This is a repository copy of Adaptive Beamforming for Vector-Sensor Arrays Based on Reweighted Zero-Attracting Quaternion-Valued LMS Algorithm. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/93756/ Version: Accepted Version Article: Jiang, M.D., Liu, W. and Li, Y. (Accepted: 2015) Adaptive Beamforming for Vector-Sensor Arrays Based on Reweighted Zero-Attracting Quaternion-Valued LMS Algorithm. IEEE Transactions on Circuits and Systems II: Express Briefs. ISSN 1558-3791 https://doi.org/10.1109/TCSII.2015.2482464 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Adaptive Beamforming for Vector-Sensor Arrays Based on Reweighted Zero-Attracting Quaternion-Valued LMS Algorithm Mengdi Jiang, Wei Liu and Yi Li Abstract—In this work, reference signal based adaptive beam- quaternion algebra is non-commutative. Very recently, prop- forming for vector sensor arrays consisting of crossed dipoles is erties and applications of a restricted HR1 gradient operator studied. In particular, we focus on how to reduce the number for quaternion-valued signal processing were provided in [19]. of sensors involved in the adaptation so that reduced system complexity and energy consumption can be achieved while an Based on these recent advances in quaternion-valued signal acceptable performance can still be maintained, which is espe- processing, we here derive a reweighted zero attracting (RZA) cially useful for large array systems. As a solution, a reweighted quaternion-valued least mean square (QLMS) algorithm by zero attracting quaternion-valued least mean square algorithm is introducing a RZA term to the cost function of the QLMS proposed. Simulation results show that the algorithm can work algorithm. Similar to the idea of the RZA least mean square effectively for beamforming while enforcing a sparse solution for the weight vector where the corresponding sensors with zero- (RZA-LMS) algorithm proposed in [20], the RZA term aims to valued coefficients can be removed from the system. have a closer approximation to the l0 norm so that the number of non-zero valued coefficients can be reduced more effec- Index Terms—vector sensor array, quaternion, adaptive beam- forming, LMS, zero attracting. tively in the adaptive beamforming process. This algorithm can be considered as an extension of our recently proposed zero-attracting QLMS (ZA-QLMS) algorithm [21], where the I. INTRODUCTION l1 norm penalty term was used in the update equation of the weight vector. We will show in our simulations that the RZA- Adaptive beamforming has a range of applications and LMS algorithm has a much better performance in terms of has been studied extensively in the past for traditional array both steady state error and the number of sensors employed systems [1], [2], [3], [4]. With the introduction of vector after convergence. sensor arrays, such as those consisting of crossed-dipoles and A review of adaptive beamforming based on vector sensor tripoles [5], [6], [7], adaptive beamforming for such an array arrays is provided in Sec. II, and the proposed RZA-QLMS system has attracted more and more attention recently [6], [8], algorithm is derived in Sec. III. Simulations are presented in [9], [10]. Sec. IV, and conclusions drawn in Sec. V. In this work, we consider the crossed-dipole array and study the problem of how to reduce the number of sensors II. ADAPTIVE BEAMFORMING BASED ON VECTOR involved in the beamforming process so that reduced system SENSOR ARRAYS complexity and energy consumption can be achieved while A. Quaternionic Array Signal Model an acceptable performance can still be maintained, which is especially useful for large array systems. In particular, we will z use the quaternion-valued steering vector model for crossed- dipole arrays [8], [9], [10], [11], [12], [13], [14], [15], [16], and propose a novel quaternion-valued adaptive algorithm for θ reference signal based beamforming. ... In the past, several quaternion-valued adaptive filtering algo- d y φ rithms have been derived in [9], [16], [17], [18]. Notwithstand- ing the advantages of the quaternionic algorithms, extra cares x have to be taken in their developments, in particular when the