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Journal of the Franklin Institute 352 (2015) 2486–2503 www.elsevier.com/locate/jfranklin

Adaptive cyclostationary array beamforming with robust capabilities$ Ju-Hong Leea,b,n, Chia-Ching Chaoa, Chia-Cheng Huanga, Wen-Chen Loa a

Graduate Institute of Communication Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan b Department of Electrical Engineering, Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan Received 29 April 2014; received in revised form 30 November 2014; accepted 17 March 2015 Available online 31 March 2015

Abstract The existing techniques exploiting the signal cyclostationarity have been shown to be effective in performing antenna array beamforming without requiring the steering information. However, these techniques suffer from performance degradation due to both cycle frequency error (CFE) for the desired signal and ﬁnite data sample (FDS) effects. To deal with the CFE, we present an iterative averaging (IA) scheme to estimate the actual cyclic correlation matrix. With regard to the effect of cycle leakage due to ﬁnite data samples, we develop a novel fully data-dependent diagonal loading (NFD-DL) scheme to estimate the actual data correlation matrix. Then, a novel method is developed based on the IA scheme in conjunction with the NFD-DL scheme to possess the robust capabilities against the CFE and the FDS effects simultaneously. The simulation results show that the proposed method outperforms the existing methods in dealing with antenna array beamforming for cyclostationary signals in the presence of both CFE and FDS effects. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

☆

This work was supported by the Ministry of Science and Technology of Taiwan under Grants MOST 100-2221-E002-200-MY3 and MOST 103-2221-E-002-123-MY3. n Corresponding author at: Department of Electrical Engineering, Graduate Institute of Communication Engineering, and Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan. E-mail addresses: [email protected] (J.-H. Lee), [email protected] (C.-C. Chao), [email protected] (C.-C. Huang), [email protected] (W.-C. Lo). http://dx.doi.org/10.1016/j.jfranklin.2015.03.029 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Beamforming using antenna array of sensors such as radio antennas is useful in the process of detecting the presence of the desired signal, estimating their directions of arrival and other parameters, and estimating the signal waveforms themselves. For conventional adaptive array beamforming, we require the prior information of either the impinging direction or the waveform of the desired signal to adapt the weights [1,2]. The adaptive weights of an antenna array beamformer under a steered-beam constraint are calculated by minimizing the beamformer's output power subject to the constraint that forces the beamformer to make a constant response in the steering direction. Hence, the beamformer's performance is very sensitive to the accuracy of the steering vector. However, the true direction vector of the desired signal may not be exactly known in some applications, e.g., the application in land mobilecellular radio systems. It has been shown in [2–6] that even a small mismatch between the true direction vector of the desired signal and the steering vector deteriorates the effectiveness of a steered-beam beamformer. In contrast, adaptive beamforming utilizing signal cyclostationarity has been widely considered due to two main reasons. Based on the cyclostationary property, we can solve the problem of blind adaptive beamforming, i.e., an adaptive beamformer can automatically preserve the desired signal while cancelling noise and interference without any prior information regarding the direction angle of the desired signal [7]. Moreover, cyclostationarity is a statistical property possessed by most of practical man-made communication signals [8,9]. Adaptive beamforming utilizing signal cyclostationarity known as cyclic adaptive beamforming has been widely considered [7,10,11]. Recently, several cyclic-related techniques have been developed to deal with different realistic scenarios and applications [12–18]. The advantage of the above techniques is that they work without requiring the direction vector or the waveform of the signal considered. Notable among them, the constrained cyclic adaptive beamforming (C-CAB) technique [10] possesses the ability of enhancing the interference suppression. Moreover, its asymptotic performance is similar to that of the conventional linearly constrained minimum variance (LCMV) beamformer [2]. The a priori information required by the aforementioned techniques is only the cycle frequency of the signals considered. Hence, their performance is sensitive to the accuracy of the presumed cycle frequency. However, the actual cycle frequency may not be known exactly in practical applications due to, e.g., the phenomenon of Doppler shift. To tackle the problem of array performance degradation due to cycle frequency error (CFE), the robust cyclic adaptive beamforming approaches in [19,20] develop an iterative procedure to ﬁnd an appropriate estimate of the cycle frequency of the desired signal. Recently, a more effective technique utilizing a compensation method in conjunction with the subspace projection method of [21] was proposed in [22] under three constraints: (1) the unknown difference between the presumed cycle frequency and the actual one for the desired signal is small. (2) The cycle frequencies of the desired signal and interference are well separated. (3) The strengths of the signal sources are somewhat close to each other. Then, a cyclic conjugate correlation matrix by using a compensation matrix to cope with the deterioration of its dominant singular value when cycle frequency error (CFE) exists is constructed. The compensation matrix is constructed by projecting the singular vector corresponding to the dominant singular value onto the signal subspace of the correlation matrix of the received array data vector. In practice, only a ﬁnite time interval is available for receiving the array data vector. The ensemble correlation matrix of the received array data vector is approximated by a sample correlation matrix computed from a ﬁnite set of array data samples taken in the time interval. Using sample correlation matrix instead of ensemble correlation matrix for performing array beamforming leads to the so-called ﬁnite data sample (FDS) effect [21]. The above

