The Generalized Sidelobe Canceller Based on Quaternion Widely Linear Processing

We investigate the problem of quaternion beamforming based on widely linear processing. First, a quaternion model of linear symmetric array with two-component electromagnetic (EM) vector sensors is presented. Based on array's quaternion model, we propose the general expression of a quaternion semiwidely linear (QSWL) beamformer. Unlike the complex widely linear beamformer, the QSWL beamformer is based on the simultaneous operation on the quaternion vector, which is composed of two jointly proper complex vectors, and its involution counterpart. Second, we propose a useful implementation of QSWL beamformer, that is, QSWL generalized sidelobe canceller (GSC), and derive the simple expressions of the weight vectors. The QSWL GSC consists of two-stage beamformers. By designing the weight vectors of two-stage beamformers, the interference is completely canceled in the output of QSWL GSC and the desired signal is not distorted. We derive the array's gain expression and analyze the performance of the QSWL GSC in the presence of one type of interference. The advantage of QSWL GSC is that the main beam can always point to the desired signal's direction and the robustness to DOA mismatch is improved. Finally, simulations are used to verify the performance of the proposed QSWL GSC.


Introduction
As an important tool of multidimensional signal processing, the quaternion algebra has been applied to parameter estimation of 2D harmonic signals [1], DOA estimation of polarized signals [2][3][4], image processing [5], space-timepolarization block codes [6], Kalman filter [7], adaptive filter [8,9], independent component analysis (ICA) algorithm [10], widely linear modeling and filtering [11][12][13][14][15], nonlinear adaptive filtering [16], and blind source separation [17]. In these applications, the interest in quaternion widely linear processing has recently increased due to the use of the full second-order statistical information in the quaternion domain. In [13], a widely linear quaternion least mean square (WL-QLMS) algorithm was presented to improve the accuracy in adaptive filtering of both second-order circular (Qproper) and second-order noncircular (Q-improper) quaternion signals. In [14], a class of variable step-size algorithms were introduced into the WL-QLMS, so as to enhance the WL-QLMS tracking ability and sensibility to dynamically changing environments. In [15], a reduced-complexity WL-QLMS algorithm was developed to reduce computational cost and speed convergence of the WL-QLMS. In [16], a nonlinear quaternion adaptive filter and its widely linear version were proposed based on locally analytic nonlinear activation function. In [17], a quaternion widely linear predictor was applied to the blind source separation to extract both Qproper and Q-improper sources. All the quaternion widely linear algorithms employ the quaternion widely linear model and the associated augmented quaternion statistics, which includes the information in both the standard covariance and the three pseudocovariances, so that their performance was enhanced. Motivated by the benefits of signal processing in quaternion domain, the quaternion beamformers [18][19][20] were recently developed. In [18], a quaternion minimum mean square error algorithm was proposed and applied to the beamforming of an airborne trimmed vector-sensor array. In [19], a quaternion-capon beamformer using a crossed dipole 2 The Scientific World Journal array was proposed to improve the robustness of Capon beamformer. In [20], an interference and noise cancellation algorithm of quaternion MVDR beamformer was proposed to cancel the uncorrelated interference. Because the only information in the standard covariance of quaternion signals is used, the performance of these beamformers is not obviously enhanced. To make full use of the information in both the standard covariance and the three pseudocovariances of quaternion signals, the quaternion widely linear model may be introduced in the quaternion beamformer. Unfortunately, the quaternion beamformer based on widely linear model has received a little attention.
In this paper, we investigate the problem of quaternion beamforming based on widely linear processing. Because the quaternion output vector of array with two-component EM vector sensors is constructed by two jointly proper complex vectors, we present firstly the general expression of a quaternion semiwidely linear (QSWL) beamformer in this paper. Unlike the complex widely linear beamformers in [21][22][23], the QSWL beamformer can handle the proper complex vectors, whereas in this case the complex widely linear beamformer degenerates to conventional linear processing due to the vanishing of pseudocovariance. Thus, the QSWL beamformer can employ more information than the complex widely linear beamformers to improve the performance. Moreover, we propose a useful implementation of QSWL beamformer, that is, QSWL generalized sidelobe canceller (GSC), using a linear symmetric array. The QSWL GSC consists of two-stage beamformers. By designing the weight vectors of two-stage beamformers, the interference is completely canceled in the output of QSWL GSC and the desired signal is not distorted.

