Spectrum sensing algorithms based on correlation statistics of polarization vector

Signal Processing ] (]]]]) ]]]–]]]

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Contents lists available at ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Spectrum sensing algorithms based on correlation statistics of polarization vector

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Cali Guo n, Xiaobin Wu

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Beijing Key Laboratory of Network System Architecture and Convergence, School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, P.O. BOX 89, No. 10 Xitucheng Road, Haidian District, Beijing 100876, China

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a r t i c l e i n f o

abstract

Article history: Received 23 November 2013 Received in revised form 4 May 2014 Accepted 7 May 2014

In this paper, we consider the problem of spectrum sensing in cognitive radios (CRs) by exploiting inherent polarization characteristics of signal. Since polarization vector can completely describe the radiation's polarization characteristics, we first derive the probability density function (PDF) of received polarization vector and its moments. Then we find the fact that both component and serial correlations of received polarization vector (i.e., the mixed polarization vector of primary signal and noise) are different from those of noise with high probability. This distinctive difference can be used to decide whether the primary signal exists or not. Therefore, component correlation sensing (CCS) algorithm and serial correlation sensing (SCS) algorithm are proposed respectively. Furthermore the closed-form expressions of probabilities of false alarm and detection are available for CCS and SCS algorithms. Simulations show that both CCS and SCS detectors achieve better performance with higher cross-polar discrimination (XPD) and polarization channel correlation coefficients. We also show that, if channel is highly depolarized, CCS performs better than SCS. Otherwise, the latter shows better performance. Compared with existing polarization based detectors, both CCS and SCS detectors perform better in the presence of noise power uncertainty. & 2014 Elsevier B.V. All rights reserved.

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Keywords: Cognitive radio (CR) Spectrum sensing Polarization vector Component correlation Serial correlation

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1. Introduction Cognitive radio (CR) is a spectrum sharing technology which enables secondary users (SUs) to access licensed spectrum without causing interference to primary users (PUs) [1]. Spectrum sensing plays a fundamental role in cognitive radio networks and a variety of spectrum sensing methods have been proposed recently in [2–9], such as energy detection (ED) [2], matched filtering detection [3], cyclostationary detection [4], eigenvalue-based detectors including maximum minimum eigenvalue (MME) [5],

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Corresponding author. Tel.: þ86 131 2038 1858. E-mail addresses: [email protected] (C. Guo), [email protected] (X. Wu).

arithmetic-to-geometric mean (AGM) [6], and generalized likelihood ratio test (GLRT) [7], and covariance-based detectors including covariance absolute value (CAV) [8] and CorrSum [9]. Most of the existing methods make use of the amplitude (power), frequency, or phase characteristics of signal to differentiate received primary signal from background noise. However, the polarization characteristics, which actually represent signals’ essential characteristics, are unexploited in previous works. The polarization characteristics are embodied in a vector which received by two branches of the dual-polarized antennas. Polarization contains not only the amplitude, frequency, and phase information of signals, but also the information of the relative relationship between the two components of the vector. Unfortunately all of the previously mentioned

http://dx.doi.org/10.1016/j.sigpro.2014.05.010 0165-1684/& 2014 Elsevier B.V. All rights reserved.

61 Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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C. Guo, X. Wu / Signal Processing ] (]]]]) ]]]–]]]

sensing methods convert the vector signals into scalar quantities at the SUs prior to processing. In other words, the polarization characteristics is thereby ignored. Furthermore, all radio transmit and receive antennas are intrinsically polarized. Hence, exploring polarization in wireless communications, such as diversity and multiplexing, has been a potential research direction [10–12]. Meanwhile dualpolarized antennas have become a promising cost-effective and space-effective configuration and have been widely used in existing commercial wireless communication systems, such as 2G/3G [10], or future commercial wireless communication systems, such as LTE/LTE-A [10,13]. Moreover, it is natural that cognitive radio would be incorporated into these commercial wireless communication systems to effectively resolve the dilemma between the increasing spectrum requirements and scarce spectrum resources [14]. Hence leveraging cognitive radio in dual-polarized commercial wireless communication systems may logically be the next step. However, to the best of the authors’ knowledge, the use of polarization for spectrum sensing has not yet been investigated extensively, and few related research works have been reported in the literature. Recently, a dual-polarized architecture [15] was designed for spectrum sensing under the assumption of known reference polarization just like that in the traditional radar systems [16]. In fact, the dual-polarized antennas enable the SUs to obtain primary signal's polarization information which can be derived from the amplitude ratio and relative phase between the two orthogonally polarized signal branches. Therefore our recent works focused on spectrum sensing based on the obtained polarization information. We first proposed a virtual polarization detection (VPD) method [17], which was regarded as energy based polarization detection since this method optimally combined signals received from two branches of dual-polarized antennas at the energy level. Sharma et al. [18] analyzed the performance of energy based polarization detection with different combining techniques, such as selection combining (SC) and equal gain combining (EGC) in AWGN channel. Sharma et al. and Xu and Lim [19,20] further extended the work in [18] to dual-polarized fading channel. As mentioned above, similar to conventional ED, all the energy based polarization detection methods suffer from the problem of noise uncertainty. To overcome this problem, we then proposed a blind polarization spectrum sensing method, generalized likelihood ratio test-polarization vector (GLRT-PV) [21], based on the resultant vector length of the sample polarization vectors. Since the length of resultant vector is independent of noise covariance and the parameters of PU, the GLRT-PV method has constant false alarm rate (CFAR) property. In this paper, we further focus on fully exploiting the inherent polarization characteristics of signal. Since polarization vector fully represents the polarization information of received signal, we first derive the probability density function (PDF) of the received polarization vector and its moments. Based on the results obtained, we find that both the component correlation and serial correlation of mixed polarization vector of signal and noise are different from those of noise only with high probability. Therefore, we propose two blind spectrum sensing algorithms based on polarization vector correlation statistics, i.e., component correlation sensing (CCS) algorithm and the serial correlation

sensing (SCS) algorithm, in which the above distinguished property is used to decide whether primary signal exists or not. We should point out another reference related to our work. Recently in [21], we have found that the orientation of received polarization vectors containing noise only follows a spherical uniform distribution, while the orientation of received polarization vectors containing both primary signal and noise follows a Fisher distribution. Therefore, we proposed a sensing algorithm by exploiting the difference between the two statistics. Using the GLRT paradigm, the final test statistic, namely, GLRT-PV detector is solely based on the resultant vector length of the sample polarization vectors. Different from [21], this paper uses two kinds of correlation statistic of polarization vector to discriminate signal from background noise. The only significance correlation with [21] is that one of the proposed detector which uses serial correlation statistic, i.e., SCS detector, is equivalent to using resultant vector length of the sample polarization vectors just as GLRT-PV detector does. So, just as this paper pointed out, directional statistic based detector is only a special case of correlation statistic based detector. For practical applications, direction statistic based detector or serial correlation statistic based detector goes for different scenarios with component correlation statistic based detector. Suitable applications are associated with the propagation conditions, which may induce low or high depolarization effects of channel between CR receiver and primary transmitter. In the case of high depolarization behavior of channel, such as rich multipath propagation caused by scattering along propagation path, component-correlation-based detector is preferred than direction-based detector. However, when there exists relatively good propagation conditions, for example, the received signals propagate through a line of sight (LOS) path or a near LOS path, direction-based detector is more applicable for its better performance, and more important point is that direction-based detector has lower implementation complexity, thus more attractive for fast spectrum sensing. The rest of this paper is organized as follows. In Section 2, the polarization vector and statistical signal model are introduced. The statistical analysis of received polarization vector including PDFs and moments is developed in Section 3. The CCS and SCS algorithms as well as their theoretical performance are presented in Sections 4 and 5, respectively. Section 6 further summarizes the proposed two algorithms and makes comparison with other detectors. Extensive simulations are illustrated in Section 7. Finally, Section 8 concludes the paper. Notations: E½ denotes the expectation operation and n represents complex conjugation. Superscripts ðÞT and ðÞH denote transpose and conjugate transpose, respectively. detðAÞ is the determinant of matrix A and jxj is the absolute value (or modulus) of x.  is the Kronecker product. I n stands for the n  n identity matrix.

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2. Polarization vector and statistical signal model

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We consider a scenario where orthogonally dualpolarized antennas are employed by a SU to detect PUs. The primary transmitter is assumed to be equipped with either uni-polarized antenna or orthogonally dual-polarized antennas, which is determined by its own system design.

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Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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Note that the orthogonally dual-polarized antennas can be two antennas with arbitrary orthogonal polarizations, e.g. vertical and horizontal (abbreviated as VH), 745-degree tilted, right-hand and left-hand circularly, and clockwise and counterclockwise elliptical polarizations. In this section, initially, the concept of polarization vector is briefly introduced, and then the statistical signal model incorporating polarization is developed.

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shown in [23] 2 3 2 3 1 G0 6 7 6 7 cos 2γ 6 G1 7 6 7 7 6 7 G¼6 6 G2 7 ¼ G0 6 sin 2γ cos ϕ 7: 4 5 4 5 sin 2γ sin ϕ G3

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63 65 ð4Þ 67

When just studying polarization properties of the electromagnetic wave, G0 is not of interest. Therefore in this paper we only focus on normalized Stokes vector

11

2.1. Polarization vector of primary signal

g ¼ ð cos 2γ; sin 2γ cos ϕ; sin 2γ sin ϕÞT

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The signal transmitted by PU from dual-polarized antennas is converted into the radiated electromagnetic wave es , which can be decomposed into an arbitrary pair of orthogonally polarized waves. Without loss of generality, here es is decomposed into horizontally and vertically polarized components and can be denoted as " # i h i h esH ¼ αsH ejϕsH h^ αsV ejϕsV v^ ¼ ðeisH þ jeqsH Þh^ ðeisV þ jeqsV Þv^ es ¼ esV

Normalized Stokes vector is also referred to as polarization vector since it one-to-one corresponds to polarization state. In other words, polarization vector can completely describe the polarization characteristics of a signal.

