Covariance-Based Spectrum Sensing for Noncircular Signal in
Cognitive Radio Networks With Uncalibrated Multiple Antennas

Cognitive radio (CR) has been recognized as a powerful means to improve the spectrum efficiency by allowing secondary users (SUs) to opportunistically utilize the licensed spectrum of primary users (PUs) [1]. One major functionality of CR is the spectrum sensing (SS), which plays an important role to sense idle spectrum bands and avoid unacceptable interference to PUs[2]. In the past decade, various detection methods based on different operational requirements were proposed for SS[3]. Among them, maximum-minimum eigenvalue (MME)[4] is a very attracting method for SS with multiple antennas, due to its ability of overcoming the noise uncertainty and fading effects. However, the MME method is susceptible to unequal per-antenna noise variances as may appear in practical scenarios due to calibration errors.

In the literature, several robust multiantenna SS (MSS) methods have been proposed for overcoming this drawback, such as the covariance absolute value (CAV) [5], locally most powerful invariant test (LMPIT)[6], Hadamard (HDM) ratio test[7], volume-based detection (VD)[8] and separating function estimation test (SFET)[9] methods can deliver desirable performance under the scenario of unequal per-antenna noise variances. Nevertheless, all these methods above were devised for circular signals as they only use the standard covariance information. But practically noncircular (NC) signals, for instance offset quadrature phase shift keying (OQPSK), pulse amplitude modulation (PAM) and binary phase shift keying (BPSK) are often encountered in modern communication systems [10]. These signals contain not only the standard covariance information but also the additional complementary covariance information, which can be used for enhancing the detection performance[11]. Therefore, there has been a recent interest in developing new methods for MSS that exploit both the standard covariance and complementary covariance information. In this case, Huang et al. [12] proposed a NC-HDM method to exploit the NC characteristic of the primary signals for MSS, and they showed that the NC-HDM method can significantly improve the performance of the original HDM method. However, the calculation of the NC-HDM test statistic involves matrix determinant computation, which incurs significant computational cost. More recently, a NC local average variance (NC-LAV) method was developed in [13]. However, the NC-LAV method relies heavily on the assumption that the noise variances at all antennas are identical, thereby its performance is also sensitive to unequal per-antenna noise variances.

In this letter, we consider the SS problem for NC signal in CR networks with multiple receive antennas, focusing on the scenario where the antennas experience different levels of noise power. To leverage the benefits of the NC signal, we propose a new robust SS method named NC covariance (NCC) for such networks. Specifically, we show that the standard covariance and complementary covariance matrices of the received signals differ between the null and the alternate hypotheses, which can be used for detecting the presence of PUs. Moreover, an asymptotic distribution of the NCC statistic under the null hypothesis is derived to acquire the theoretical formula for the decision threshold. In addition, we also analyze the computational complexity of the proposed NCC method. Simulation results show that NCC outperforms the existing state-of-the-art methods.

Notation: Throughout this letter, we use boldface lowercase and uppercase letters to designate vectors and matrices, respectively. Superscripts $(\cdot)^{\ast}$, $(\cdot)^{T}$ and $(\cdot)^{H}$ indicate the complex conjugate, transpose, and Hermitian transpose, respectively. While $|\cdot|$ denotes the absolute value operation, $\mathbb{E}[x]$ is the expected value of the complex-valued random variable $x$ and $\mathbb{D}[x]=\mathbb{E}[|x|^{2}]-|\mathbb{E}[x]|^{2}$ denotes its variance. The symbols $\mathbf{I}_{M}$ and $\mathbf{0}_{M}$ refer to the $M\times M$ identity and all-zero matrices, respectively. The trace of a square matrix $\mathbf{X}$ is denoted by $\text{tr}(\mathbf{X})$. A random variable $a$ follows the real Gaussian distribution with mean $\mu_{r}$ and covariance $\sigma_{r}^{2}$ is shown by $a\sim\mathcal{N}(\mu_{r},\sigma_{r}^{2})$, while a complex-valued random variable $b$ follows the circularly symmetric complex Gaussian distribution (CSCG) with mean $\mu_{c}$ and covariance $\sigma_{c}^{2}$ is denoted by $b\sim\mathcal{N}(\mu_{c},\sigma_{c}^{2})$.

Let us first consider the standard covariance and complementary covariance matrices of the received signal vector $\mathbf{y}_{k}$, given as

Under the null hypothesis $\mathcal{H}_{0}$, $\mathbf{R}_{\mathbf{y}}$ and $\mathbf{C}_{\mathbf{y}}$ are respectively given as