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three constraints may make the compensation method of [22] with limited capabilities in dealing with the CFE and FDS problems simultaneously. In this paper, we consider the problem of simultaneously solving the CFE and the FDS effects. First, to efﬁciently tackle the CFE effect, an iterative averaging (IA) scheme to estimate the actual cyclic correlation matrix is presented. The IA scheme searches an appropriate range of cycle frequencies including the true one iteratively for averaging the presumed cyclic correlation matrix. The averaged cyclic correlation matrix is then utilized as an estimate of the actual cyclic correlation matrix to alleviate the CFE effect. Next, we develop a novel fully data-dependent diagonal loading (NFD-DL) to cope with the FDS effect. A sample data correlation matrix with loading factors automatically generated from the received array data vector is created. Using the NFD-DL scheme instead of the conventional loading approaches achieves signiﬁcant robustness against the FDS effect. Finally, we propose a method based on the IA scheme in conjunction with the NDF-DL scheme to compute an appropriate weight vector for performing cyclic adaptive array beamforming. The computer simulation examples illustrate the effectiveness of the proposed method in curing the impairment of cyclic adaptive array beamforming due to the CFE and FDS effects. This paper is organized as follows. In Section 2, we brieﬂy review the theory of adaptive beamforming for cyclostationary signals and the conventional C-CAB algorithm. We present a method based on the IA scheme in conjunction with the NFD-DL scheme for dealing with both the CFE and FDS effects in Section 3. Several simulation examples are provided in Section 4 for illustration and comparison. Finally, we conclude the paper in Section 5.

2. Adaptive beamforming for cyclostationary signals 2.1. Signal cyclostationarity Cyclostationarity is a statistical property possessed by most of practical man-made communication signals. For a signal s(t), its cyclic correlation function (CCF) and cyclic conjugate correlation function (CCCF) are given by the following inﬁnite time averages: r ss ðα; τÞ ¼ 〈sðtÞsn ðt τÞe j2παt 〉1

and

r ssn ðα; τÞ ¼ 〈sðtÞsðt τÞe j2παt 〉1

ð1Þ

respectively, where the superscript “n” denotes the complex conjugate. We say that r(t) is cyclostationary if its CCF or CCCF is not equal to zero at some time delay τ and cycle frequency α. It has been shown in [8,9] that many modulated signals exhibit cyclostationarity with cycle frequency equal to the twice of the carrier frequency or multiples of the baud rate or combinations of these. For the data vector xðtÞ received by an antenna array with N isotropic array elements, its cyclic correlation matrix (CCM) and cyclic conjugate correlation matrix (CCCM) are given by Rxx ðα; τÞ ¼ 〈xðtÞxH ðt τÞe j2παt 〉1

and

Rxxn ðα; τÞ ¼ 〈xðtÞxT ðt τÞe j2παt 〉1 ;

ð2Þ

respectively, where the superscripts “H” and “T” denote R T=2the conjugate transpose and the complex conjugate, respectively. Moreover, 〈〉1 ¼ limT-1 T=2 dt.

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2.2. Conventional adaptive beamforming using signal cyclostationarity Consider an adaptive beamformer using an M-element antenna array excited by a desired signal, J interferers, and spatially white noise. The received data vector xðtÞ is given by xðtÞ ¼ sðtÞ þ aðθd Þ þ

J X

sj ðtÞaðθj Þ þ nðtÞ ¼ sðtÞaðθd Þ þ iðtÞ;

ð3Þ

j¼1

where s(t) and sj(t) denotes the waveforms of the desired signal and the jth interferer, aðθd Þ and aðθj Þ are the direction vectors of the desired signal with direction angle θd and the jth interferer with direction angle θj, respectively, and nðtÞ is the background noise vector. The array output is given by yðtÞ ¼ wH xðtÞ, where w denotes the weight vector of the beamformer. Assume that s(t) is cyclostationary and has a cycle frequency α, but iðtÞ is not cyclostationary at α and is temporally uncorrelated with s(t). According to the original C-CAB algorithm of [10], the weight vector wCCAB required for cyclic beamforming is the solution of the following constrained minimization problem: wH Rxx w

Minimize

Subject to wH cab w ¼ 1;

ð4Þ

where wcab is the dominant left singular vector of the CCCM of Eq. (2), i.e., the left singular vector of the CCCM corresponding to the maximum singular value. Rxx ¼ EfxðtÞxðtÞH g denotes the ensemble autocorrelation matrix of xðtÞ. The solution of (4) is given by [10] wCCAB ¼

1 Rxx wcab : 1 H wcab Rxx wcab

ð5Þ

From Eqs. (2) and (3), we note that Rxxn ðα; τÞ ¼ r ssn ðα; τÞaðθd Þaðθd ÞT is a rank-one and nonHermitian matrix. Moreover, we observe that the columns of Rxxn ðα; τÞ span the same subspace as that spanned by aðθd Þ. Therefore, the left singular vector corresponding to the maximum 1 singular value of Rxxn ðα; τÞ is proportional to aðθd Þ [10] and wCCAB is proportional to Rxx aðθd Þ. As a result, the C-CAB solution wCCAB is proportional to the weight vector of the well-known 1 1 LCMV beamformer [2] which is given by wLCMV ¼ Rxx aðθd Þ=ðaðθd ÞH Rxx aðθd ÞÞ. This indicates that the C-CAB algorithm can perform as well as the LCMV algorithm under the environments without CFE. However, the performance analysis presented in [22] has shown that the C-CAB algorithm suffers from severe performance degradation even if there is a small mismatch in the cycle frequency of the desired signal. In practice, only a time interval ½0; T is available for receiving the data vector xðtÞ, where T is ﬁnite. We resort to a sample average instead of an ensemble average for approximation. Let the sampling interval be Ts. The optimal weight vector wCCAB of Eq. (5) with n data snapshots (i.e., T ¼ nT s ) used is given by ^ CCAB ðnÞ ¼ w