derivatives of quaternion-valued functions are involved, since Fig. 1. A ULA with crossed-dipoles. This work is partially funded by National Grid UK. A general structure for a uniform linear array (ULA) with M. Jiang and W. Liu are with the Department of Electronic and Elec- trical Engineering, University of Sheffield, Sheffield, S1 3JD, UK (email: M crossed-dipole pairs is shown in Fig. 1, where these pairs {mjiang3,w.liu}@sheffield.ac.uk). are located along the y-axis with an adjacent distance d, and at Y. Li is with the School of Mathematics and Statistics, University of each location the two crossed components are parallel to the Sheffield, Sheffield, S3 7RH, UK (email:yili@sheffield.ac.uk). Copyright (c) 2015 IEEE. Personal use of this material is permitted. x-axis and y-axis, respectively. For a far-field incident signal However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected]. 1Here H (Hamilton) denotes the quaternion domain and R the real domain. 2 e[n] with a direction of arrival (DOA) defined by the angles θ and d[n] + φ, its spatial steering vector is given by − y [n] −j2πd sin θ sin φ/λ x [n] w [n] Sc(θ,φ) = [1,e , 1 1 − − . ,e j2π(M 1)d sin θ sin φ/λ]T (1) . ··· . T [ ] where λ is the wavelength of the incident signal and xM n wM [n] denotes the transpose operation. For a crossed dipole{·} the spatial-polarization coherent vector is given by [22], [23] Fig. 2. Reference signal based adaptive beamforming. [ cos γ, cos θ sin γejη] for φ = π S 2 (2) p(θ,φ,γ,η)= − jη −π {[cos γ, cos θ sin γe ] for φ = 2 − B. Reference Signal Based Adaptive Beamforming where γ is the auxiliary polarization angle with γ [0, π/2], and η [ π, π] is the polarization phase difference.∈ The aim of beamforming is to receive the desired signal The∈ array− structure can be divided into two sub-arrays: one while suppressing interferences at the beamformer output. parallel to the x-axis and one to the y-axis. The complex- When a reference signal d[n] is available, adaptive beam- valued steering vector of the x-axis sub-array is given by forming can be implemented by the standard adaptive filtering structure, as shown in Fig. 2, where xm[n], m = 1, ,M π ··· cos γSc(θ,φ) for φ = 2 are the received quaternion-valued input signals through the M Sx(θ,φ,γ,η)= − −π (3) {cos γSc(θ,φ) for φ = 2 pairs of crossed-dipoles, and wm[n]= am +bmi+cmj +dmk, and for the y-axis it is expressed as m = 1, ,M are the corresponding quaternion-valued weight coefficients··· with a, b, c and d being real-valued. y[n] jη π cos θ sin γe Sc(θ,φ) φ = is the beamformer output and e[n] is the error signal S 2 (4) y(θ,φ,γ,η)= jη −π cos θ sin γe S (θ,φ) φ = T T {− c 2 y[n]= w [n]x[n], e[n]= d[n] w [n]x[n] , (11) Before we present the quaternion-valued steering vector − model, we first very briefly review some basics about quater- where nion. A quaternion can be described as T q w[n] = [w1[n],w2[n], ,wM [n]] ··· T q = q1 +(q2i + q3j + q4k), (5) x[n] = [x [n],x [n], ,x [n]] . (12) 1 2 ··· M ∗ where q1, q2, q3, and q4 are real-valued [24], [25]. In this The conjugate form of the error signal is , given by ∗ e [n] paper, we consider the conjugate operator of q as q = q1 ∗ ∗ ∗ − e [n]= d [n] xH [n]w [n], (13) q2i q3j q4k. The three imaginary units i, j, and k satisfy − − − ∗ ij = k, jk = i, ki = j, where H is the combination of both T and opera- {·} {·} {·} 2 2 2 tions for a quaternion. Then w can be updated by minimizing ijk = i = j = k = 1; (6) ∗ − the instantaneous square error J0[n]= e[n]e [n]. where the exchange of any two elements in their order gives For a general quaternion-valued function f(w), the differ- a different result. For example, we have ji = ij rather than entiation with respect to the vector w and w∗ is ji = ij. For a general quaternion-valued function− f(q), the df(q) ∂f ∂f ∂f ∂f derivative with respect to q can be expressed as [19], i j k ∂a1 − ∂b1 − ∂c1 − ∂d1 dq ∂f 1 . [21], [26] = . (14) ∂w 4 . df(q) 1 ∂f(q) ∂f(q) ∂f(q) ∂f(q) ∂f ∂f ∂f ∂f (7) = ( i j k) , i j k dq 4 ∂q1 − ∂q2 − ∂q3 − ∂q4 ∂aM − ∂bM − ∂cM − ∂dM while the derivative of f(q) with respect to q∗ is given by ∂f ∂f ∂f ∂f df(q) 1 ∂f(q) ∂f(q) ∂f(q) ∂f(q) + i + j + k ∗ = ( + i + j + k) .