15 17 19 21

ð1Þ 23 25 27 29 31 33 35

where αsH , ϕsH and αsV , ϕsV are the amplitude-phase pairs of the orthogonal h^ and v^ electric-field components (the unit vectors h^ and v^ are chosen to represent horizontal and vertical polarization basis), i and q are the in-phase and quadrature components for both h^ and v^ components, respectively. Note that if PU is equipped with uni-polarized antenna, (1) can also be used as esH ¼ 0 or esV ¼ 0 representing vertical or horizontal polarization, respectively. As written in (1), the polarization state of the PU is completely defined by the constant amplitude ratio and relative phase of h^ and v^ electric-field components. Conveniently, amplitudephase descriptor ðγ; ϕÞ is adopted to represent polarization state, where

37

8 < γ ¼ arctan½αsV =αsH ;

39

: ϕ ¼ ϕ  ϕ ¼ arctan½eq =ei  arctan½eq =ei ; sV sH sV sH sV sH

h πi γ A 0; 2 ϕ A ½0; 2π:

ð2Þ 41 43 45 47 49

Since the descriptor ðγ; ϕÞ is not easy to visualize vividly, an alternative and mathematically convenient description of polarization state can be defined in terms of four real measurable quantities known as the Stokes vector as follows [22]: 3 3 2 2 3 2 α2sV þα2sH G0 jesH j2 þ jesV j2 6 7 6 7 6 2 2 7 7 6 G1 7 6 jesH j  jesV j 7 6 α2sV α2sH 7 7¼6 7¼6 S¼6 n 6 G2 7 6 2ReðesH e Þ 7 6 2αs αs cos ½ϕ  ϕ  7 sV sH 5 5 4 4 5 4 sV V H G3  2ImðesH ensV Þ 2αsV αsH sin ½ϕsV  ϕsH 

51

ð3Þ

53

It should be noted that G0 in (3) is a physically recognizable quantity which corresponds to the sum of the power in the h and v electric-field components and thus represents the total power (i.e., energy information) of the received signal. Extracting energy information G0 from each Stokes variables, i.e., elements of Stokes vector, and using the relationship between amplitude-phase descriptor ðγ; ϕÞ and amplitudephase pairs αsH , ϕsH , and αsV , ϕsV shown in (2), we can link the Stokes vector with energy and polarization state as

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ð5Þ

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2.2. Statistical signal model 81 In order to properly investigate the use of polarization information to detect PU signals, the statistical model of received signal that reflects polarization characteristics must be developed. The received electromagnetic wave ex captured by orthogonally dual-polarized antennas at SU can be written as ( H0 : ex ¼ n ð6Þ H1 : ex ¼ hes þ n where H0 and H1 are two hypotheses indicating the PU's absence and presence, respectively, n is received electric-field vector of Additive White Gaussian Noise (AWGN), and h is the channel response between the primary transmitter and the secondary receiver. Under hypothesis H0 , the received signal vector is noise vector n which can be denoted as n ¼ ½nh ; nv T . The noise model proposed here consists of a large number of independent and randomly oriented electric dipoles. For the purpose of brevity, only the statistically significant details of this model will be presented. Using the central limit theorem, nh and nv can be assumed as independent zeromean Gaussian random processes with identical power just in [24]. Hence,the noise covariance matrix Rn should be a scalar matrix δ2 I 2 , where δ2 is the power of nh and nv. Then consider hypothesis H1 . Eq. (2) shows that the polarization state of PU is deterministic and time-invariant. However, after transmission in channel, the specific polarization state is corrupted by the noise and channel fading. This effect is defined as channel depolarization which would change not only the polarization state but also the correlation between polarization components to some extent. This change of polarization characteristics can be completely described by the elements of a 2  2 dualpolarized channel matrix h. Assume that the channel is Rayleigh fading, then we model h using the method described in [11,25] as h ¼ ghp :

83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117

ð7Þ

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where g is a scalar complex Gaussian term representing Rayleigh fading and hp models the channel depolarization effect. For a HV to HV situation (transmit and receive antennas

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Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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are HV polarized), we write the channel matrix hp as ! hVV hVH hp ¼ hHV hHH

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where χ is the inverse of the cross-polar discrimination (XPD), which considers the cross-polar transmissions, e.g., transmission from a vertically (abbreviated as V) polarized transmit antenna to a horizontally (abbreviated as H) polarized receive antenna, and is usually defined as XPDðdBÞ ¼ 10log10 ð1=χÞ ¼ 10log10 ðjhVV =j2 jhHV j2 Þ ð10Þ

where the same assumption that the couplings from V to H and H to V have the same power on average was made as in [25] and it was also validated by the measurements reported in [26]. In other words, 0 o χ r1 depends on the XPD. Good XPD yields small χ and vice versa. Here we need to point out that employing two physical antennas on each side of the link with the same polarization, which is a typical 2  2 MultipleInput Multiple-Output (MIMO) uni-polarized system, can be interpreted as a special case of χ ¼ 1. We furthermore define the following polarization channel correlation coefficients: 8 n n > E½hVV hVH  E½hHV hHH  > > tp ¼ ¼ pffiffiffi pffiffiffi > > χ χ > > > > n n > > h  E½hVH hHH  > < r p ¼ E½hVV pffiffiffi HV ¼ pffiffiffi χ χ > > > pp ¼ E½hVV hnHH  > > > > n > > E½hVH hHV  > > > : qp ¼ χ

ð11Þ

Most experimental results [27,28] suggest that transmit correlation (i.e., the correlation coefficient between VV and VH, and HH and HV) and receive correlation (i.e., the correlation coefficient between VV and HV, and HH and VH) are typically very small. For the sake of simplicity, throughout the paper, we therefore assume that tp and rp are equal to zero just as [25]. As far as pp and qp are concerned, since experimental results are really scarce, according to the ray-tracing simulation results in [29], we also assume that pp ¼ qp . Then when there exists primary signal, the received electromagnetic wave ex at SU is not generally constantly polarized but time-dependent. Again with the aid of central limit theorem, the horizontal component exH and vertical component exV can be assumed statistically as correlated rather than independent zero-mean Gaussian random processes. Therefore, the received electromagnetic wave ex is distributed as CN ð0; Rx Þ, where covariance matrix Rx can be obtained as "

Rx ¼ E½ex  eH x¼

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2

¼ 10 log10 ðjhHH =j jhVH j Þ:

57 59

E½G0  ¼ r 11 þ r 22 ; E½G1  ¼ r 11 r 22 ; E½G2  ¼ r 12 þr 21 ; E½G3  ¼ jðr 12 r 21 Þ:

where hXY represents the channel gain between polarization X at the receive side and Y at the transmit side. We assume [11] ( E½jhVV j2  ¼ E½jhHH j2  ¼ 1 ð9Þ E½jhHV j2  ¼ E½jhVH j2  ¼ χ

2

19

ð8Þ

E½exH  enxH  E½exV  enxH 

# " E½exH  enxV  r 11 9 n E½exV  exV  r 21

r 12 r 22

E½G2 ; E½G3 ÞT , where E½Gi  for i ¼ 0; 1; 2; 3 is well known as Stokes parameter. These Stokes parameters can be obtained from the covariance matrix Rx as

#

ð12Þ

Instead of covariance matrix Rx , a much more convenient alternative is statistical Stokes vector E½G ¼ ðE½G0 ; E½G1 ;

ð13Þ In this case, the degree of polarization P, a useful quantity used to describe the extent of depolarization of primary polarization state after channel by using the ratio of the average energy of the polarized component of the received vector signal to the total average energy, is given by [30] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE½G1 Þ2 þ ðE½G2 Þ2 þðE½G3 Þ2 4 det ðRx Þ P ¼ 1 ð14Þ ¼ E½G0  ½trðRx Þ2 where 0 r P r 1, meaning that ðE½G0 Þ2 Z ðE½G1 Þ2 þ ðE½G2 Þ2 þ ðE½G3 Þ2 . The equality holds if the signal is constantly polarized or fully polarized [30]. Namely, the degree of polarization of primary transmitted signal satisfies P ¼1. However, the degree of polarization of received signal is small compared to 1 as result of the depolarization channel. So, to some extent, the value of P of received signal can describe the depolarization effect of channel between CR receiver and primary transmitter. Good channel condition yields high P and vice versa. In practical applications, when only considering polarization characteristics of signal, we can obtain the discrete polarization vector sample gðnÞ as g ðnÞ ¼ where 2

GðnÞ RL ex ðnÞ  enx ðnÞ ¼ ¼ ½g 1 ðnÞ; g 2 ðnÞ; g 3 ðnÞT G0 ðnÞ ex ðnÞeH x ðnÞ

1 6 RL ¼ 4 0 0

0

0

1

1

j

j

1

ð15Þ

3

0 7 5: 0

Hereafter, it would be natural to exploit the feature of polarization vector gðnÞ for the detection problem. 3. The statistics of polarization vector In this section, we develop statistical properties of polarization vector g under both hypotheses H1 and H0 . 3.1. Statistics of polarization vector under H1 When primary signal exists, the observation ex consists of the signal and noise. Assuming Gaussian statistics, as in the general case shown by Goodman [31], the probability density function (PDF) of received electric-field vector is     1 1 ð16Þ exp  eH f ex =H1 ¼ 2 x Rx ex : π det ðRx Þ Then the joint PDF of the in-phase and quadrature components for both h^ and v^ electric-field components can be written as   1 f eixH ; eqxH ; eixV ; eqxV =H1 ¼ 2 π det ðRx Þ

  1  r 22 α2xH þr 11 α2xV  2Re r 12 enxH exV : exp  detðRx Þ ð17Þ

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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According to the Jacobian determinant JðÞ, we have ! ! eixH ; eqxH ; eixV ; eqxV ∂ðeixH ; eqxH ; eixV ; eqxV Þ J ¼ det ¼ αxH αxV αxH ; ϕxH ; αxV ; ϕxV ∂ðαxH ; ϕxH ; αxV ; ϕxV Þ ð18Þ