Quaternion Model of Vector-Sensor Array.
Consider that a scenario with one narrowband, completely polarized source, which is traveling in an isotropic and homogeneous medium, impinges on a uniform linear symmetric array from direction ( , ). This uniform linear symmetric array consists of 2 two-component vector sensors, which is depicted in Figure 1, and the spacing between the adjacent two vector sensors is assumed to be half wavelength. The array's reference point is at the array centroid and all vector sensors are indexed by − , . . . , −1, 1, . . . , from left to right. Two complex series 1 ( ) and 2 ( ) are obtained from first and second components of the th two-component vector sensor, respectively. The complex series 1 ( ) and 2 ( ) in the planes spanned by {1, }, where denotes a pure unit imaginary, are given by where and denote the incidence source's elevation angle measured from the positive -axis and azimuth angle measured from the positive -axis, respectively. represents −M M · · · · · · −2 −1 1 2 x y z Figure 1: A uniform linear symmetric array.
the orientation angle, and signifies the ellipticity angle.
( , ) is the spatial phase factor describing wavefield propagation along an array, and ( , ) = * − ( , ) due to the symmetric structure of uniform linear array. ( ) is the complex envelope of the waveform, assumed to be a stationary stochastic process with zero mean and second-order circularity. Thus, 1 ( ) and 2 ( ) are also a stationary stochastic process with zero mean and secondorder circularity.
where denotes a pure unit imaginary in a quaternion domain 1 . ( , , , ) = 1 ( , , , ) + 2 ( , , , ) is the quaternion-valued response on a two-component vector sensor. This transformation maps the complex signal 1 ( ) on scalar and imaginary fields of a quaternion, and the complex signal − 2 ( ) is simultaneously mapped to the and imaginary fields of a quaternion. It is noted that ( ) is -proper 2 because two complex series 1 ( ) and − 2 ( ) are second-order circularity. When the quaternion-valued additive noise is considered, ( ) can be rewritten as where ( ) = 1 ( ) + − 2 ( ). 1 ( ) and − 2 ( ) are, respectively, the complex-valued additive noises at the first component of the th vector sensor and the second component of the − th vector sensor, which are assumed to be zero mean, Gaussian noise with identical covariance 2 . And it is assumed that ( ) and ( ), where ̸ = , are uncorrelated.
In order to extend this model to the multisource case, we assume that uncorrelated, completely polarized plane waves impinge on this array. One is the desired signal characterized by its arrival angles ( , ) and polarization parameters ( , ); the others are the interference characterized by its arrival angles ( , ) and polarization parameters

The Quaternion Semiwidely Linear Beamformer.
The most general linear processing is full widely linear processing, which consists in the simultaneous operation on the quaternion vector and its three involutions. Then, a quaternion widely linear beamformer can be written as where W, G, H, and F are the quaternion-valued weight vectors. Superscript (⋅) ⊲ denotes the quaternion conjugate and transpose operator. x ( ) ( ), x ( ) ( ), and x ( ) ( ) denote the quaternion involution of x( ) over a pure unit imaginary , , and , respectively.
The full widely linear processing is optimal processing for the Q-improper quaternion vector. Since the quaternionvalued vector x( ) is -proper vector, the optimal processing reduces to semiwidely linear processing 3 . Because the semiwidely linear processing consists only in the simultaneous operation on the quaternion vector and its involution over , the general expression of a quaternion semiwidely linear (QSWL) beamformer can be written as where x ( ) ( ) is given by [11] x ( ) ( ) = − x ( ) Moreover, we can write the quaternion-valued output series ( ) in the following Cayley-Dickson representation: where [ ] 1 and [ ] 2 denote, respectively, the first and the second complex-valued components of a quaternion ; that is, Thus, the QSWL beamformer has two complex-valued output series 1 ( ) and 2 ( ) in the planes spanned by {1, }, Since the conventional "long vector" beamformer has only one complex-valued output series 1 ( ), the QSWL beamformer can obtain more information than the conventional "long vector" beamformer. The increase of information results in the improvement of QSWL beamformer's performance. In addition, we incorporate the information on both x( ) and x ( ) ( ), so that the QSWL beamformer with different characteristics may be obtained by designing two weight vectors W and G under some different criterions.