^ 1 ðnÞw ^ cab ðnÞ R xx 1 ^ cab ðnÞ ^ cab ðnÞR^ ðnÞw w

ð6Þ

xx

^ ^ where Pn R xx ðnÞ is H the sample autocorrelation matrix of xðtÞ and is given by R xx ðnÞ ¼ k ¼ 1 xðt k Þxðt k Þ , where xðt k Þ denotes the sample of xðtÞ taken at the kth time instant. ^ cab ðnÞ is the dominant left singular vector of the sample CCCM given by w n X ^ xxn ðα; τ; nÞ ¼ 〈xðtÞxT ðt τÞe j2παt 〉T ¼ R xðt k Þxðt k τÞT e j2παtk : ð7Þ

1 n

k¼1

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^ cab ðnÞ is a consistent estimate of aðθd Þ and can be expressed It has been shown in [10] that w asymptotically as aðθd Þ κaðθd Þ, where κ denotes a proportional constant. Based on the assumption of ergodicity, we note from Eq. (6) that w^ CCAB ðnÞ converges to wCCAB as the number n of the array data samples approaches inﬁnite. Nevertheless, the C-CAB algorithm suffers from the performance deterioration if the ﬁnite data sample (FDS) effect exists, i.e., n is not sufﬁciently large. Moreover, it has been shown in [19] that the energy of the SOIs exists in the result of the convolution of the sinc function and the cyclic conjugate autocorrelation matrix when a cycle frequency mismatch occurs, i.e., there is a difference Δα between the presumed cycle frequency αp and the actual cycle frequency α. The signal energy in cyclic conjugate autocorrelation will leak out as sinc function due to the CFE Δα. 3. The proposed method To deal with the CFE and FDS effects simultaneously, we present a method based on an iterative averaging (IA) scheme in conjunction with a novel fully data-dependent diagonal loading (NFD-DL) as follows. 3.1. An iterative averaging (IA) scheme An appropriate manner to reduce the CFE effect is to take the average of the sample CCCM given by Eq. (7) in a presumed range of the cycle frequencies including the actual cycle frequency α of the desired signal. The averaged cyclic conjugate correlation matrix (ACCCM) algorithm was originally proposed by [23] for alleviating the impairment due to CFE in estimating the bearings of cyclostationary signals. The ACCCM is computed as follows [23]: Z αb 1 ^ ^ xxn ðα; τ; nÞ dα R R xxn ðαa ; αb ; τ; nÞ ¼ αb αa αa 1 Rðαb ; τ; nÞ Rðαa ; τ; nÞ ; ð8Þ ¼ αb αa where Rðαb ; τ; nÞ ¼ 〈 j2πtxðtÞxT ðt τÞe j2παb t 〉T ; Rðαa ; τ; nÞ ¼ 〈 j2πtxðtÞxT ðt τÞe j2παa t 〉T

ð9Þ

α a and αb denote the lower and upper bounds of the actual cycle frequency α, respectively. We note from Eq. (8) that the smaller the difference αb αa is, the more accurately R^ xxn ðαa ; αb ; τ; nÞ approaches R^ xxn ðα; τ; nÞ, i.e., we have the following result in the limit case: lim R^ xxn ðαa ; αb ; τ; nÞ ¼ R^ xxn ðα; τ; nÞ

αb -αþ αa -α

ð10Þ

However, the main difﬁculty in using the original ACCCM algorithm is that it provides an operation like a windowing processing on the time-domain data and, hence will also enhance the effects due to interference and noise on array performance. Moreover, the larger the difference αb αa is, the more performance degradation the array beamformer suffers. Therefore, determining an appropriate range of the cycle frequencies including the actual cycle frequency α of the desired signal becomes a crucial issue. In the following, we present an iterative averaging (IA) scheme to alleviate the above difﬁculty. Assume that αl ¼ α^ Δαa and αu ¼ α^ þ Δαa denote the lower and

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^ respectively. Moreover, Δαa is positive and greater upper bounds of a presumed cycle frequency α, than the CFE Δα. Then, we create the following matrix from Eq. (8): Aðαl ; αu ; τ; nÞ ¼ R^ xxn ðαl ; αu ; τ; nÞ R^ xxn ðαl ; αu ; τ; nÞH

ð11Þ

and ﬁnd the dominant eigenvalue λmax fAðαl ; αu ; τ; nÞg of Aðαl ; αu ; τ; nÞ as follows. First, compute a vector gmax ðnÞ from Aðαl ; αu ; τ; nÞ by taking the product Aðαl ; αu ; τ; nÞ ½1; 1; …; 1T . Then, λmax fAðαl ; αu ; τ; nÞg can be found by g ðnÞH Aðαl ; αu ; τ; nÞgmax ðnÞ λmax A ðαl ; αu ; τ; nÞ ¼ max gmax ðnÞH gmax ðnÞ