5

αxH ; αxV ; ϕx



∂ðαxH ; αxV ; ϕx Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4G0 G22 þG23

7

J

9

where the phase difference ϕx ¼ ϕxV  ϕxH . With (17)–(19), we can get the statistics of Stokes sub-vector

11 13 15

G

¼ det

∂ðGÞ

ð19Þ

 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  exp   2det ðRx Þ 2 2 2 4π det ðRx Þ G1 þ G2 þG3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  E½G0  G21 þ G22 þG23  E½G1 G1 E½G2 G2 E½G3 G3 :

  f G=H1 ¼

ð20Þ 17 19 21 23 25 27 29 31 33 35

  1  P2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΘ 1 g 2i þ g 2j 2 2 4π 1  g i g j 0

B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ ð1  ηi g i  ηj g j  ηk 1  g 2i  g 2j Þ2 1 þ

43

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ð26Þ

2

 3ηi ð1  P ÞðP

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C A ð1 ηi g i ηj g j þ ηk 1 g 2i  g 2j Þ2 1

ð1 ηi g i Þð1 P 2 Þ 2½ð1  ηi g i Þ2 ðη2j þη2k Þð1 g 2i Þ3=2

 Θ 1  g i jÞ:

3ð1  P 2 Þð3P 6  35η4i þ 15P 2 η2i ð2 þη2i Þ 3P 4 ð1 þ6η2i ÞÞ 12P 9

69

73

77

81

Δ:

83

Proof of Theorem 1. See Appendix A.

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Next we derive the joint moments of random variables gi, gj, and gk. According to the definition of joint moments and using the Eqs. (22) and (23), we can get Theorem 2.

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Theorem 2. The joint moments of gi, gj, and gk for i aj a k are  1  P 2ηj π   I i1 þ Q i1 2ηi Q i2 þη2i Q i3 4π B2

89 91

2

ð29Þ

93

4ðη2j  η2k Þπ 2πðη2j  η2k Þ 1  P2  þ Q i1  3ηi Q i2 4π B4 B4 !  2η2k π  2 3 þ3ηi Q i3  ηi Q i4 Þ þ 2 Q i1  ηi Q i2  Q i3 þ ηi Q i4 ð30Þ B

95

 1  P 2 4πðηj ηk Þ 2 8ðηj ηk πÞ  I i2  ηi I i3 4 4 3 4π B B  4πηj ηk  2 3 Q 3η Q þ3η þ i2 i i3 i Q i4 ηi Q i5 B4

 2ηj ηk π   Q η Q  Q þ η Q ð32Þ i2 i3 i4 i5 i i B2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Q i1  Q i5 are given in (1.2) (1.6), B ¼ η2j þ η2k , and I in ði ¼ 1; 2; 3Þ are E½g 2i g j g k =H1  ¼

I i2 ¼

2ηi P2

þ

ηi P3

P 2 3η2i P4

Δ

þ

ð33Þ

P 4 P 2 P 2 η2i þ3η2i 2P 5

99

103

2πη2j  η2k   þ Q i2  3ηi Q i3 þ 3η2i Q i4  η3i Q i5 B4 !  2η2 π  ð31Þ  k2 Q i2  ηi Q i3  Q i4 þηi Q i5 B

I i1 ¼

97

101

2 2 2 2  1  P 2 2πðηj ηk Þ 2 4ðηj  ηk πÞ  I i2 ηi I i3 4 4 3 4π B B

ð23Þ

In most practical cases, to get PDFs as (22)–(24) is difficult, even impossible. So we attempt to study the moments of these PDFs which can describe the characteristic of their distributions. Moreover, from mathematical statistics [32], since moments of the distribution, i.e., population moments, can be extremely well estimated by sample moments, these moments are easily obtained for further analysis by averaging measured samples drawn from the population in practice. We first consider the moments of random variable gi. Brosseau [33] has given the first and second order moments of gi. Here we develop them in a more general

67

79

ð28Þ

E½g 2i g 2j =H1  ¼

ð24Þ

ð27Þ

5P 2 η2i ð18 þ 23η2i Þ þ 2P 5 ð9 þ114η2i þ 16η4i ÞÞ=ð12P 9 Þ þ

65

75

þ5η2i  P 2 ð3 þ η2i ÞÞΔÞ=ð4P 7 Þ

  E g 4i =H1 ¼ ð2P½6P 8 þ 105η4i  3P 6 ð5þ 8η2i Þ

where ΘðÞ is the step function, i, j, k ¼ 1; 2; 3 and i a ja k. Similarly, the individual marginal PDF is   f g i =H1 ¼

4

63

71

E½g 3i =H1  ¼ ðηi ½18P 5 þ 30Pη2i  2P 3 ð9 þ13η2i Þ

E½g i g 2j =H1  ¼

41

51

  1 η2 ð3  2P 2 Þ ðP 2  3η2i Þð1  P 2 Þ þ Δ E g 2i =H1 ¼ 1  2 þ i 2P 5 P P4

  f g i g j =H1 ¼

39

49

ð25Þ

E½g i g j =H1  ¼

1

47

Theorem 1.   η η ð1  P 2 Þ Δ E g i =H1 ¼ 2i  i 2P 3 P

where ηi is calculated as ηi ¼ E½Gi =E½G0 . Using (14), we can denote η21 þη22 þη23 ¼ P 2 . Integrating (22) on one of the components of g gives

37

45

case. The similar results can also be found in [34]. For completeness, the essential steps of the derivation are outlined in Appendix A. According to the definition of moments and Eq. (24), we can get the n-th (n ¼ 1; 2; 3; 4; 5) moment of gi in Theorem 1.

According to the conditional probability of multi-variant distribution f ðGÞ ¼ f ðG0 jGÞf ðGÞ, the joint PDF of the Stokes vector G is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðG=H1 Þ ¼ δðG0  G21 þ G22 þ G23 Þf ðG=H1 Þ ð21Þ where δðtÞ is Dirac delta function. Then we have the joint PDF of polarization vector g as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

  1 f g=H1 ¼ δ 1  g 21 þ g 22 þ g 23 4π 1  η21  η22  η23  ð22Þ ð1  η1 g 1  η2 g 2  η3 g 3 Þ2

5

105 107 109 111 113 115 117 119 121

Δ

ð34Þ

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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6

1

I i3 ¼

6P 4 ηi þ P 2 ηi ð9 þ 4η2i Þ 15η3i 3P 6

þ

ηi ð3P 4 3P 2 ð1 þ η2i Þ þ 5η2i Þ 2P 7

Δ:

symmetry property of C 1 is considered, we define the component correlation coefficient as

ð35Þ

3

ρc ¼ 5

Proof of Theorem 2. See Appendix B.

7

3.2. Statistics of polarization vector under H0

9

15

When there is no primary signal, the received observation ex consists of noise only. Based on the previous results E½G1  ¼ E½G2  ¼ E½G3  ¼ 0, we have ηi ¼ 0. The probability distributions expressed in (22)–(24) reduce to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

  1 ð36Þ f g=H0 ¼ δ 1  g 21 þg 22 þ g 23 4π

17

  f g i g j =H0 ¼

11 13

19 21 23 25 27 29 31 33 35 37 39 41 43 45 47



2π

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΘ 1  g 2i g 2j 1  g 2i  g 2j



ð37Þ

  1 f g i =H0 ¼ : 2

ð38Þ

Then the following results, under hypothesis H0 , are given   E g i =H0 ¼ 0

  1 E g 2i =H0 ¼ 3

  E g 3i =H0 ¼ 0

  1 E g 4i =H0 ¼ 5 ð39Þ

h i h i E g i g j =H0 ¼ 0 E g i g 2j =H0 ¼ 0 h i h i 1 E g 2i g j g k =H0 ¼ 0: E g 2i g 2j =H0 ¼ 15

ð40Þ

Thus far, the moments of polarization vector under hypotheses H1 and H0 are obtained, from which some meaningful properties can be obtained to differentiate between the signal and noise in the next two sections. 4. Spectrum sensing algorithm based on component correlation of polarization vector Comparing joint moment E½g i g j  in (40) under H0 hypothesis with that in (29) under H1 hypothesis, we can conclude that any two components of received polarization vector are uncorrelated without primary signal, yet correlated in general with the existence of primary signal. This distinctive difference of component correlation above lays the mathematical ground for the sensing algorithm proposed in this Section.

49

4.1. Component correlation sensing (CCS) algorithm

51

The statistical autocorrelation matrix of the received polarization vector can be defined as 2 3 c1;1 c1;2 c1;3 6 7 C 1 ¼ E½gg T  ¼ 4 c2;1 c2;2 c2;3 5 ð41Þ c3;1 c3;2 c3;3

53 55 57 59 61

where ci;j ¼ E½g i g j . According to (29) and (40), if primary signal is absent, the off-diagonal elements of C 1 are all zeros; otherwise, if there is a signal, C 1 is not a diagonal matrix. Since the off-diagonal elements of C 1 represent the component correlation of polarization vector, when

3

∑

i o j;i;j ¼ 1

jci;j j ¼

3

∑

i o j;i;j ¼ 1

63 65

jE½g i g j j:

ð42Þ 67

If the primary signal is present, ρc will be generally larger than that if there is noise only. Hence, the component correlation coefficient can be used to detect the presence of primary signal. Substituting E½g i g j  from (29) and (42), result shows that ρc is not an invariant test due to its dependency on η ¼ ðη1 ; η2 ; η3 ÞT . At worst, if any two elements of η are zeros, the test statistic given in (42) is invalid to detect the presence of signal. Thus a question is asked: Does there exist an optimal η which makes the largest test statistic? Let η0 represent the optimal η satisfying the following constrained optimization: 8 ρc < max η ð43Þ : s:t: 0 r η21 þ η22 þ η23 r 1: In terms of definitional domain of ηi A ½  1; 1 for i ¼ 1; 2; 3, it is easy to see that ρc has symmetric property. In view of this fact, the constraint condition of optimization problem in (43) now is reasonably limited to 0 r ηi r 1. In this case, the value of E½g i g j  should always be positive. Then, ρc in (43) can be converted into an equivalent form ρc ¼ E½g 1 g 2  þE½g 1 g 3  þ E½g 2 g 3 :