The QSWL Generalized Sidelobe Canceller
In this section, a useful implementation of the QSWL beamformer, that is, QSWL generalized sidelobe canceller (GSC), is proposed. The QSWL GSC, which is depicted in Figure 2, consists of two-stage beamformers. In the first-stage beamformer (weight vector is W), we attempt to extract a desired signal without any distortion from observed data. To cancel interference, we attempt to estimate interference in second-stage beamformer (weight vector is G). By employing the output of the second-stage beamformer to cancel interference in the output of the first-stage beamformer, there is no interference in the output of the QSWL GSC. Compared with the conventional "long vector" beamformer, the advantages of two-stage beamformers are that the main beam can always point to desired signal's direction, even if the separation between the DOAs of the desired signal and interference is less, and the robustness to DOA mismatch is improved.
In the following, we derive the expressions of quaternionvalued weight vectors W in the first-stage beamformer and G in the second-stage beamformer. It is assumed that two ( = 2) uncorrelated, completely polarized plane waves, whose waveform is unknown but whose DOA and polarization may be priorly estimated from techniques presented in [2,4,[25][26][27][28][29], impinge on an array depicted in Figure 1. One plane wave is the desired signal and its complex envelope is denoted by ( ); the other plane wave is the interference and its complex envelope is denoted by ( ). Since ( ) and ( ) are complex series, the output of the QSWL GSC must be complex series. Because the quaternion-valued output has two complex-valued components in the planes spanned by {1, }, we define the first complex-valued output component as the output of the QSWL GSC. Thus, the complex-valued output of the QSWL GSC is written as where ( ) is the complex-valued output of the first-stage beamformer; that is, In the first-stage beamformer, we attempt to minimize the interference-plus-noise energy in ( ), subject to the constraint [W ⊲ k ] 1 = 1.
Since the Cayley-Dickson representation of W, k , k , and N( ) is, respectively, where Superscript (⋅) denotes the complex conjugate and transpose operator. Thus, (10) can be rewritten as Then, W can be derived by solving the following constrained optimization problem: is the covariance matrix and x ( ) = k ( ) + N( ) is the measurement vector of array in the absence of the desired signal. The solution of this constrained optimization problem is obtained by using Lagrange multipliers; that is, If the interference is uncorrelated with the additive noise, W can be written in the simple form (the proof is in Appendix A) where where denotes the input interference-to-noise ratio (INR). Moreover, the quaternion-valued optimal weight vector W may be given by where J 1 = [I 2 ×2 , 0 2 ×2 ] and J 2 = [0 2 ×2 , I 2 ×2 ] are two selection matrices. It is noted that in some applications, such as Radar, R may be estimated in intervals of no transmitted signal. But, R is not obtained in other applications, such as Communications. In these applications, we may replace R by R , where is the covariance matrix and x( ) = k ( ) + k ( ) + N( ) is the measurement vector of array. When the distortionless constraint is perfectly matched with the desired signal, the weight vector W is identical in both R and R [30]. By using the optimal weight vector W, the complex output of the first-stage beamformer can be given by 3.2. The Second-Stage Beamformer. From (7), we have In the second-stage beamformer, we attempt to minimize the noise energy in ( ), subject to the constraints [G ⊲ k ( ) ] 1 = 0 and [G ⊲ k ( ) ] 1 = [W ⊲ k ] 1 . In the following, two schemes are presented to implement this aim.
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Scheme 1, That Is, Combined QPMC and MVDR.
Let G = w w , where w is a quaternion-valued diagonal weight matrix and w is a complex weight vector. In this scheme, the first step is to achieve the constraint [G ⊲ k ( ) ] 1 = 0 by designing w , and this step is referred to quaternion polarization matched cancellation (QPMC); the second step is to minimize the noise energy in ( ) subject to the constraint [G ⊲ k ( ) ] 1 = [W ⊲ k ] 1 by designing w , and this step is referred to MVDR.
Let w = diag{ (− ), . . . , ( )}; then we have where superscript (⋅) * denotes the quaternion conjugate operator. From (25) where diag(q ) = diag{ − ( , ), . . . , ( , )}. Inserting G = w w into (24), ( ) can be rewritten as (27) Then, w can be derived by solving the following constrained optimization problem: The solution of this constrained optimization problem is obtained by using Lagrange multipliers; that is, If the desired signal and interference are uncorrelated with the additive noise, w can be written in the simple form (the proof is in Appendix B) where =ṼṼ = 2 ( 2 where R(⋅) denotes the real part of a complex number. is given by (20).