ð12Þ

according to the Rayleigh theory of [24, pp. 216–217]. Using Eq. (12) instead of directly taking the eigenvalue decomposition of the matrix Aðαl ; αu ; τ; nÞ to obtain λmax fAðαl ; αu ; τ; nÞg can reduce the required computational complexity from OðM 3 Þ to OðM 2 Þ. From Eq. (11), we note that λmax fAðα1 ; α2 ; τ; nÞg4λmax fAðα3 ; α4 ; τ; nÞg if α A ½α1 ; α2 and α2 = ½α3 ; α4 . Based on this result, we present an iterative scheme to ﬁnd appropriate lower and upper bounds for Eq. (8). First, we take the average of the initial lower and upper bounds, i.e., αave ¼ ðαl þ αu Þ=2. Then, we compute Aðαl ; αave ; τ; nÞ and Aðαave ; αu ; τ; nÞ by using Eq. (11). Hence, we adjust the lower and upper bounds of α for Eq. (8) according to the following rule: ( αl ¼ αave if λmax fAðαave ; αu ; τ; nÞg4λmax fAðαl ; αu ; τ; nÞg ð13Þ αu ¼ αave if λmax fAðαl ; αave ; τ; nÞg4λmax fAðαl ; αu ; τ; nÞg The above process is repeated until the difference of the resulting lower and upper bounds, αnl and αnu of α is less than a preset threshold ε, i.e., αnu αnl r ε. Finally, we compute Aðαnl ; αnu ; τ; nÞ from Eq. (11) by using αnl and αnu .

3.2. A novel fully data-dependent diagonal loading (NFD-DL) The FD-DL is originally developed in [25] for improving the robust capabilities of conventional LCMV beamforming against steering mismatch and FDS effect. The principle of the conventional DL techniques is to add a scaled identity matrix to the sample autocorrelation matrix R^ xx ðnÞ to obtain R^ xx ðnÞ þ γI. Then, the optimal LCMV weight vector with n data snapshots used is given by 1

^ DL ðnÞ ¼ w

^ ðnÞ þ γIgaðθd Þ fR xx ^ 1 ðnÞ þ γIgaðθd Þ aðθd ÞH fR

ð14Þ

xx

1 1 instead of wLCMV ¼ Rxx ðnÞaðθd Þ=ðaðθd ÞH Rxx ðnÞaðθd ÞÞ. The main drawback with Eq. (14) is that there is no easy and reliable scheme for choosing an appropriate loading factor γ. To tackle this P ^ k ðnÞ to replace γI and then problem, the FD-DL proposed a general loading matrix Pk ¼ 1 R xx deﬁne a matrix

^ FDDL ðnÞ ¼ R ^ xx ðnÞ þ R

P X k¼1

k R^ xx ðnÞ

ð15Þ

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to replace R^ xx ðnÞ to create the following weight vector for LCMV beamforming: P ^ k ðnÞÞ 1 aðθd Þ ðR^ xx ðnÞ þ Pk ¼ 1 R xx w^ FDDL ðnÞ ¼ PP k H ^ aðθd Þ ðR xx ðnÞ þ k ¼ 1 R^ xx ðnÞÞ 1 aðθd Þ

ð16Þ

The effectiveness of the FD-DL and how to determine the parameter P have been analyzed in detail in [25]. The main advantages of using the FD-DL over the existing DL techniques are that the required loading factor can be completely obtained from the received array data without requiring any additional sophisticated scheme and achieves more signiﬁcant robustness against steering vector error and FDS effect [25]. However, the FD-DL cannot be used for the beamforming problem without steering angle information. In the following, we develop a novel FD-DL (NFD-DL) to deal with the FDS effect for cyclostationary array beamforming. Although the matrix Rxxn ðα; τÞ ¼ r ssn ðα; τÞaðθd Þaðθd ÞT is a rank-one and non-Hermitian matrix, its columns span the same subspace as that spanned by aðθd Þ. Taking the product of Rxxn ðα; τÞH and a weight vector w, we have Rxxn ðα; τÞH w ¼ r ssn ðα; τÞn aðθd Þn aðθd ÞH w:

ð17Þ

The Euclidean norm of Eq. (17) is given by ‖Rxxn ðα; τÞH w‖2 ¼ Mjr ssn ðα; τÞj2 jwH aðθd Þj2 :

ð18Þ

Accordingly, the following conventional LCMV beamforming problem Minimize

wH Rxx w

Subject to wH aðθd Þ ¼ 1;

ð19Þ

is equivalent to the following beamforming problem: Minimize

wH Rxx w

Subject to ‖Rxxn ðα; τÞH w‖2 ¼ Mjr ssn ðα; τÞj2 :

ð20Þ

The optimal solution to Eq. (20) can be found by solving the unconstrained minimization problem given by Minimize

JðwÞ ¼ wH Rxx w þ λðMjr ssn ðα; τÞj2 ‖Rxxn ðα; τÞH w‖2 Þ;

ð21Þ

where λ represents the Lagrange multiplier. Differentiating JðwÞ with respective to w and setting it to zero provides the optimal solution for Eq. (21) as follows: w 1 ¼ Rxx Rxx w ¼ λRd w or Rd w; ð22Þ λ where Rd ¼ Rxxn ðα; τÞ Rxxn ðα; τÞH ¼ Mjr ssn ðα; τÞj2 aðθd Þaðθd ÞH . From Eq. (22), we note that the 1 optimal weight vector wo is the dominant eigenvector of the matrix Rxx Rd and is designated by 1 wo ¼ RfRxx Rd g:

ð23Þ 1 Rxx aðθd Þ.