ð44Þ   According E g1 g2 þ    to the  1 well  2  known  2  inequality,  2 E g 1 g 3 þ E g 2 g 3 r 2fE g 1 þE g 2 þ E g 3 g, it can be easily verified that ρc achieves maximum when       1 E g1 g2 ¼ E g1 g3 ¼ E g2 g3 ¼ : 3

ð45Þ

Substituting (45) into (29), the solution gives η01 ¼ η02 ¼ pffiffiffi η03 ¼ 3=3. Thus the degree of polarization P ¼ ðη01 Þ2 þ ðη02 Þ2 þ ðη03 Þ2 ¼ 1. According to (29), the corresponding optimal mean polarization vector turns out to be g 0 ¼ E½g ¼ pffiffiffi pffiffiffi pffiffiffi ð 3=3; 3=3; 3=3ÞT with optimal polarization state represented by the amplitude-phase descriptor ðγ 0 ; ϕ0 Þ ¼ ð0:15π; π=4Þ. Thus far, we can conclude that there exists a maximal value of test statistic at the optimal mean polarization state. So if we obtain received mean polarization state from the statistical autocorrelation matrix and rotate it to the optimal mean polarization state, the maximal and stable detection performance can be expected. Inspired by this idea, another appropriate test statistic will be discussed in the following. The first step is to obtain the received mean polarization state. Since the received statistical autocorrelation matrix C 1 is a full-rank symmetric matrix, its eigendecomposition is expressed as

69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115

ð46Þ

117

where λ1, λ2, and λ3 are eigenvalues of C 1 , λ1 Z λ2 Zλ3 Z0, and ν1 ; ν2 , and ν3 are corresponding eigenvectors. Calling upon well-known results of matrix theory, the eigenvector ν1 just corresponds to the received mean polarization vector g with polarization state ðγ ; ϕÞ.

119

H H C 1 ¼ λ1 ν1 νH 1 þλ2 ν2 ν2 þ λ3 ν3 ν3

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1 3 5

Algorithm algorithm.

Then it is necessary to find a matrix which rotates the received mean polarization vector from g to g 0 . Note that the rotation of polarization vector is equivalent to unitary transformation of the received electric-field vector ex . Let e~ x denotes the electric-field vector after rotation, then we have e~ x ¼ Uex , where U is an unitary matrix expressed as

7

1. Component

correlation

sensing

(CCS)

65 Step 1. Initialize: solve constrained optimization problem in (43) to obtain the optimal polarization state ðγ 0 ; ϕ0 Þ.

"

9

U¼

0

cos γ cos γ 0 þ sin γ sin γ 0 ejðϕ  ϕ Þ 0  jϕ0

cos γ sin γ e

11

sin γ cos γ 0 e  jϕ  cos γ sin γ 0 e  jϕ 0 jðϕ0  ϕÞ

0  jϕ

 sin γ cos γ e

sin γ sin γ e

þ cos γ cos γ

0

0

# :

ð47Þ

19 21 23 25 27

75 According to (15), after rotation the polarization vector g~ can be expressed as n g~ ¼ RL e~ x  e~ x n H R ðUe Þ  ðUe x xÞ L e~ x e~ x ¼ H

Uex ðUex Þ

RL ðU  U n ÞRL 1 RL ðex  enx Þ

¼

Uex ðUex ÞH

We recognize RL ðU  U Q which is written as 2 cos 2γ 0 6 0 Q ¼ 4 sin 2γ 0 cos ϕ sin 2γ 0 sin ϕ0

31 33 35 37 39 41 43 45 47 49 51 53 55

cos 2γ 6 6 4 sin 2γ cos ϕ sin 2γ sin ϕ

n

ÞRL 1

  ¼ RL U  U n RL 1 g:

ð48Þ

as rotation matrix denoted as 3

 sin 2γ 0 cos ϕ0

 sin 2γ 0 sin ϕ0

cos 2 γ 0  sin 2 γ 0 cos 2ϕ0

 sin 2 γ 0 sin 2ϕ0

 sin γ 0 sin 2ϕ0

cos 2 γ 0 þ sin γ 0 cos 2ϕ0

2

2

29

 sin 2γ cos ϕ 2

7 5

2

3

 sin 2γ sin ϕ 2

cos 2 γ  sin γ cos 2ϕ

 sin γ sin 2ϕ

 sin 2 γ sin 2ϕ

cos 2 γ þ sin 2 γ cos 2ϕ

Step 2. Sample the received polarization vectors, as previously described in (15). Step 3. Estimate mean polarization vector using (52), ^ and obtain the estimated mean polarization state ð^γ ; ϕÞ, then construct estimated rotation matrix Q^ as (49). Step 4. Compute the sample autocorrelation matrix C 1 ðNÞ of the received signal. Then rotate the sample autocorrelation matrix to obtain C~ 1 ðNÞ as shown in (51). Step 5. Calculate test statistic T 1 ðNÞ using (53). Step 6. Make the Decision: if T 1 ðNÞ 4 γ 1 , signal exists (“yes” decision); Otherwise, signal does not exist (“no” decision) where γ1 is the threshold given in Section 4.2 which is chosen to meet the requirement of the probability of false alarm.

7 7: 5

Now the new test statistic can be written in the form ρ~c ¼

3

∑

i o j;i;j ¼ 1

E½g~ i g~ j :

ð50Þ

In practical detection, given the rotated polarization vector samples g~ ðnÞ, the sample autocorrelation matrix after such rotation can be calculated as T 1 N 1 N ^ T ∑ g~ ðnÞg~ ðnÞ ¼ ∑ Q g ðnÞg T ðnÞQ^ C~ 1 ðNÞ ¼ Nn¼1 Nn¼1  T 1 N ∑ C 1 ðN Þ Q^ ¼ Q^ Nn¼1

ð51Þ

where N is sample number, Q^ is the estimated rotation matrix which has the same form as Q in (49) with the ^ The latter is generated by maxsubstitution ðγ ; ϕÞ ¼ ð^γ ; ϕÞ. imum likelihood estimate (MLE) of the received sample polarization vector g^ ¼

1 N RL ex ðnÞ  enx ðnÞ ∑ : N n ¼ 1 ex ðnÞeH x ðnÞ

ð52Þ

In the end the final test statistic of component correlation sensing (CCS) algorithm is 3

∑

c~ i;j ðNÞ

T 1 ðNÞ ¼

59

~ ~ ~ where c~ i;j ðNÞ ¼ 1=N∑N n ¼ 1 g i ðnÞg j ðnÞ is the element of C 1 ðNÞ. Thus far, the outline of the proposed component correlation sensing (CCS) algorithm is summarized as follows.

i o j;i;j ¼ 1

ð53Þ

77 79 81 83 85 87 89 91

4.2. Performance analysis of CCS algorithm

93

The performance of the proposed CCS detector is evaluated in terms of the probability of false alarm P f 1 and the probability of detection P d1 , which depend on the distributions of T 1 ðNÞ under H0 and H1 hypotheses. According to the central limit theorem, T 1 ðNÞ follows asymptotically normal distribution when the number of samples N is large enough

95

ð49Þ

57

61

71 73

13

17

67 69

7

15

63

T 1 ðNÞ  NðμT 1 ; s2T 1 Þ

μT 1 ¼

3

∑

i o j;i;j ¼ 1

E½g~ i g~ j 

s2T 1

of T 1 ðNÞ are

ð55Þ

101 103 105 107 109 111

and s2T 1 ¼

99

ð54Þ

where μT 1 and s2T 1 are the mean and variance of T 1 ðNÞ, respectively. Theorem 3. The mean μT 1 and variance given by

97

1 h 2 2 i N 1 E g~ i g~ j þ ðE½g~ i g~ j Þ2 N N i o j;i;j ¼ 1

h i N 1 h i  3  1 E g~ 2i g~ j g~ k þ E g~ i g~ j E g~ i g~ k μ2T 1 : þ2 ∑ N i o j;i o k;i;j;k ¼ 1 N 3



∑

ð56Þ Proof of Theorem 3. See Appendix C. From (55) and (56), the exact expressions for μT 1 and s2T 1 have explicitly shown the dependency on statistical parameters E½g~ i g~ j , E½g~ 2i g~ 2j , and E½g~ 2i g~ j g~ k . Let R~ x denote the

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8

1 3 5 7 9 11

covariance matrix of the received field vector e~ x after rotation. Just as mentioned in (12), covariance matrix R~ x can be written as   ð57Þ R~ x ¼ URx U H 9 r~ 11 r~ 12 r~ 21 r~ 22 where U is the unitary matrix defined in (47). Then denote η~ ¼ ðr~ 11  r~ 22 =r~ 11 þ r~ 22 ; r~ 12 þ r~ 21 =r~ 11 þ r~ 22 ; jðr~ 12  r~ 21 Þ=r~ 11 þ r~ 22 ÞT . Hence, analytical expressions for E½g~ i g~ j , E½g~ 2i g~ 2j  and E½g~ 2i g~ j g~ k  can be directly derived from ~ (29), (31), and (32) through simple substitution η ¼ η. As in the noise case, we have

13

μT 1 =H0 ¼ 0

15

s2T 1 =H0 ¼

17

where the special case R~ n ¼ URn U H ¼ Rn is used, because the covariance matrix of noise Rn is in scalar form. The false alarm occurs when T 1 ðNÞ is larger than decision threshold γ1 at hypothesis H0 . Therefore the probability of false alarm can be computed from

19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

1 5N

P f 1 ¼ PfT 1 ðN Þ 4 γ 1 =H0 g ( ) T 1 ðNÞ γ1 ¼ P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=H0 1=ð5NÞ 1=ð5NÞ ! γ1 ¼ Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ð5NÞ