Scheme 2, That Is, LCMV.
In this scheme, we employ the LCMV beamformer as the second-stage beamformer.
Since the Cayley-Dickson representation of G, k ( ) , k ( ) , and N ( ) ( ) are, respectively, where Thus, (24) can be rewritten as Then, G can be derived by solving the following constrained optimization problem: is the covariance matrix and x ( ) = k ( ) + N ( ) is the , where 1 is given by (31). The solution of (37) is given by [30] If the desired signal and interference are uncorrelated with the additive noise, G can be written in the simple form (the proof is in Appendix C) where The Scientific World Journal where is given by (20). Moreover, the quaternion-valued optimal weight vector G may be given by where J 1 = [I 2 ×2 , 0 2 ×2 ] and J 2 = [0 2 ×2 , I 2 ×2 ] are two selection matrices. By using the optimal weight vector G, the complex output of second-stage beamformer can be given by Thus, the complex output of QSWL GSC may be rewritten as From the above equation, we see that the interference component is completely canceled in the output GSC ( ).

The Performance Analysis.
Since the QSWL GSC can totally remove the interference, its output signal-tointerference ratio (SIR) tends to infinite. Thus, we focus our attention on the output signal-to-noise ratio (SNR) and array's gain. (13) and (34), we have Then, p can be written as When the combined QPMC and MVDR are adopted in the second-stage beamformer, p can be written in the simple form (the proof is in Appendix D) where is given by (31). From (44), the expression of output SNR and array's gain may be written as where denotes the input signal-to-noise ratio (SNR) and is given by (20). When the LCMV is adopted in the second-stage beamformer, p can be written in the simple form (the proof is in Appendix E) where ] is given by (41). Then, the expression of output SNR and array's gain may be written as From (49) and (52), we can see that the output SNR and array's gain depend on not only separation between the DOA's of the desired signal and interference (i.e., |q q |), but also difference between the polarization of the desired signal and interference (i.e., |[ ⊲ ] 1 |). The dependency of them on |q q | and |[ ⊲ ] 1 | is shown in the following consequences.
(1) When |q q | = 0, the separation between the DOAs of the desired signal and interference reaches to the maximum. In this case, = = 2 | | 2 . Further, |q q | increases with the decrease of the DOA's separation. Thus, the array's gain of and will reduce if | | 2 is a constant. When q = q , |q q | = 2 . This implies that there is no separation between the DOAs of the desired signal and interference. In this case, the array's gain is given by The Scientific World Journal Further, = if = and = . Thus, the array's gain = = 0 because = ∞. This implies that the QSWL GSC fails.
In addition, the output SNR and array's gain depend also on the input INR , the array's element number 2 , the interference response's power | | 2 , and the desired signal response's power | | 2 .

Performance for One Type of Interference.
In the presence of a single interference, we illustrate the performance of the proposed QSWL GSC by employing two experiments. We assume that = 6, = = 60 ∘ , = = 30 ∘ , = = 30 ∘ , and = 1 ∘ . In the first experiment, we investigate the effect of the angular separation Δ between the desired signal and the interference, where Δ = − . Figure 3 displays the output SINR as a function of Δ , where = Δ , = 0 ∘ . From Figure 3, it is seen that the output SINR of the proposed QSWL GSC is larger than that of the CLCMV in small angular separation (i.e., Δ < 40 ∘ ). Compared with the QSWL-QPMC-MVDR, the QSWL-LCMV has a little increase of the output SINR. This implies that the QSWL GSC has the better performance in small angular separation.
In the second experiment, we assume that the covariance matrix R , instead of R , is available. Figure 4 displays the power patterns for three values of |Δ |: 60 ∘ , 20 ∘ , and 10 ∘ , where = |Δ |, = 0 ∘ . From Figure 4, it is seen that the power is almost zero towards the interference's DOA (located at 0 ∘ ) in all cases. When |Δ | decreases, the main lobe of the QSWL GSC points almost to the source location, but the main lobe of the CLCMV is away from the source location. This implies that the QSWL GSC outperforms obviously CLCMV as the desired signal moves towards the interference. In addition, the side lobes are amplified with a decrease of |Δ |. These side lobes lead the beamformer to capture the white noise which spans the whole space, so that the performance of beamformer degrades.