As a result, It is easy to show that the optimal weight vector wo is proportional to adaptive beamforming using the optimal weight vector wo yields the same performance as that of the LCMV beamforming without requiring the a priori information aðθd Þ. Again, we have the optimal weight vector given by 1

^ ðnÞR ^ d ðnÞg w^ o ðnÞ ¼ RfR xx

ð24Þ

when only n data snapshots are available in practice, where the M M matrix R^ d ðnÞ ¼ ^ xxn ðα; τ; nÞ R^ xxn ðα; τ; nÞH . Moreover, the term R^ 1 ðnÞ in Eq. (24) is replaced by R ^ FDDL ðnÞ R xx of Eq. (15) to enhance the robustness against the FDS effect. Accordingly, the proposed

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NFD-DL produces an optimal weight vector as follows: 1

^ ^ ^ NDFDL ðnÞ ¼ RfR w FDDL ðnÞR d ðnÞg

ð25Þ

Finally, we present a novel method based on the IA scheme of Section 3.1 in conjunction with ^ CF ðnÞ to tackle the CFE and the NFD-DL of Section 3.2 to create an appropriate weight vector w FDS effects simultaneously as follows: 1

^ CF ðnÞ ¼ RfR^ FDDL ðnÞAðαnl ; αnu ; τ; nÞg w

ð26Þ

A block diagram for showing how the proposed method works is given in Fig. 1. In the following, we describe a procedure for summarizing the proposed method step-by-step: Step 1: Set an appropriate range ½αl ¼ α^ Δαa ; αu ¼ α^ þ Δαa for a presumed cycle frequency ^ the parameter P required by Eq. (16), and a preset threshold ε. α, Step 2: Compute R^ xxn ðαl ; αu ; τ; nÞ from Eq. (8) and Aðαl ; αu ; τ; nÞ from Eq. (11). Step 3: Find the dominant eigenvalue λmax fAðαl ; αu ; τ; nÞg of Aðαl ; αu ; τ; nÞ from Eq. (12). Step 4: Perform the iterative process according to Eq. (13) to determine αnl and αnu of α^ satisfying αnu αnl r ε. Step 5: Compute the matrix Aðαnl ; αnu ; τ; nÞ corresponding to αnl and αnu . Step 6: Compute the matrix R^ FDFL ðnÞ according to Eq. (15) and the product ^ FDFL ðnÞ 1 Aðαn ; αn ; τ; nÞ. R l u Step 7: Carry out the required eigen-decomposition to ﬁnd the dominant eigenvector of the ^ FDFL ðnÞ 1 Aðαn ; αn ; τ; nÞ and set it to be the resulting optimal weight matrix R l u 1 ^ CF ðnÞ ¼ RfR^ FDDL ðnÞAðαnl ; αnu ; τ; nÞg according to Eq. (26). vector w

3.3. Computational complexity Here, we compare the computational complexity required by each of the above mentioned techniques. The C-CAB algorithm [10] requires roughly a complexity of OðM 3 Þ for performing the singular value decomposition (SVD) on the corresponding M M cyclic correlation matrices [26], where M is the number of array elements. To apply the method of [20], we have to compute the maximum singular value of the M M data correlation matrix. The required computational complexity is also about OðM 3 Þ. On the other hand, the method of [22] requires a computational ^ xx ðnÞ and the SVD of the M M cyclic complexity of OðM 3 Þ for performing the SVD of R correlation matrices after implementing the compensation procedure. As for the proposed

Fig. 1. The block diagram of the proposed method.

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method, we note that the main computational complexity required is the SVD of Aðαl ; αu ; τ; nÞ ^ 1 ðnÞAðαn ; αn ; τ; nÞ. Hence, the proposed method also needs a complexity and the SVD of R l u FDDL 3 of OðM Þ. As a result, the computational complexity required by each of the aforementioned techniques is roughly OðM 3 Þ. 4. Computer simulation examples Here, we present several computer simulation examples for illustration and comparison. For all simulations, the desired signal and the interference signals are narrowband and far ﬁeld binary phase-shift keying (BPSK) signals with rectangular pulse shape. For simplicity, the time delay τ is set to zero. The background noise is spatially white Gaussian noise with mean zero and variance equal to one. Moreover, the number of Monte Carlo runs is 50. Example 1. In this example, a uniform linear antenna array (ULA) with 10 array elements and λs =2 inter-element spacing is used, where λs is the wavelength of the desired signal, the desired signal with cycle frequency α ¼ 2 and an interference signal with cycle frequency α ¼ 2.4 are impinging on the array from 01 and 401 off array broadside, respectively. The sampling frequency and the baud rate are set to 10 and 5=11, respectively. The signal-to-noise power ratio (SNR) is 5 dB for the desired signal and 10 dB for the interference signal. Moreover, the CFE Δα ¼ 0:02, i.e., α^ ¼ α þ Δα ¼ 1:98. The parameters Δαa ; ε, and P are set to 0.1, 0.01 and 5, respectively. Fig. 2 depicts the array output signal-to-interference plus noise power ratio (SINR) versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Fig. 3 plots the output SINR versus the CFE for the above mentioned methods. Fig. 4 shows the resulting array beam patterns. We observe from the simulation results that the proposed method outperforms the other methods. Example 2. In this example, the desired signal with cycle frequency α ¼ 2 and an interference signal with cycle frequency α ¼ 7:8 are impinging on the array from 01 and 401 off array broadside, respectively. The SNR is 0 dB for the desired signal and 10 dB for the interference signal. Moreover, the CFE Δα ¼ 0:02, i.e., α^ ¼ α þ Δα ¼ 1:98. The parameters Δαa ; ε, and P are set to 0.1, 0.01 and 5, respectively. Fig. 5 depicts the array output SINR versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the 15