ð58Þ ð59Þ

ð60Þ

where Q ðÞ is the standard Q-function, and γ1 is the decision threshold evaluated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð61Þ γ 1 ¼ Q  1 ðP f 1 Þ 1=ð5NÞ where Q  1 ðÞ is the inverse function of standard Q-function. After setting threshold, we now calculate the probability of detection. For the given threshold γ1, when the signal is present, the probability of detection in close-form is 0 1 P d1

Bγ 1 μT C ¼ PfT 1 ðNÞ 4γ 1 =H1 g ¼ Q @ qffiffiffiffiffiffiffi1 A s2T 1

ð62Þ

where μT 1 and s2T 1 are given by (55) and (56), respectively. The mapping between P f 1 and P d1 is the so-called receiver operating characteristics (ROC). Thus using (61) to eliminate γ1 from (62), an analytical ROC expression for the proposed CCS detector can be obtained as 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 BQ ðP f 1 Þ 1=ð5NÞ  μT 1 C qffiffiffiffiffiffiffi ð63Þ P d1 ¼ Q @ A: s2T 1 5. Spectrum sensing algorithms based on serial correlation of polarization vector

53 55 57 59 61

To further exploit the correlation of polarization vector, this section considers serial correlation. The serial correlation test was first suggested by Beran and Watson [35] and used originally to test whether an ordered sequence of unit vectors was correlated or not. Motivated by their work, here we use this test to determine whether there is correlation among the received polarization vector sequences gð1Þ, gð2Þ; …; gðnÞ.

We define the serial correlation coefficient of polarization vector gðnÞ and gðmÞ as ρs ¼ g T ðnÞgðmÞ. The average serial correlation coefficient is 3

E½ρs  ¼ ðE½gÞT E½g ¼ ∑ ðE½g i Þ2 :

63 65

ð64Þ

67

When there is no primary signal, according to E½g i =H0  ¼ 0, the serial correlation coefficient

69

i¼1

E½ρs =H0  ¼ 0:

ð65Þ

It indicates that the polarization vector sequences of noise are independent, in other words, the polarization states represented by these polarization vector sequences are randomly distributed. When there is signal and noise: E½ρs =H1  ¼ ðE½g 1 =H1 Þ2 þ ðE½g 2 =H1 Þ2 þ ðE½g 3 =H1 Þ2 :

ð66Þ

Substituting previous result from (25) into (66), then we have " #2   1 ð1  P 2 Þ Δ : ð67Þ E ρs =H1 ¼ 2 1  2P P

71 73 75 77 79 81 83

Recalling the concept of degree of polarization P defined in (14) and the fact that 0 o P r1 under hypothesis H1 , (67) implies that there exists correlation among polarization states and the value of correlation mainly depends on P rather than received mean polarization state. This result is an immediate consequence of the fact that the serial correlation coefficient of a transmitted primary signal satisfies E½ρs  ¼ 1 due to P¼1. In other words, the polarization vector gðnÞ is perfectly correlated with gðmÞ since all samples have the same constant polarization state. However, after the primary signal is transmitted though channel, the correlation of received polarization vector samples decreases with the depolarization effect of channel. In summary, the serial correlation of polarization vector sequence can also be used to detect signal.

Stacking the received polarization vector sequences gð1Þ; gð2Þ; gð3Þ; …; gðnÞ yields the following n  3 matrix: XðnÞ ¼ ½gð1Þ; gð2Þ; …; gðnÞT . The serial correlation matrix of the received polarization vector is defined as C 2 ¼ XðnÞX T ðnÞ ¼ ½gð1Þ; gð2Þ; …; gðnÞT ½gð1Þ; gð2Þ; …; gðnÞ 2 3 cð1; 1Þ cð1; 2Þ ⋯ cð1; nÞ 6 7 6 cð2; 1Þ cð2; 2Þ ⋯ cð2; nÞ 7 7 ¼6 ð68Þ 6 ⋮ ⋮ ⋯ ⋮ 7 4 5 cðn; 1Þ cðn; 2Þ ⋯ cðn; nÞ where cðn; mÞ ¼ gðnÞT gðmÞ. For the special case n¼m, we have cðn; nÞ ¼ 1, thus ∑n cðn; nÞ ¼ n. If signal is not present, we get ∑n;m cðn; mÞ ¼ 0 from (65). In this case C 2 ¼ I n . Otherwise, if there is a signal, the sum of the overall elements of matrix C 2 will be larger than n in general according to (67). Therefore, the test statistic of serial correlation sensing (SCS) algorithm can be reasonably defined as the average sample serial correlation coefficient, that is, 1

N

∑

N

∑ cðn; mÞ:

N2 n ¼ 1 m ¼ 1

87 89 91 93 95 97 99

5.1. Serial correlation sensing (SCS) algorithm

T 2 ðN Þ ¼

85

101 103 105 107 109 111 113 115 117 119 121

ð69Þ

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123

C. Guo, X. Wu / Signal Processing ] (]]]]) ]]]–]]]

1 3 5 7 9 11 13 15 17

If there is noise only, T 2 ðNÞ-1=N. Otherwise, when signal is present with high SNR, T 2 ðNÞ-1. The outline of the proposed serial correlation sensing (SCS) algorithm is described as follows.

5.2. Performance analysis of SCS algorithm

21

In this section, we will derive the theoretical expressions for the false alarm probability P f 2 , detection probability P d2 and threshold γ2 of SCS algorithm. Using the central limit theorem, T 2 ðNÞ can also be approximated as Gaussian distribution with mean μT 2 and variance s2T 2 , i.e.,

25 27 29 31 33 35 37

T 2 ðNÞ  NðμT 2 ; s2T 2 Þ:

ð70Þ

Firstly, μT 2 and s2T 2 are derived under H1 hypothesis: h i N 1 N μT 2 =H1 ¼ E ½T 2 ðNÞ ¼ 2 ∑ ∑ E gðnÞT g ðmÞ N n¼1m¼1 " #     3 3 1 2 ∑ ∑ ∑ðg i ðnÞÞ þ ∑g i ðnÞ∑g j ðmÞ ¼ 2E N i ¼ j;i;j ¼ 1 i a j;i;j ¼ 1 " # 3 1 1 N 1 3 ∑ ðE½g i Þ2 ¼ 2 N þNðN  1Þ ∑ ðE½g i Þ2 ¼ þ N N i¼1 N i¼1 ð71Þ

39 41 43 45 47

h i s2T 2 =H1 ¼ E T 22 ðN Þ ðE½T 2 ðNÞÞ2 h i N N N 1 N ¼ 4 ∑ ∑ ∑ ∑ E gðnÞT g ðmÞgðkÞT g ðlÞ  ðE½T 2 ðNÞÞ2 N n¼1m¼1k¼1l¼1 " # 3   1 ¼ 4E ∑ ∑g i ðnÞ∑g i ðmÞ∑g i ðkÞ∑g i ðlÞ N i ¼ j;i ¼ 1 þ

"

2 N4

3

E

∑

i a j;i;j ¼ 1

 ∑g i ðnÞ∑g i ðmÞ∑g j ðkÞ∑g j ðlÞ

# 

ðE½T 2 ðNÞÞ2

49 51 53 55 57 59 61

h

E ∑g i ðnÞ∑g i ðmÞ∑g j ðkÞ∑g j ðlÞ

ð72Þ where   E ∑g i ðnÞ∑g i ðmÞ∑g i ðkÞ∑g i ðlÞ " " ¼∑ n

∑

∑

m a n k a m;n

∑ E½g i ðnÞg i ðmÞg i ðkÞg i ðlÞ

i a m;n;k

!

þ E½g i ðnÞg i ðmÞg 2i ðkÞ

#

þ E½g i ðnÞg 3i ðmÞ

# þ E½g 4i ðnÞ

¼ NðN  1ÞðN  2ÞðN 3ÞðE½g i Þ4 þ 6NðN  1ÞðN  2ÞE½g 2i ðE½g i Þ2 þ3NðN  1ÞðE½g 2i Þ2 þ NðN 1ÞE½g i E½g 3i  þNE½g 4i 

ð73Þ

63

i

65

¼ NðN  1ÞðN  2ÞðN 3ÞðE½g i Þ2 ðE½g j Þ2 þNðN  1ÞðN 2Þ ½E½g 2i ðE½g j Þ2 þ 4E½g i E½g j E½g i g j  þ ðE½g i Þ2 E½g 2j  þ NðN  1Þ

67

 ½ðE½g 2i ÞðE½g 2j Þ þ 2E½g i g 2j  þ NðN  1Þ½2E½g i E½g i g 2j 

69

þ 2E½g j E½g 2i g j  þ NE½g 2i g 2j :

ð74Þ

Under hypothesis H0 , applying all the terms of (39) and (40) to (71) and (72), μT 2 and s2T 2 can be obtained as μT 2 =H0 ¼ 1=N

ð75Þ

s2T 2 =H0 ¼ 2ðN  1Þ=ð3N3 Þ:

ð76Þ

71 73 75

Then the probability of false alarm, decision threshold and the probability of detection are 0 1 γ 2  1=N B C P f 2 ¼ Q @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 3 2ðN 1Þ=ð3N Þ

19

23

and

Algorithm 2. Serial correlation sensing (SCS) algorithm. Step 1. Sample the received polarization vectors, as previously described in (15). Step 2. Compute serial correlation matrix as in (68). Step 3. Calculate the decision value T 2 ðNÞ defined in (69). Step 4. Make the decision: if T 2 ðNÞ 4 γ 2 , signal exists (“yes” decision); otherwise, signal does not exist (“no” decision) where γ2 is the threshold given in Section 5.2 which is chosen to meet the requirement of the probability of false alarm.