Meanwhile, under the alternative hypothesis $\mathcal{H}_{1}$, we can derive $\mathbf{R}_{\mathbf{y}}|\mathcal{H}_{1}$ and $\mathbf{C}_{\mathbf{y}}|\mathcal{H}_{1}$ as

where $\mathbf{R}_{\mathbf{s}}$ and $\mathbf{C}_{\mathbf{s}}$ are respectively the standard covariance and complementary covariance matrices of the transmitted signal $\mathbf{s}_{k}$. It is clear from the above derivation that $(r_{mm}\in\mathbb{R})>0$, $(r_{mn}=r^{\ast}_{nm})\in\mathbb{C}$ and $(c_{mn}=c_{nm})\in\mathbb{C}$, where $r_{mn}$ and $c_{mn}$ denote the $(m,n)$th element of $\mathbf{R}_{\mathbf{y}}$ and $\mathbf{C}_{\mathbf{y}}$, respectively. Meanwhile, we can notice that under $\mathcal{H}_{0}$, $\mathbf{R}_{\mathbf{y}}|\mathcal{H}_{0}$ is a diagonal matrix and $\mathbf{C}_{\mathbf{y}}|\mathcal{H}_{0}$ is a zero matrix. However, under the alternative hypothesis $\mathcal{H}_{1}$, the off-diagonal elements of $\mathbf{R}_{\mathbf{y}}|\mathcal{H}_{1}$ and all the elements of $\mathbf{C}_{\mathbf{y}}|\mathcal{H}_{1}$ are nonzeros. Combing all above observations, we let

If the signal $\mathbf{s}_{k}$ is absent, $T=0$. While if the signal $\mathbf{s}_{k}$ is present, $T>0$. Hence, $T$ can be leveraged to distinguish between hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$.

It should be noted that, in practical scenarios, the standard covariance and complementary covariance matrices (i.e., $\mathbf{R}_{\mathbf{y}}$ and $\mathbf{C}_{\mathbf{y}}$) can only be computed via a limited number of samples, as given by

Therefore, we propose a NCC method based on (5) as

where $T_{N}$ is the test statistic of the proposed NCC method, $\hat{r}_{mn}$ and $\hat{c}_{mn}$ denote the elements of $\hat{\mathbf{R}}_{\mathbf{y}}$ and $\hat{\mathbf{C}}_{\mathbf{y}}$ at the $m$th row and the $n$th column, respectively. Given a predefined decision threshold $\lambda_{N}$, if $T_{N}\geq\lambda_{N}$, the signal $\mathbf{s}_{k}$ is present (“$\mathcal{H}_{1}$” decision); otherwise, signal $\mathbf{s}_{k}$ is not present (“$\mathcal{H}_{0}$” decision).

We now conduct experiments to compare the performance of the proposed NCC method with the following seven methods: CAV [5], LMPIT[6], HDM[7], VD[8], SFET[9], NC-HDM[12], and NC-LAV[13]. At each run, the elements in $\mathbf{H}$ are drawn independent and identically distributed from $\mathcal{CN}(0,1)$, and the noise powers $\{\sigma^{2}_{m}\}_{m=1}^{M}$ are independently drawn from the uniform distribution within $[-\alpha,\alpha]$ dB, while the noise parameter is set as $\alpha=1$ in the simulations. Additionally, the $i$th primary signal $s_{i}(k)(i=1,\ldots,q)$ is assumed to be independent BPSK constellation with different power $\gamma_{i}(i=1,\ldots,q)$, and the average signal-to-noise ratio (SNR) is defined as 10log$\Big{(}\frac{\text{tr}(\mathbf{H}\mathbf{R}_{\mathbf{s}}\mathbf{H}^{H})}{\text{tr}(\mathbf{R}_{n})}\Big{)}$, each simulation result is averaged over $10^{5}$ independent Monte Carlo runs.

Figs. 1(a) and (b) plot the detection probability as a function of SNR for the eight methods under the scenario of unequal per-antenna noise variances when $P_{f}=0.05$, where in Fig. 1(a), the simulation parameters are set as $q=1$, $M=4$ and $K=100$; Meanwhile, in Fig. 1(b), the simulation parameters are set as $q=3$, $M=8$ and $K=200$. From both figures, we can make the following observations. First, one can clearly observe that the proposed method is superior to all other methods in terms of detection probability. For example, in Fig. 1(a), the proposed NCC method achieves at least 0.5 dB gain compared with the other methods at detection probability of 80%. Moreover, in Fig. 1(b), when SNR $=-11$ dB, the proposed method offers higher detection probability than its counterparts by at least 10%. Second, it is observed that increasing the SNR can produce a performance gain to all compared methods. Figs. 2(a) and (b) display the receiver operating characteristic (ROC) curves for all compared methods (i.e., detection probability versus false alarm probability), where SNR $=-9$ dB and SNR $=-11$ dB are set in Figs. 2(a) and (b), respectively. There we first see that increasing the false alarm probability is beneficial for all considered methods, i.e., as $P_{f}$ increases, the detection probabilities of all methods progressively go to one. Next we see that the detection performance of NCC is noticeably better than that of other methods. For example, in both figures, to attain a detection probability of 0.8, the proposed NCC method requires lower $P_{f}$ than its counterparts.

Discussion on simulation results: From Figs. 1(a)-1(b) and 2(a)-2(b), we can see that the NCC and NC-HDM have better detection performance than that of other methods, this is because both NCC and NC-HDM methods can exploit the additional information contained in the complementary covariance matrix of received samples, leads to considerable performance gain. Although the NC-LAV method is devised to exploit the NC characteristic of the primary signals. However, this method is developed under the assumption that the noise variances at all antennas are identical, thereby it cannot provide a desirable performance in the scenario of unequal per-antenna noise variances.

In this letter, the spectrum sensing problem in multiantenna CR networks with NC primary signals was addressed under the scenario of unequal per-antenna noise variances, and a low-complexity NCC method was proposed to exploit the NC characteristic of the primary signals. Numerical results show that the NCC method is superior to the state-of-the-art detection methods.