Output SINR (dB)

10 5 0 −5

LCMV without CFE and FDS Method of [10]

−10 Method of [20] Method of [22]

−15

Proposed method

−20 0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 2. The output SINR versus the number of data snapshots for Example 1.

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15

Output SINR (dB)

10 5 0 −5

LCMV without CFE and FDS Method of [10]

−10 −15

Method of [20] Method of [22] Proposed method

−20 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 3. The output SINR versus the CFE under 150 data snapshots for Example 1.

0

Power Gain (dB)

−10

−20

−30

−40

−50

LCMV without CFE and FDS Method of [10] Method of [20] Method of [22]

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 4. The array beam patterns under 150 data snapshots for Example 1.

results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Fig. 6 plots the output SINR versus the CFE for the above mentioned methods. Fig. 7 shows the resulting array beam patterns. We observe from the simulation results that the proposed method outperforms the other methods and provides array performance very close to that of the LCMV algorithm when the number of data snapshots is larger than 100. Example 3. The example is the same as Example 1 except that the SNR is 0 dB for the desired signal, i.e., the desired signal is weaker in this case. Fig. 8 depicts the array output SINR versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Fig. 9 plots the output SINR versus the CFE for the above mentioned methods. Fig. 10 depicts the resulting array beam patterns. Again, we observe from the simulation results that the proposed method provides more robustness against the CFE and FDS effects than the other methods. Example 4. Here, we consider amore general realistic scenario with a ULA of 10 array elements and 3λs =2 inter-element spacing, where λs is the wavelength of the desired signal. The sampling frequency and the baud rate are set to 1 GHz and 5 108 =11, respectively. The desired signal with carrier frequency ¼ 100 MHz and cycle frequency α¼ 200 MHz, and an interference signal with

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Output SINR (dB)

5 0 −5 −10

LCMV without CFE and FDS Method of [10]

−15 Method of [20] Method of [22]

−20

Proposed method

−25 0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 5. The output SINR versus the number of data snapshots for Example 2.

10

Output SINR (dB)

5 0 −5 −10

LCMV without CFE and FDS Method of [10]

−15 −20

Method of [20] Method of [22] Proposed method

−25 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 6. The output SINR versus the CFE under 150 data snapshots for Example 2.

0

Power Gain (dB)

−10

−20

−30

−40

−50

LCMV without CFE and FDS Method of [10] Method of [20] Method of [22]

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 7. The array beam patterns under 150 data snapshots for Example 2.

carrier frequency ¼ 110 MHz and cycle frequency α¼ 220 MHz are impinging on the array from 101 and 401 off array broadside, respectively. The SNR is 5 dB for the desired signal and 10 dB for the interference signal. Moreover, the CFE Δα ¼ 10 MHz, i.e., α^ ¼ α þ Δα ¼ 210 MHz. The

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10

Output SINR (dB)

5 0 −5 LCMV without CFE and FDS Method of [10]

−10 −15

Method of [20] Method of [22]

−20 Proposed method

−25 0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 8. The output SINR versus the number of data snapshots for Example 3. 10

Output SINR (dB)

5 0 −5 −10 −15

LCMV without CFE and FDS Method of [10] Method of [20] Method of [22]

−20 Proposed method

−25 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 9. The output SINR versus the CFE under 150 data snapshots for Example 3.

0

Power Gain (dB)

−10

−20

−30

−40

−50

LCMV without CFE and FDS Method of [10] Method of [20] Method of [22]

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 10. The array beam patterns under 150 data snapshots for Example 3.

parameters Δαa , ε, and P are set to 0.1, 0.01 and 5, respectively. Fig. 11 depicts the array output SINR versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using LCMV algorithm with the known signal direction

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angles 101 and 401 are also shown in this ﬁgure. Figs. 12 and 13 plot the output SINR versus the CFE and the resulting array beam patterns, respectively, for the above mentioned methods. For this more general realistic scenario, we also observe from the simulation results that the proposed method provides more robustness against the CFE and FDS effects than the other methods. Example 5. To evaluate the capability of the proposed method in dealing with strong situation under negative SNR, we repeat Example 2 except that the SNRs are set to 5 dB and 10 dB for the desired signal and the interference signal, respectively. Fig. 14 shows the array output SINR versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Figs. 15 and 16 plot the output SINR versus the CFE and the resulting array beam patterns, respectively, for the above mentioned methods. From the simulation results, we see that the proposed method works satisfactorily as compared to the existing methods. Example 6. To evaluate the performance of the proposed method in dealing with the situation where the array sensors are very close, we repeat Example 2 except that the inter-element spacing is set to 0.15 λs instead of 0.5 λs. Fig. 17 shows the array output SINR versus the number of data 15

Output SINR (dB)

10 5 0

LCMV without CFE and FDS Method of [10]

−5 −10

Method of [20] Method of [22]

−15

Proposed method −20

0

200

400

600

800

1000

Number of Data Snapshots

Fig. 11. The output SINR versus the number of data snapshots for Example 4.