9

γ2 ¼ Q

1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðP f 2 Þ 2ðN  1Þ=ð3N3 Þ þ1=N

79 81 83

ð78Þ

85

0

P d2

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 Q ðP Þ 2ðN 1Þ=ð3N Þ þ 1=N  μ =H 1 f T B C 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Q@ A s2T 2 =H1

ð77Þ

77

ð79Þ

where μT 2 =H1 and are given by (71) and (72), respectively. It should be noted that T 2 ðNÞ shown in (69) can also be written as s2T 2 =H1

N 1 N ∑ gðnÞT ∑ g ðmÞ N2 n ¼ 1 m¼1 "

2 N

2 N

2 # N 1 ∑ g 1 ðnÞ þ ∑ g 2 ðnÞ þ ∑ g 3 ðnÞ ¼ 2 : N n¼1 n¼1 n¼1

T 2 ðN Þ ¼

Recall that ð∑Nn ¼ 1 g1 ðnÞÞ2 þ ð∑Nn ¼ 1 g 2 ðnÞÞ2 þð∑Nn ¼ 1 g3 ðnÞÞ2 ¼ R2 , where R is the resultant vector length of the sample polarization vectors and once was used as a good measurement of the dispersion of polarization state on the unit Poincare sphere in our previous work [21]. Therefore, the decision variable presented in (69) becomes ðR=NÞ2 . Hence, the proposed SCS algorithm is actually also a resultantvector-length-based method. The test statistic is mainly determined by the degree of dispersion of polarization state on the unit Poincare sphere. Hence SCS is an invariant test statistic irrespective of received mean polarization state.

87 89 91 93 95 97 99 101 103 105 107 109 111

6. Discussions

113

By observing (61) and (78), we can see that the threshold γ1 of CCS algorithm and γ2 of SCS algorithm can be precomputed based only on sample number and the probability of false alarm, irrespective of the information of signal, noise and channel. Therefore, both the CCS algorithm and SCS algorithm are CFAR detectors. As for the detection performance, substituting (25)–(32) into related term of (55), (56), (71), and (72), and then into detection probability expressions of (63) and (79), we can

115

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10

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

C. Guo, X. Wu / Signal Processing ] (]]]]) ]]]–]]]

see that the expressions of detection probability for CCS and SCS detectors both finally depend on the number of samples N, SNR, and polarization of the received signal. It is known that the signal polarization at the receiver can be different from the original polarization at the transmitter as a result of the channel depolarization behavior, which can be unpredictable in some situations. In order to guarantee a minimum detection performance (therefore a maximum interference level to the primary system) in practical design and configuration of the detectors, we should qualitative discuss the performance with channel depolarization effect. Since the detection probability expressions are too cumbersome to further analysis, we choose degree of polarization P as a measure to evaluate channel depolarization effect. Take SNR ¼ 10 dB as example, Fig. 1 plots the numerical results of Pd against P with different number of samples N. Fig. 1 shows that the number of samples N can be chosen to achieve a target Pd required by practical requirement according to the value of P. In practical detection, the value of P can be estimated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 det ðR^ x Þ ð80Þ P^ ¼ 1  ½trðR^ x Þ2 where R^ x ðNÞ is the sampled covariance matrix of received signal and can be pre-computed as

1 N ∑ eðnÞeðnÞH ð81Þ R^ x ðNÞ ¼ N n¼1 In terms of the computational complexity, the major complexity of the proposed two methods consists of (1) transformation of the received electric-field vector samples into the polarization vector samples: generally 4N multiplications and 3N divisions; (2) computation of the test statistic: the CCS algorithm requires about 6N multiplications and additions for computing the autocorrelation matrix and 81 multiplications and additions for matrix rotation in addition, while the SCS algorithm needs about 3 multiplications and 3N additions. Here we compare the proposed two methods with four existing detectors: the traditional ED, MME, CAV, and polarization-based GLRT-PV detector. Assumed that ED, MME, and CAV detectors have M receive antennas. The ED, which enjoys the highest computational efficiency, only requires MN multiplications and ðM 1ÞðN  1Þ additions. The complexity of MME and CAV is about ML times and L times of that of ED respectively, where L is the smooth factor defined in [5]. According to the analysis above, the complexity of the proposed two methods is almost the same as that of ED when the number of receive antenna is the same, i.e., M ¼2, yet much lower than that of MME and CAV methods especially when L is large. Compared with GLRT-PV detector, the total computational complexity of SCS is the same, while CCS is slightly higher. Considering the hardware complexity, the only hardware requirement to perform the two proposed detectors is dual-polarized antennas, which is one kind of antenna structures for the deployment of multiple antenna systems and has been widely used in practical wireless systems, especially in Time Division-Synchronization Code Division Multiple Access (TD-SCDMA) and Time Division-Long

Term Evolution (TD-LTE). Here note that the proposed CCS and SCS detectors take place within the baseband processor rather than in the antenna hardware, thus adding no additional hardware complexity. It should be noted that the proposed two algorithms are both based on the assumption that the h and v electricfield components of noise are independent and have identical variance δ2. However, the filter of receiver will generate polarization-dependent gain that can affect the polarization of noise [36]. Since the effect of the filter on noise input is involved by the convolution of the impulse response with the polarization vector of noise, it cannot induce correlation between the two orthogonal polarization components of noise electric-field vector. However, since the gains of filter for two orthogonal polarization components are different in general, the output power of each component is not identical, thus leading to power imbalance between the h and v components after filter. Let g h and g v are gains of h and v components of filters, respectively. Suppose that g v ¼ μg h , where μrepresents power imbalance factor. Then the covariance matrix of the noise after filter becomes " # 1 0 2 Rn ¼ s g h : ð82Þ 0 μ Using (13), we get E½G0  ¼ ð1 þ μÞs g h , E½G1  ¼ ð1  μÞs g h , E½G2  ¼ E½G3  ¼ 0, thus η1 ¼ ð1  μÞ=ð1 þμÞ, η2 ¼ η3 ¼ 0. So the expressions in (39) and (40) under hypothesis H0 are changed and the moments and statistical distributions of T 1 ðNÞ and T 2 ðNÞ are altered comparing to (54) and (70). Consequently, both the performance of proposed CCS algorithm and SCS algorithm suffers from the power imbalance, which mainly depends on the value of power imbalance factor μ. To deal with this problem, a simple method is multiplying one of the h^ and v^ electric-field branches with a scale factor 1=μ. The value of μ is determined by the filter property and thus can be precomputed by the receiver. 2

63 65 67 69 71 73 75 77 79 81 83 85 87

2

89 91 93 95 97 99 101

7. Simulation results

103

In this section, we present some simulation results to confirm the theoretical analysis and demonstrate the effectiveness of the proposed two sensing methods through comparison with other methods. In these simulations, all the results are averaged over 10,000 Monte Carlo realizations. For each realization, random channel, random noise, and random modulated signal of PU are generated.

105

7.1. Comparison between simulation and analytical results

113

First, we compare the simulation and analytical results of the false alarm probability for the two proposed methods. The thresholds as function of Pf in (61) for CCS algorithm and in (78) for SCS algorithm are demonstrated in Fig. 2. Invoking the central limit theorem, we assume that the two test statistics are Gaussian distributed. To verify the accuracy of this assumption for small sample number, the false alarm probability when N¼100 is presented. From Fig. 2, it can be seen that there is no significant deviation between the

115

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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1

9 11

CCS N=100 CCS, N=1000 CCS, N=10000 SCS, N=100 SCS, N=1000 SCS, N=10000

0.9

Probability of Detection

7

63

1

3 5

11

0.8 0.7

65 67 69

0.6 0.5

71

0.4

73

0.3

75

13 0.2

15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

0.1

77 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Degree of Polarization Fig. 1. Probability of detection versus degree of polarization: Pf ¼ 0.1 and SNR ¼ 10 dB.

numerical results obtained from analysis (the curves with mark “-analytical”) under Gaussian assumption and those obtained from simulation (the curves with mark “-simulation”), even for N as small as 100. We also plot the ROC curves of the proposed two algorithms with different sample number N for fixed Pf ¼0.1 and SNR ¼  4 dB in Fig. 3. In order to observe the detection performance, we take a typical Urban NLOS microcell scenario as an example, where the parameters of dual-polarized Rayleigh fading channel including crosspolar discrimination XPD and polarization channel correlation coefficients pp and qp are set accordingly. We choose XPD ¼ 7:5 dB, i.e., χ ¼ 0:18, according to the measurement results given by [37] and pp ¼ qp ¼ 0:8 as suggested in [29], respectively. More analysis about the effect of those channel parameters are made in later simulation. Fig. 3 shows that the theoretical formulae for Pd match their simulation results quite well for the proposed two algorithms under N ¼10,000. However when N ¼100, there exists deviation due to Gaussian approximation under small sample number. In addition, both the analytical and simulation results improve with the increase of sample number N. It can be explained by the more accurate estimate of correlations of polarization vector with larger sample number. In order to test the impact of the dual-polarized channel parameters XPD, pp, and qp, we choose fixed SNR ¼  8 dB, N ¼1000 and Pf ¼0.1, and vary the value of χ, pp, and qp all ranging from 0 to 1. Fig. 4 shows that the Pd of both CCS and SCS methods become minimum for unity value of χ and they become maximum when χ equals zero. The simulation results can be explained by the fact that the smaller χ, i.e., the larger XPD, the less energy is coupled between the cross-polarized channels. In other words, the higher value of XPD leads to less depolarization effect on channel, and thus higher correlation of both component and serial of polarization vector. The simulation results for pp and qp in Fig. 4 show that the polarization channel correlation coefficients have similar effect as XPD on the detection performance. In summary, we can conclude that high XPD and polarization channel correlation coefficients play positive roles in improving detection performance.

