On the detection probability of the standard condition number detector in finite-dimensional cognitive radio context

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 non-central 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 non-centrality matrix [32].

Denoting by t the decision threshold, then the detection probability (P _d), defined as the probability of correctly detecting the presence of PU, and the false alarm probability (P _fa), defined as the probability of detecting the presence of PU while it does not exist, are, respectively, given by (5) and (6) and shown in Fig. 1.

$$\begin{array}{*{20}l} &P_{\mathrm{d}} = P(\text{SCN} \geq t / \mathcal{H}_{1}) \end{array} $$

(5)

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 Nuttall-Q function which could be replaced by Marcum Q-function 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 non-central/central approximation that approximates the distribution of the non-central uncorrelated Wishart matrix by the distribution of the central semi-correlated Wishart matrix. This approximation is presented in the next section. The exact general form of the distribution of the SCN of central semi-correlated Wishart matrix is provided by [30] and is used to approximate F ₁(t) for K=2 in [18, 30].

The non-central/central approximation is recalled in Lemma 1.

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 semi-correlated 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 non-central uncorrelated Wishart matrix with arbitrary non-centrality 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 Nuttall-Q functions or hypergiometric functions. Therefore, by considering the non-central/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.

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 semi-correlated 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 line-of-sight (LOS) component of the channel matrix approaches the scaled all-ones matrix [37].

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].

In this section, we will derive the general form of the CDF of the SCN of central semi-correlated Wishart matrix for arbitrary K and N. Then, we emphasize by considering the three-dimensional case (K=3). Given (8) and the results of this section, P _d can then be directly computed.

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 non-central/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.

Figures 2 and 3 validate the analytical form of the distribution of the SCN of the central semi-correlated Wishart matrix derived in Section 5. A three-dimensional system is considered, and the receiver matrix is given in (28). The correlation matrix, Σ _K , is modeled according to the normalization of Lemma 1; thus, σ ₁=1+K ρ and σ ₂=σ ₃=1. Figure 2 shows the impact of varying the number of received samples (N) while the SNR (ρ) variation is considered in Fig. 3. Results show a perfect match between the empirical SCN distribution of the central semi-correlated Wishart matrix and the analytical form in Section 5. Moreover, these figures show a validation of the joint distribution given in Theorem 1.

Central/non-central approximation accuracy is shown in Fig. 4. In this case, we consider a BPSK modulated signal and a complex Gaussian noise with unit variance (\(\sigma _{\eta }^{2} = 1\)). Since we have considered a flat fading channel, we suppose that the signal will experience a unity fading magnitude. Figure 4 gives the empirical distribution of the SCN of non-central uncorrelated Wishart matrices (under \(\mathcal {H}_{1}\) hypothesis) and its corresponding normalized analytical approximation using Lemma 1 for different SNR values (ρ). Results show a perfect match between the empirical results and the analytical approximation at low SNR values, however, approximation accuracy degrades as SNR increases (ρ>−2 dB).

Figures 5 and 6 present the impact of the system parameters (K, N, and ρ) on the approximation in terms of the performance of the SCN detector for a preset target P _fa. Figure 5 shows the P _d of the SCN detector of three-dimensional CR as a function of SNR for different number of samples (N={10,30,50,100}) while in Fig. 6, we vary K (K={2,3}) with the false alarm being set to to P _fa=0.1. Both figures show perfect accuracy for low SNR values (ρ<−2 dB); however, as SNR increases the approximation starts to show dissimilarity with the empirical results until both probabilities reach 1 (both distributions, the empirical and the analytical, are totally to the right of the considered threshold).

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 non-central/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.

To stand on the previous results and to validate the approximation accuracy obtained at low SNR for different values of N and K, we present in Fig. 7 the P _d of SCN metric for different N and K with SNR fixed at ρ=−10 dB and P _fa=0.1. The figure shows a perfect match between the empirical and the analytical results for all values of K and N. As it could be seen from the figure, the distribution under \(\mathcal {H}_{1}\) hypothesis partially exceeds the threshold (P _d<1), therefore, the approximation perfectly fits the empirical distribution. Thus, at an acceptable SNR value, the approximation does not show any degradation as N and K values change.

The performance comparison between the SCN detector, the LE detector, and the ED [5] is illustrated in Fig. 8. In this case, we set K=3, N=500, and P _fa=0.1. If noise variance is perfectly known, results show that the LE detector outperforms the SCN detector as it is the optimal detector under this condition. However, noise variance is not known in practice. Accordingly, we considered a 0.3 dB noise uncertainty and the results are shown in Fig. 8. Results show that the SCN detector outperforms both the LE and ED as noise uncertainty is considered.

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.