Study of spectrum sensing exploiting polarization: From optimal LRT to practical detectors

Abstract

Spectrum sensing exploiting polarization can be leveraged to improve detection performance. In this paper, we present an optimal likelihood ratio test (LRT) which serves as the upper bound on the performance of all methods that exploit polarization under Gaussian assumption. However, the performance of LRT for realistic blind sensing is unclear since the practical implementation of polarization sensing has to face several challenges, which is the focus of this paper. Four practical sensing algorithms utilizing polarization are considered in this work, that is, a generalized likelihood ratio test (GLRT) for polarization sensing using maximum likelihood estimate (MLE) method when any prior knowledge is unknown, an optimal test when the noise of channel between primary user and secondary user is partially polarized, a realistic detector considering that the received polarization wave arrived with a specific direction rather than is almost perpendicular to the receiver dual-polarized antennas, and a more practical blind detector for polarization channel exhibiting polarization mode dispersion (PMD) phenomenon especially in wideband system. The simulation results show the performance gains possible with proposed detectors and how well the proposed detectors are expected to perform in practice.

Introduction

Cognitive radio (CR) has emerged as one of the most promising candidate solutions to improve spectrum utilization [1], in which secondary users (SUs) opportunistically utilize the frequency spectrum originally assigned to licensed primary users (PUs). One fundamental requirement of CR is the ability to identify the white space in the spectrum of interest by SUs. Hence, SUs should be able to independently detect spectral opportunities without any assistance from PUs. This ability is called spectrum sensing, which is considered one of the most critical tasks in CR networks.

Quite a few spectrum sensing algorithms have been proposed. These methods exploit different parameters to detect PUs while having different requirements for implementation, hence they have their different application scenarios. Among these methods, matched filter (MF) is optimal [2] in the sense of known prior information of the PU signals, such as modulation type, pulse shaping, and so on, which may not be available to SUs in practice. Since most PU signals show periodic patterns related to symbol rate, chip rate, channel code, or cyclic prefix, such periodic signals can be appropriately modeled as second-order cyclostationary random processes [3]. Knowing some of the cyclic characteristics of a signal, one can construct detectors that exploit the cyclostationarity [4], [5] at the cost of increased complexity, long latency and high sensitivity to sampling error. The energy detector (ED) [6] is one of the most commonly employed spectrum sensing schemes, since it does not require any prior knowledge about the PU's signal and it performs with low computational and implementation costs. However, in the presence of large noise uncertainty, the high probability of false alarm and low probability of detection will make ED invalid [7]. To overcome this difficulty, current research focuses on exploiting the signal structure in time or space dimensions. Examples are eigenvalue-based detection methods including maximum minimum eigenvalue (MME) [8], arithmetic-to-geometric mean (AGM) [9], generalized likelihood ratio test (GLRT) based detection [10], and covariance based detection methods including covariance absolute value (CAV) [11], CorrSum [12], F-test [13], and function of matrix based detection (FMD) [14]. These detectors require a relatively large number of signal samples to reach satisfactory performance, especially in low signal-to-noise ratio (SNR), resulting in long sensing time. Moreover, their performances degrade substantially in the presence of non-Gaussian noise. The Goodness of fit test (GFT), such as Anderson–Darling (AD) sensing [15], order-statistic (OS) based sensing [16], blind AD [17], Kolmogorov–Smirnov (K–S) test [18], calculates the discrepancy between the distribution of the observed samples and the distribution of the samples expected under noise conditions, which is empirically estimated. Hence, the GFT detector is robust to non-Gaussian noise.

The above mentioned detectors make use of amplitude (energy), frequency, or space dimensions of the received primary signal to differentiate signal from background noise. In fact, wireless communications exploit the electromagnetic medium, which is intrinsically polarization-sensitive, thus exploring polarization has become a potential research direction of wireless communications [19], [20], [21]. The polarization of electromagnetic wave is defined to be the instantaneous position of the electric-field vector perpendicular to the direction of propagation [22]. It is completely specified by the amplitude ratio and phase difference of two orthogonal electric-field components. In past decades much wireless technology, however, has been devised without explicit consideration of polarization, and so does sensing methods mentioned previously, in which the vector signals of PUs are converted to scalar quantities at the SUs prior to processing. In other words, it means that the contribution of the polarization characteristics as vector information elements of the PU signals is thereby lost. Nowadays, dual-polarized antennas have become a promising cost-effective and space-effective configuration and have widely used in practical deployed wireless communication systems [23], [24]. In dual-polarized CR systems, dual-polarized antennas can enable the SUs to obtain polarization information of the primary signal derived from the amplitude ratio and relative phase between the two orthogonally-polarized branch signals. Note that polarization provides an additional dimension besides time, frequency, and space, hence will bring new opportunities to enhance detection performance. Therefore leveraging polarization to further develop spectrum sensing algorithms in CR system may be a logical next step.

