence, that the n ^{th} derivative of the function f(x) of the form \(e^{\frac {a}{x}}\) is given by (29).
with \(e(\lambda _{i},\sigma _{j}) = e^{\frac {\lambda _{i}}{\sigma _{j}}}\), the required derivatives are evaluated using (29) and \(\Gamma _{s}(l)=\prod _{i=1}^{s}(li)!\). Then, the result is obtained by substituting (29) and (31) into (30) and simplifying.
In the case, we have several coincidences (e.g., σ _{ p,q }=σ _{ p }, σ _{ t,s }=σ _{ s }), the same analysis could be applied and the limit must be taken over all coincidence cases.
9.2 Appendix 2: the CDF and PDF of SCN
In the case where Ω _{3} is not a rank 1 matrix (we have more than one PU), then the eigenvalues of Σ _{3} are not equal anymore. For sake of completeness of threedimensional system case, we give in Theorem 3 and Corollary 2 the CDF and PDF of the SCN metric when all the correlation eigenvalues are distinct.
Theorem 3 (CDF of SCN of Central SemiCorrelated Wishart Matrix when Σ _{3} has distinct eigenvalues).
Proof.
Then, the result comes after simplification.
Corollary 2.
Proof.
Obtained by differentiating Eq. (32).
Declarations
Acknowledgements
This work is supported by a program of cooperation between the Lebanese University and the Azem & Saada social foundation (LUAZM) and by CentraleSupélec (France).
Competing interests
The authors declare that they have no competing interests.
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
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