Performance for Two Types of Interference.
In the presence of several types of interference, the performance analysis of the proposed QSWL GSC is not so easy in theory. Thus, we employ two experiments to illustrate the performance of the proposed QSWL GSC. We assume that there are two types of interference with identical input INR, The covariance matrix R is used. In the first experiment, we investigate the effect of DOA mismatch, where is used to denote the real presumed DOA of the desired signal. Figures  5 and 6 display, respectively, the output SINR as a function of at = 3 and = 6, where = 0 ∘ , 1 = −20 ∘ , and 2 = 20 ∘ . From Figures 5 and 6, it is seen that the QSWL-LCMV beamformer outperforms obviously both the QSWL-QPMC-MVDR and the CLCMV. In the case of no DOA mismatch and = 6, the output SINR of the QSWL-LCMV is 25 dB more than that of the CLCMV, but the output SINR of the CLCMV is larger than that of the QSWL-QPMC-MVDR. When the array's element number reduces, the performance of the CLCMV degrades. At = 3, the output SINR of the QSWL-QPMC-MVDR are 12 dB more than that of the CLCMV. The cause of this behavior is that the QSWL GSC has larger degrees of freedom than the CLCMV. Thus, the QSWL GSC has the better performance when many constraints are simultaneously imposed and/or the array's element number is reduced. In addition, when the DOA mismatch is little, such as ≤ 10 ∘ , the output SINR of three beamformers do not obviously degrade.
In the second experiment, we investigate the effect of snapshots , where the covariance matrix R is replaced by the sample covariance matrixR . Figure 7 displays the output SINR as a function of at = 6, where = 0 ∘ , 1 = −30 ∘ , and 2 = 30 ∘ . From Figure 7, it is seen that the QSWL GSC has slower convergence than the CLCMV. But the output SINR of the QSWL-LCMV are obviously more than that of the CLCMV.

Conclusion
In this paper, we propose a quaternion semiwidely linear beamformer and its useful implementation based on a quaternion model of linear symmetric array with 2 twocomponent EM vector sensors. Since the QSWL GSC consists of two-stage beamformers, it has more information than the complex "long vector" beamformer. The increase of information results in the improvement of the beamformer's performance. By designing the weight vectors of two-stage beamformers, the interference is completely canceled in the output of QSWL GSC and the desired signal is not distorted. Simulation results reveal that the proposed QSWL GSC has the better performance in small angular separation and the robustness to DOA mismatch.

Appendices
A. Derivation for (19) If the interference and additive noise are uncorrelated, we have where 2 and 2 are the covariance of interference and additive noise, respectively. By making use of the matrix inversion lemma (see Appendix in [30]), we find (18), we obtain .

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The Scientific World Journal is the determinant of 2 × 2 matrix C R −1 C. Plugging (C.1) into (39) and after some manipulation, we obtain By making use of the matrix inversion lemma, we find Using (19) and (26), we can easily obtain According to the definition in [11], x is -proper if it can be represented by means of two jointly proper complex vectors in the plane spanned by {1, }.
3. According to the definition in [11], the augmented covariance matrix of a -proper quaternion vector x can be written as where x = [x , x ( ) , x ( ) , x ( ) ] is the augmented quaternion vector;x = [x , x ( ) ] is the semiaugmented quaternion vector; and Rx ,x = {xx ⊲ } is the semiaugmented covariance matrix of quaternion vector x. In comparison with the semiaugmented covariance matrix Rx ,x , the augmented covariance matrix R x,x has no more extra information. In other words, the full widely linear processing is equivalent to the semiwidely linear processing in handling the -proper quaternion vector. We should not expect that the performance is improved by replacing semiwidely linear processing with full widely linear processing.