15

Output SINR (dB)

10

5

LCMV without CFE and FDS Method of [10]

0

Method of [20] Method of [22]

−5

−10 −10

Proposed method −5

0

5

10

CFE (Δα) (MHz)

Fig. 12. The output SINR versus the CFE under 150 data snapshots for Example 4.

J.-H. Lee et al. / Journal of the Franklin Institute 352 (2015) 2486–2503 0

Power Gain (dB)

−10

−20

−30

LCMV without CFE and FDS Method of [10]

−40

Method of [20] Method of [22]

−50

Proposed method

−60 −90 −80 −70 −60 −50 −40 −30 −20 −10

0

10 20 30 40 50 60 70 80 90

Direction Angle (Degree)

Fig. 13. The array beam patterns under 150 data snapshots for Example 4.

6 4

Output SINR (dB)

2 0 −2 LCMV without CFE and FDS Method of [10]

−4 −6

Method of [20] Method of [22]

−8

Proposed method

−10 0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 14. The output SINR versus the number of data snapshots for Example 5.

5

Output SINR (dB)

0

−5

−10

LCMV without CFE and FDS Method of [10]

−15

Method of [20] Method of [22] Proposed method

−20 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 15. The output SINR versus the CFE under 150 data snapshots for Example 5.

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Power Gain (dB)

−10

−20

−30

LCMV without CFE and FDS Method of [10]

−40

Method of [20] Method of [22]

−50

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 16. The array beam patterns under 150 data snapshots for Example 5.

10

Output SINR (dB)

5

0

−5

−10

LCMV without CFE and FDS Method of [10]

−15

Method of [20] Method of [22] Proposed method

−20

0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 17. The output SINR versus the number of data snapshots for Example 6.

snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Figs. 18 and 19 plot the output SINR versus the CFE and the resulting array beam patterns, respectively, for the above mentioned methods. From the simulation results, we note that the proposed method also outperforms the existing methods in this situation. Example 7. Here, we investigate the performance of the proposed method in the presence of a strong SNR in the desired signal and the interference signal. The example is the same as Example 2 except that the SNRs are set to 10 dB and 20 dB for the desired signal and the interference signal, respectively. Fig. 20 shows the array output SINR versus the number of data snapshots by using the proposed method, the methods of [10,20,22]. For comparison, the results of using the LCMV algorithm with the known signal direction angles 01 and 401 are also shown in this ﬁgure. Figs. 21 and 22 depict the output SINR versus the CFE and the resulting array beam patterns, respectively, for the above mentioned methods. Again, we observe from the simulation results that the proposed method deals with the strong situation more effectively as compared to the existing methods.

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15

Output SINR (dB)

10 5 0 −5

LCMV without CFE and FDS Method of [10]

−10 Method of [20] Method of [22]

−15 −20 −0.1

Proposed method

−0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 18. The output SINR versus the CFE under 150 data snapshots for Example 6.

0

Power Gain (dB)

−10

−20

−30

LCMV without CFE and FDS Method of [10]

−40

Method of [20] Method of [22]

−50

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 19. The array beam patterns under 150 data snapshots for Example 6.

25 20

Output SINR (dB)

15 10 5

0 LCMV without CFE and FDS Method of [10]

−5 −10

Method of [20] Method of [22]

−15 −20

Proposed method

0

100

200

300

400

500

600

700

800

900

1000

Number of Data Snapshots

Fig. 20. The output SINR versus the number of data snapshots for Example 7.

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Output SINR (dB)

15 10 5 0 LCMV without CFE and FDS Method of [10]

−5 −10 −15

Method of [20] Method of [22]

−20

Proposed method

−0.1 −0.08 −0.06 −0.04 −0.02

0

0.02

0.04

0.06

0.08

0.1

CFE (Δα)

Fig. 21. The output SINR versus the CFE under 150 data snapshots for Example 7.

0

Power Gain (dB)

−10

−20

−30

−40

LCMV without CFE and FDS Method of [10]

−50

Method of [20] Method of [22]

Proposed method −60 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70 80 90

Directional Angle (Degree)

Fig. 22. The array beam patterns under 150 data snapshots for Example 7.