Nowadays, a few polarization spectrum sensing algorithms have been investigated recently. The first study [25] is the design of dual-polarized architecture for signal sensing, which requires the reference polarization as a priori just like the assumption made in the traditional radar systems [26]. The recent work focused on polarization sensing using polarization information obtained by dual-polarized antennas. The authors in [27] studied a Reconfigurable Polarization Detection (RPD) method using virtual polarization adaption by which the polarization state of receive antenna adaptively changes to match that of transmit antenna so that the average received power of wanted signal gives a maximum and unwanted signal gives a minimum. The method of adapting the polarization at transmit and receive is referred to as “virtual adaptation” in recognition of the fact that the adaptation takes place within the processor, not at the antenna. However, such polarization adaption process requires an iterate search with step size as small as possible to find the matched polarization state with high computation complexity. The RPD method in [27] was used for spectrum sensing by optimizing the receiving polarization according to received polarization state of signal to achieve the maximum received signal to noise ratio (SNR). Since this method optimally combines two signals received from branches of antennas at the energy level, here we call it energy based polarization detector. Ref. [28] analyzed the performances of energy based polarization detectors with different combining schemes, such as Selection Combining (SC), Equal Gain Combining (EGC), and Optimum Polarization Based Combining (OPBC) (the test statistic of which is the same as that of RPD) in AWGN channel and then [29] and [30] extended the work in [28] to dual-polarized fading channel. As mentioned above, similar to conventional ED, all the energy based polarization detectors suffer from the noise power uncertainty. To overcome this difficulty, then we proposed a blind polarization spectrum sensing method, generalized likelihood ratio test-polarization vector (GLRT-PV) [31], based on the sum of observed samples of polarization vector that describes polarization state information of received vector signal. Compared with energy based polarization detectors, GLRT-PV has constant false alarm rate (CFAR) property which allows one to find a detection threshold that achieves a fixed probability of false alarm irrespective of intensity changes in the noise background. However, energy information of received signal unfortunately failed to be utilized by GLRT-PV.

The motivation of this work is two-fold. The one is to provide an optimal spectrum sensing algorithm exploiting polarization given by LRT, which provides an upperbound performance allowing to be precisely assessed under Gaussian assumption, that is, the received signal is assumed to follow Gaussian distribution. The other one is to address detectors when considering their realization and performance in practical CR applications. Four practical sensing algorithms using polarization are considered in this work: (i) a generalized likelihood ratio test (GLRT) with finite signal samples in the Neyman–Pearson (NP) sense when prior knowledge is unknown. (ii) an optimal test when the noise of channel between PU and SU is part-polarized. (iii) a realistic detector considering that the received polarization wave arrived with a specific direction rather than is almost perpendicular to the receive dual-polarized antennas. (iv) a more practical blind detector for dual-polarized channel exhibiting polarization mode dispersion (PMD) phenomenon especially in wideband system.

The rest of this paper is organized as follows. The signal model incorporated polarization information for sensing is given in Section 2. The optimal likelihood ratio test (LRT) based polarization sensing algorithm using the NP theorem is presented in Section 3. The generalized LRT (GLRT) for polarization sensing using maximum likelihood estimate (MLE) method is discussed in Section 4. To build a practical sensing device, many factors should be considered. Sections 5, 6, and 7 present some detectors based on practical considerations with special emphasis on situations such as the noise is partially polarized, the polarization wave arrives with arrival angle, and the channel exhibits PMD phenomenon, respectively. Finally, concluding remarks are given in Section 8.

$\mathbf{E}[\cdot ]$ denotes the expectation operation and ⁎ represents complex conjugation. Superscripts ${(\cdot )}^T$ and ${(\cdot )}^H$ denote transpose and conjugate transpose, respectively. $\det (\mathit{A})$ is the determinant of matrix A and $|x|$ is the absolute value (or modulus) of x. ⊗ is the Kronecker product. ${\mathit{I}}_n$ stands for the $n\times n$ identity matrix. $\Vert \cdot \Vert$ represents the Euclidean norm and $\mathit{tr}\left(\mathit{A}\right)$ denotes the trace of matrix A. $\mathit{diag}(A)$ denotes a diagonal matrix with diagonal elements expressed in A.