Note that, from Fig. 4, when χ o 0:85 and pp ¼ qp 4 0:3, i.e., the depolarization effect of channel is weak, the SCS obtains a much better detection performance than the CCS scheme. However, the opposite result is observed when channel is highly depolarized, i.e., χ Z 0:85 and pp ¼ qp r 0:3 in Fig. 4. In order to explain the above observations, we resort to the statistic distributions of T 1 ðNÞ and T 2 ðNÞ under hypotheses H0 and H1 with strong and weak depolarization effect of channel. These comparisons are displayed in Figs. 5 and 6, respectively, for SNR ¼  8 dB, N¼1000. Comparing Fig. 5(a) with (b), obviously, the proposed SCS algorithm has higher detection performance than CCS algorithm when χ ¼ 0:1 and pp ¼ qp ¼ 0:9. Otherwise, when χ ¼ 0:9 and pp ¼ qp ¼ 0:1, the SCS algorithm shown in Fig. 6(a) and (b) has lower performance than CCS. Hence, the conclusion can be drawn that component correlation is more robust to depolarization effect than vector correlation.

79 81 83 85 87 89 91 93 95 97 99

7.2. Comparison with other sensing algorithms

101

Next, comparison between the two proposed algorithms and other existing sensing algorithms is presented. Note that the proposed two methods require two channels per sample by using one pair of dual-polarized antennas. To ensure comparability, we have to compare with existing multi-antenna detectors such as MME, AGM, and CAV with two identical polarized channels, i.e., two uni-polarized antennas employed. This case is the same as the comparison between previously proposed GLRT-PV method and these multi-antenna-based detectors shown in [21], in which the results have shown that the detector with dual-polarized antennas has clear advantage in case that polarization of PU is unknown. So here we mainly compare the proposed two methods with other existing polarization based detectors. Recently, a few polarization-based detectors have been proposed. As the simplest dual-polarized detector, Equal Gain Combining based polarization detector (EGC-PD) utilizes the energy received by the two branches, i.e., the vertical and horizontal branches of dual-polarized antennas, with equal gain. The virtual polarization detection

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12

1

9 11 13

Probability of False Alarm

7

CCS−analytical N=100

0.9

3 5

63

1

65

CCS−simulation N=100

0.8

SCS−analytical N=100

0.7

SCS−simulation N=100

67

CCS−analytical N=10000

0.6

69

CCS−simulation N=10000

0.5

SCS−analytical N=10000

0.4

SCS−simulation N=10000

71 73

0.3

75 0.2

15

77

0.1 0 −0.2

17

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Threshold

19

79 81

Fig. 2. Pf versus threshold.

21

83

23

85 1

25

29 31 33 35

Probability of Detection

27

87

CCS−analytical N=100 CCS−simulation N=100 SCS−analytical N=100 SCS−simulation N=100 CCS−analytical N=10000 CCS−simulation N=10000 SCS−analytical N=10000 SCS−simulation N=10000

0.9 0.8 0.7 0.6 0.5

89 91 93

0.4

95

0.3

97

0.2

37

99

0.1

39

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of False Alarm

41

101 103

Fig. 3. ROC curves: Pf ¼0.1, SNR ¼  4 dB, χ ¼ 0:18, and pp ¼ qp ¼ 0:8.

43

105

45

107

47

109 111

51 53 55 57

Probability of Detection

49 1

113 115

0.5 1 0.8 0 1

0.6 0.8

0.4

0.6

59

χ

0.4

,q p pp

0.2 0.2 0

61

117

0

Fig. 4. Impact of dual-polarized channel parameters XPD, pp, and qp: SNR ¼  8 dB, N ¼1000 and Pf ¼0.1.

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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13

63

1

5 7 9 11 13

65

f(T (N)/H )

Probability Density Function

3

f(T1(N)/H0) 1

1

Threshold γ

1

Pf1 P

69

d1

71 73 75 77

15 −0.5

−0.4

−0.3

−0.2

−0.1

17

0

0.1

0.2

0.3

0.4

0.5

79

Statistic T1(N)

81

19 f(T2(N)/H0)

21

25 27 29 31

83

f(T2(N)/H1)

Probability Density Function

23

Threshold γ2 Pf2 Pd2

37

89 91 93 95

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Statistic T2(N) Fig. 5. Comparison of the T 1 ðNÞ and T 2 ðNÞ distribution with weak depolarization effect: χ ¼ 0:1 and pp ¼ qp ¼ 0:9; SNR ¼  8 dB, N ¼ 1000.

43 45 47 49 51 53 55 57 59 61

97 99 101

39 41

85 87

33 35

67

(VPD) detector achieves better performance using optimal energy combining technology based on the adaptation of received polarization state. However, since it is not practical to change the hardware structure of antenna to adapt to different polarization states, virtual polarization adaptation scheme should be used in VPD detector, resulting a high cost of computational complexity. Instead of using energy information, generalized likelihood ratio test-polarization vector (GLRT-PV) detector exploits resultant vector length of the sample polarization vectors. To compare the performance of these polarization based detectors, Pd is plotted against the average SNR for EGC-PD, VPD, GLRT-PV, CCS and SCS methods with Pf ¼0.1 andN ¼1000 in Fig. 7. We also consider noise uncertainty's effect on these detectors using the approach from [38]. Here, we adopted the “robust statistics” method presented in [39] to model noise uncertainty in order to describe the “worst case”, which can easily show the SNR wall for detectors. We assume x denotes the degree of noise uncertainty in dB, the actual noise power thus falls in the interval Ω ¼ ½δ2 =η; ηδ2 , where η ¼ 10x=10 . In Fig. 7, “-x dB” means that the noise

uncertainty is x dB. From Fig. 7, we can see that the performance of the GLRT-PV method is the same as SCS method. This confirms previous analysis in Section 5.2 that the SCS and GLRT-PV methods both use the length of resultant vector as test statistic. Fig. 7 also shows that EGCPD and VPD perform better than CCS and SCS under the assumption of perfect knowledge of the noise variance. However, in the presence of noise uncertainty with 3 dB, the performance of EGC-PD and VPD degrades severely. The SNR wall for VPD and EGC-PD are found to be 4.1 dB and 6.5 dB, respectively. On the other hand, the SNR wall does not exist for GLRT-PV, CCS, and SCS. Hence the GLRT-PV, CCS and SCS detectors have the favorite noise-robust property as mentioned above. Moreover, when χ ¼ 0:1, and pp ¼ qp ¼ 0:9, the SCS and GLRT-PV schemes outperform the CCS scheme while achieving a slightly lower performance than CCS scheme when χ ¼ 0:9, and pp ¼ qp ¼ 0:1, which confirms again the observations in Fig. 4. In order to show a overall effect of channel parameters, i.e., XPD and channel correlation coefficients pp, and qp, on the proposed two methods and other existing polarization

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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1

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3

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5

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7

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9

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13

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15

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17

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81

21

83

23

85

25

87

27

89

29

91

31

93

33

95

35

97

37 39

Fig. 6. Comparison of the T 1 ðNÞ and T 2 ðNÞ distribution with strong depolarization effect: χ ¼ 0:9 and pp ¼ qp ¼ 0:1;SNR ¼  8 dB, N ¼ 1000. (a) T 1 ðNÞ distribution. (b) T 2 ðNÞ distribution.

45 47 49 51 53 55 57 59 61

101 103

41 43

99

based detectors, we fix SNR ¼-10 dB, Pf ¼0.1 and N ¼1000 and plot the probability of detection with varied value of χ, pp, and qp ranging from 0 to 1 as shown in Fig. 8. We see that the decreased χ and increased pp and qp improve the performance of GLRT-PV, CCS, and SCS, while VPD and EGC-PD performance remains almost unchanged. This phenomenon can be easily explained that both VPD and EGC-PD use total energy of two branches of received signal. Though the energy of two branches coupled from each other and the correlation between them are changed by channel depolarization behavior, the overall energy of two branches of received signal does not change, thus the detection performance does not affected by channel depolarization. As for our proposed CCS and SCS in practical detection, in order to guarantee target detection performance, the detection of received signal propagated through channel with high depolarization effect requires large number of samples N while small N does well for weak depolarization effect case. The value of N can be set according to estimated degree of polarization P^ in (82).

8. Conclusion 105 In this work, the inherent polarization characteristics of signal were exploited for spectrum sensing with dualpolarized antennas. We firstly derived the statistical distribution of the received polarization vector and its moments, from which we found that the component correlation and serial correlation statistics of the received polarization vector in hypothesis H0 are different from those in hypothesis H1 . Hence we proposed two algorithms based on polarization vector correlation: component correlation sensing (CCS) and serial correlation sensing (SCS). This paper then provided the theoretical performance of the two detectors in terms of closed-form expressions for the probability of false alarm and the probability of detection. Simulations results showed that both CCS and SCS detectors achieved better performance with higher XPD and polarization channel correlation coefficients. It was also showed that, if channel is highly depolarized, CCS performed better than SCS. Otherwise,

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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1

15

63

1 0.9

3 5 7 9 11

Probability of Detection

0.8

VPD EGC−PD CCS χ=0.1 p =q =0.9

65

SCS χ=0.1 p =q =0.9

67

p

p

p

0.7

p

CCS χ=0.9 p =q =0.1 p

p

0.6

SCS χ=0.9 p =q =0.1

0.5

GLRT−PV χ=0.1 pp=qp=0.9

0.4

GLRT−PV χ=0.9 p =q =0.1

0.3

VPD−3dB EGC−PD−3dB

p

69

p

p

71

p

73

0.2

13

75

0.1

15

77

0 −15

−10

−5

0

17

5

10

15

79

SNR(dB) Fig. 7. Comparison with existing polarization based detectors: Pf ¼ 0.1 and N¼ 1000.

19

81 83

21

85

23 25

29 31 33

1

Probability of Detection

27

0.8

37

87 89

0.6

91

0.4

93

0.2 0 1

35

39

VPD EGC−PD CCS SCS VPD−3dB EGC−PD−3dB

95 1 0.8

97

0.8 0.6

χ

0.4

0.2

0.2 0

0.6 0.4 , pp qp

99

0

Fig. 8. Comparison with existing polarization based detectors with varying dual-polarized channel parameters χ, pp, and qp: Pf ¼0.1, SNR ¼  10 dB, and N ¼ 1000.