5. Conclusion This paper has presented a novel method to deal with the cycle frequency error (CFE) and ﬁnite data sample (FDS) effects simultaneously for adaptive cyclostationary beamforming. An iterative averaging (IA) scheme is developed to ﬁnd an appropriate averaged cyclic correlation matrix which is then utilized as an estimate of the actual cyclic correlation matrix to alleviate the CFE effect. To tackle the FDS effect, we propose a novel fully data-dependent diagonal loading (NFD-DL) approach to create a sample data correlation matrix with loading factors automatically generated from the received array data. The IA scheme in conjunction with the NFD-DL approach has made the proposed method very robust against the CFE and FDS effects. The computer simulation results have shown the effectiveness of the proposed method. References [1] H.L. Van Trees, Detection, Estimation, and Modulation Theory, Optimum Array Processing (Part IV), John Wiley and Sons, Ltd., NY, 2002. [2] R.T. Compton Jr., Adaptive Antennas, Concepts, and Performance, Prentice Hall, Englewood Cliff, NJ, 1989.

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[3] J. Li, P. Stoica, Z. Wang, On robust capon beamforming and diagonal loading, IEEE Trans. Signal Process. 51 (7) (2003) 1702–1715. [4] J.-H. Lee, K.-P. Cheng, Adaptive array beamforming with robust capabilities under random phase perturbations, IEEE Trans. Signal Process. 53 (1) (2005) 365–371. [5] J.-H. Lee, C.-C. Wang, Adaptive array beamforming with robust capabilities under random sensor position errors, IEE Proc.—Radar Sonar Navig. 152 (6) (2005) 383–390. [6] J.-H. Lee, C.-C. Wang, K.-P. Cheng, Adaptive array beamforming with robust capabilities under steering angle mismatch, Signal Process. 86 (2) (2006) 296–309. [7] B.G. Agee, S.V. Schell, W.A. Gardner, Spectral self-coherence restoral: a new approach to blind adaptive signal extraction using antenna arrays, Proc. IEEE 78 (4) (1990) 753–767. [8] W.A. Gardner, Spectral correlation of modulated signals: Part I—analog modulation, IEEE Trans. Commun. 35 (6) (1987) 584–594. [9] W.A. Gardner, W.A. Brown III, C.-K. Chen, Spectral correlation of modulated signals: Part II—digital modulation, IEEE Trans. Commun. 35 (6) (1987) 595–601. [10] Q. Wu, K.M. Wong, Blind adaptive beamforming for cyclostationary signals, IEEE Trans. Signal Process. 44 (11) (1996) 2757–2767. [11] K.-L. Du, M.N.S. Swamy, A class of adaptive cyclostationary beamforming algorithms, Circuits Syst. Signal Process. 27 (1) (2008) 35–63. [12] J.Y. Huang, P. Wang, Q. Wan, Sidelobe suppression for blind adaptive beamforming with sparse constraint, IEEE Commun. Lett. 15 (3) (2011) 343–345. [13] C.Y. Liu, Y.F. Chen, C.P. Li, Blind beamforming schemes in SC-FDMA systems with insufﬁcient cyclic preﬁx and carrier frequency offset, IEEE Trans. Veh. Technol. 58 (9) (2009) 4848–4859. [14] R. Wang, C. Hou, An adaptive beamforming method based on properties of cyclostationary signals, in: Proceedings of the IEEE International Conference on Signal Processing, October, 2010, Beijing, China, pp. 4848–4859. [15] R. Wang, C. Hou, J. Yang, Blind adaptive beamforming algorithm based on cyclostationary signals, in: Proceedings of the IEEE International Conference on Image Analysis and Signal Processing, April, 2010, Zhejing, China, pp. 481–491. [16] W. Zhang, W. Liu, Low-complexity blind beamforming based on cyclostationarity, in: Proceedings of the European Conference on Signal Processing, August, 2012, Bucharest, Romania, pp. 1–5. [17] J. Li, G. Wei, Y. Ding, Adaptive beamforming based on covariance matrix reconstruction by exploiting interferences' cyclostationarity, Signal Process. 93 (9) (2013) 2543–2547. [18] C. Capdessus, A.K. Nandi, Extraction of a cyclostationary source using a new cost function without pre-whitening, Signal Process. 91 (11) (2011) 2497–2505. [19] J.-H. Lee, Y.-T. Lee, Robust adaptive array beamforming for cyclostationary signals under cycle frequency error, IEEE Trans. Antennas Propag. 47 (2) (1999) 233–241. [20] J.-H. Lee, Y.-T. Lee, W.-H. Shih, Efﬁcient robust adaptive beamforming for cyclostationary signals, IEEE Trans. Signal Process. 48 (7) (2000) 1893–1901. [21] J.-H. Lee, C.-C. Huang, Blind adaptive beamforming for cyclostationary signals: a subspace projection approach, IEEE Antennas Wirel. Propag. Lett. 8 (2009) 1406–1409. [22] J.-H. Lee, C.-C. Huang, Robust cyclic adaptive beamforming using a compensation method, Signal Process. 92 (4) (2012) 954–962. [23] J. Zhang, G. Liao, J. Wang, Robust direction ﬁnding for cyclostationary signals with cycle frequency error, Signal Process. 85 (12) (2005) 2386–2393. [24] D.W. Lewis, Matrix Theory, World Scientiﬁc, Singapore, 1991. [25] C.-C. Huang, J.-H. Lee, Robust adaptive beamforming using a fully data-dependent loading technique, Prog. Electromagn. Res. B 37 (2012) 307–325. [26] D.S. Watkins, Fundamentals of Matrix Computations, 3dr ed., John Wiley & Sons, Inc., 2010.