103

41

105

43 45 47 49 51

the latter shows better performance. Moreover, in this paper we also verified that the proposed SCS was equivalent to the GLRT-PV detector. Compared with existing polarization based detectors, the proposed two detectors had superior performance and moderate computational complexity. So CCS and SCS are attractive spectrum sensing approaches and can be easily applied in many practical wireless systems with dual-polarized antennas, such as TDSCDMA, and TD-LTE system.

53 55 57 59 61

101

Acknowledgments This work was supported in part by the Chinese National Nature Science Foundation under Grants 61271177 and 60902047, the Fundamental Research Funds for the Central University under Grant 2013RC0108, and Beijing Higher Education Young Elite Teacher Project under Grant 96254006.

Appendix A 107 Proof of Theorem 1. According to the definition of n-order moment and using (24), we can get   1  P2 E g ni =H1 ¼ 2

Z

111

g ni ð1  ηi g i Þ  Θ 1  g i jÞ dg i 3=2 1 Ωi 1

where Ωi ¼ P 2 g 2i  2ηi g i þ1  P 2 þη2i . Denote Q in ¼ 3=2 Ωi dgi , then   1  P2   Q in  ηi Q iðn þ 1Þ : E g ni =H1 ¼ 2

109

113 R1 1

g ni =

115 117

ðA:1Þ 119

Evaluation of Qin for n ¼ 1, 2, 3, 4, 5 gives Q i1 ¼

2ηi ð1  P 2 Þð1  η2i Þ

121 ðA:2Þ

Please cite this article as: C. Guo, X. Wu, Spectrum sensing algorithms based on correlation statistics of polarization vector, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.05.010i

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16

1

Q i2 ¼ 

3 5

2ð1  P 2  η2i Þ 2

P ð1  P

2

Þð1  η2i Þ

Q i4 ¼

þ

11

17

ðA:4Þ

P 6 ð1  P 2 Þð1  η2i Þ 3ðP

4

þ 5η2i P 2 ð1 þ η2i ÞÞΔ 7 2P

þ

5ηi ð3P 4 þ7η2i  3P 2 ð1 þ η2i ÞÞΔ 2P 9

ðA:5Þ

ðA:6Þ

where Δ ¼ ln 1 P=1 þ P. By substituting (A.2)–(A.6) into (A.1), we arrive at (25)–(28). Appendix B

25

29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

2ηj u 2 u2 r r 2 BðA BuÞ

where r ¼ Z π=2 M¼ ¼

2ηj r B

Proof of Theorem 2. Here we mainly confine ourselves to consider the derivation of the statistic of E½g i g j . The derivation of the remaining statistics of E½g i g 2j , E½g 2i g 2j , and E½g i g j g 2k  is similar and, in the interest of brevity, is left out. Using (23), the first order joint moment of gi and gj can be expressed as h i Z E gi gj ¼ g i g j f ðg Þ dgi dg j dg k R3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z g g δð1  g 2 þ g 2 þ g 2 Þ i j i j k ¼c dg i dg j dgk 3 ð1  η g  η g  η g Þ2 R i i j j k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z g δð1  g 2 þg 2 þ g 2 Þ j i j k ¼ c gi dg j dg k dg i ðB:1Þ 2 R R2 ð1  ηi g i  ηj g j ηk g k Þ

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du r 2  u2

ðB:2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1  g 2i . Let u ¼ r sin x, then

65

2ηj r sin x 1 r cos x dx BðA  Br sin xÞ2 r cos x

67

Z

π=2

sin x

 π=2

ðA  Br sin xÞ

2

69 ðB:3Þ

dx:

71

ðA BrxÞ

¼ 2

73

1 1 A a2 þ Br A  Brx Br ðA BrxÞ2

then

75

# 1 1 A 1 þ  dx Br A Br sin x Br ðA Br sin xÞ2  π=2 " # 2ηj r π A2 π : ðB:4Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ 2 2 3=2 2 2 B Br A B r 2 BrðA  B r 2 Þ

2ηj r M¼ B

Z

"

π=2

Substituting (B.4) into (B.1) results in " # Z h i 2ηj r π A2 π dg i  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ E gi gj ¼ c gi 2 2 3=2 B Br A2 B2 r 2 BrðA  B r 2 Þ 2 Z 2ηi π 1 6 gi ffi ¼c 2 4  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B 1 ð1 ηi g i Þ  B2 ð1  g 2i Þ 3 g i ð1  ηi g i Þ2

þ

2

2

3=2

ðð1 ηi g i Þ B ð1  g 2i ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 Defining I in ¼  1 g ni = ð1  ηi g i Þ2  B2 ð1  g 2i Þ dg i , we have Z I i1 ¼

ηk g j  ηj g k v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi η2j þ η2k

then ηj u þηk v ηk u ηj v g j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi; g k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi: η2j þ η2k η2j þ η2k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R gj δð1  g2i þ g2j þ g2k Þ Denote M ¼ R2 ð1  η g  η g  η g Þ2 dg j dgk , then i i j j k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z δð1  g 2 þ u2 þ v2 Þ ηj u þ ηk v i qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi du dv: M¼ 2 2 2 R2 ð1  η g  ηj þ ηk uÞ η2j þη2k i i qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let A ¼ 1  ηi g i , B ¼ η2j þη2k , then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z δð1  g 2 þu2 þ v2 Þ ηj u þ ηk v i M¼ du dv B ðA BuÞ2 R2 Z

Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ηj u þηk v 2 þ u2 þv2 Þ dv du δð1  g ¼ i 2 R BðA BuÞ R

77 79 81 83 85 87 89 91

7 5 dg i :

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2η η g i = ð1  ηi g i Þ2  B2 ð1 g 2i Þ dg i ¼ 2 i þ 3i Δ: P P 1

93 95

1

97 99

Thus

101

h i 1 P 2 2η π   j  I i1 þQ i1  2ηi Q i2 þ η2i Q i3 E gi gj ¼ 4π B2

103

where Q i1  Q i3 are given by (A.2)–(A.4) respectively.

105

where c ¼ ð1  P 2 Þ=ð4πÞ. Let ηj g j þηk g k u ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; η2j þ η2k

63

Since x

23

27

¼

 π=2

 h    i Q i5 ¼ ηi 6P 8 þ 24P 6 1 η2i  105η2i 1  η2i h  i þ ηi P 2 45 þ70η2i  115η4i h   i  8   þ ηi P 4  75 þ 59η2i þ 16η4i = 3P 1  P 2 1  η2i

19 21

P

3

Z ðA:3Þ

2P 6  5P 4 ð1  η2i Þ  15η2i ð1 η2i Þ þP 2 ð3 þ 10η2i  13η4i Þ

9

15

Δ

2η ð3η2  2η2 P 2 þ P 4 þ 2P 2  3Þ 3ηi Δ þ 5 Q i3 ¼ i i 4 i 2 P P ð1  P Þð1  η2i Þ

7

13

þ

107 Appendix C 109 Proof of Theorem 3. It can be easily obtained that h i 3 1 N ∑ ∑ E g~ i ðnÞg~ j ðnÞ μT 1 ¼ E ½T 1 ðNÞ ¼ N n ¼ 1 i o j;i;j ¼ 1 ¼

3

∑

i o j;i;j ¼ 1

E½g~ i g~ j :

111 113 ðC:1Þ

Now, efforts are paid to find the variance of T 1 ðNÞ. Since 2

!2 3 g~ i ðnÞg~ j ðnÞ 5

h i 3 1 N ∑ E ðT 1 ðNÞÞ2 ¼ E 4 ∑ N n ¼ 1 i o j;i;j ¼ 1 " ! !# 3 3 1 N 1 N ∑ ∑ ∑ g~ i ðnÞg~ j ðnÞ ∑ g~ k ðmÞg~ l ðmÞ ¼E N n ¼ 1 i o j;i;j ¼ 1 N m ¼ 1 l o k;l;k ¼ 1

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115 117 119 121 123

C. Guo, X. Wu / Signal Processing ] (]]]]) ]]]–]]]

1 3



1 N 1 N ∑ g~ i ðnÞg~ j ðnÞ ∑ g~ i ðmÞg~ j ðmÞ Nn¼1 Nm¼1 i o j;i;j ¼ 1 



3 1 N 1 N ∑ g~ i ðnÞg~ j ðnÞ ∑ g~ i ðmÞg~ k ðmÞ E þ2 ∑ Nn¼1 Nm¼1 i o j;i o k;i;j;k ¼ 1 3

¼

∑

E

5 7 9 11 13 15 17 19 21 23 25 27 29

ðC:2Þ where  1 N 1 N ∑ g~ i ðnÞg~ j ðnÞ ∑ g~ i ðmÞg~ j ðmÞ E Nn¼1 Nm¼1 " # 2 1 N 2 ~ ~ ¼ E 2 ∑ g ðnÞg j ðnÞ N n¼1 i " # N 1 N ∑ g~ i ðmÞg~ j ðmÞ þE 2 ∑ g~ i ðnÞg~ j ðnÞ N n¼1 m a n;m ¼ 1 1 h 2 2i N  1 ðE½g~ i g~ j Þ2 ¼ E g~ i g~ j þ N N and 



1 N 1 N ∑ g~ i ðnÞg~ j ðnÞ ∑ g~ i ðmÞg~ k ðmÞ E Nn¼1 Nm¼1  N 1 ¼ 2 E ∑ g~ i ðnÞg~ j ðnÞg~ i ðnÞg~ k ðnÞ N n¼1 # þ

N

∑

n a m;n;m ¼ 1

g~ i ðnÞg~ j ðnÞg~ i ðmÞg~ k ðmÞ

i N1 h i   1 h E g~ i g~ j E g~ i g~ k : ¼ E g~ 2i g~ j g~ k þ N N

33 35 37 39 41 43 45 47 49 51 53 55 57 59

ðC:4Þ

To calculate the variance, we have s2T 1 ¼ E½ðT 1 ðNÞÞ2   ðE½T 1 ðNÞÞ2 :

31

ðC:3Þ

ðC:5Þ

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63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123

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