This paper reviews and compares three different linear algebraic signal subspace techniques for broadband direction of arrival estimation --- (i) the coherent signal subspace approach, (ii) eigenanalysis of the parameterised spatial correlation matrix, and (iii) a polynomial version of the multiple signal classification algorithm. Simulation results comparing the accuracy of these methods are presented

Alrmah, Mohamed Abubaker

Hussin, Mohammed Nuri

Lambotharan, Sangarapillai

Weiss, Stephan

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryAlrmah, Mohamed Abubaker and Hussin, Mohammed Nuri and Weiss, Stephan and Lambotharan,Sangarapillai (2012) Comparative study for broadband direction of arrival estimation techniques. In:9th IMA International Conference on Mathematics in Signal Processing, 2012-12-17 - 2012-12-20,Birmingham.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/COMPARISON OF BROADBAND DIRECTION OF ARRIVAL ESTIMATION ALGORITHMSMohamed A. Alrmah1, Mohamed Nuri Hussin1, Stephan Weiss1 and Sangarapillai Lambotharan21Department of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, Scotland, UK2 Advanced Signal Processing Group, Dept. of EEE, Loughborough University, Loughborough, UK{mohamed.alrmah,mohamed.hussin,stephan.weiss}@eee.strath.ac.uk; s.lambotharan@lboro.ac.ukABSTRACTThis paper reviews and compares three different linear al-gebraic signal subspace techniques for broadband directionof arrival estimation — (i) the coherent signal subspace ap-proach, (ii) eigenanalysis of the parameterised spatial corre-lation matrix, and (iii) a polynomial version of the multiplesignal classification algorithm. Simulation results comparingthe accuracy of these methods are presented.1. INTRODUCTIONFor broadband direction of arrival (DoA) estimation, power-ful narrowband methods such as the multiple signal classifica-tion (MUSIC) algorithm [4], are not directly applicable, andapproaches e.g. based on performing MUSIC in independentfrequency bins are likely to result in poor performance, par-ticularly if signal frequencies do not coincide with frequencybins [1].Amongst dedicated broadband DoA estimation algo-rithms, the coherent signal subspace method (CSSM) [6]combines covariance matrices at different frequency binscoherently by means of focussing matrices whose determi-nation has most recently been address by an auto-focussingapproach in [7]. A parameterised spatial covariance (PSC)approach [2, 5] scans for possible DoAs using what will laterin this paper be termed broadband steering vectors. In [1], wehave exploited a polynomial matrix decomposition in [3] togeneralise MUSIC to the case of spatio-temporal polynomialcovariance matrices.In this paper, we want to analyse the above three ap-proaches and compare them for a number of example scenar-ios. To accomplish this, Sec. 2 introduces the data model,with narrow- and broadband approaches to DoA approachesoutlined in Secs. 3 and 4. A comparison between CCSM,PSC and polynomial MUSIC (P-MUSIC) is then performedin Sec. 5.Notation. Matrix and vector quantities are represented byupper- and lowercase bold variables, e.g. A and a. The Her-mitian transpose of A is denoted as AH. Polynomial vectorsand matrices are written as a(z) and A(z), with the para-hermitian A˜(z) = AH(z−1). A transform pair a[n] andA(z) =∑∞n=−∞ a[n]z−n is abbreviated as a[n] ◦—• A(z).2. BROADBAND ARRAY DATAMODELThis section aims to describe the model behind multichanneldata collected in a vector x[n] ∈ CM by an M -element array.We assume that K far-field sources illuminate the array andcontribute to x[n] in addition to isotropic white noise v[n],x[n] =K∑k=1sk[n] +v[n] =K∑k=1ak[n] ∗ sk[n] + v[n] , (1)where sk[n] is the kth source signal, ak[n] the correspond-ing broadband steering vector, and ∗ the convolution opera-tor, which thereby forms the contribution of the kth source tothe array. The model including the steering vector in (5) onlyconsiders the angle of arrival, but neglects any attenuation inthe medium.For an arbirary array configuration, whereby rm describesthe coordinates of themth array element, the broadband steer-ing vector consists of delaysak[n] = [δ[n− τk,0] . . . δ[n− τk,M−1]]T, (2)with the normalised delayτk,m =tHk rmcTs, (3)whereby tk is the normal vector to the planar wavefront em-anating from the kth source, c the propagation speed in themedium, and Ts the sampling period.For sk[n] in (1), describing the contribution from the kthsource to x[n], the first sensor signal can be taken as refer-ence, and the relative delays of the remaining sensor signalscan be characterised assk[n] =sk[n]sk[n−∆τk,1]...sk[n−∆τk,M−1] , (4)with ∆τk,m = τk,m−τk,0. For a narrowband source with nor-malised angular frequency Ω, with a reference signal sk[n] =ejΩn, the time delays ∆τk,m collapse to simple phase shiftssk[n] =1e−jΩ∆τk,1...e−jΩ∆τk,M−1 ejΩ = aΩ,ϑkejΩ , (5)where aΩ,ϑk is termed the narrowband steering vector. Forfurther detail, the reader is referred to [1].As a specific case of (1), for a narrowband scenario withL narrowband sources sl[n] characterised by pairs {Ωl, ϑl}the array vector is given byx[n] =L∑l=1aΩl,ϑlsl[n] + v[n] , (6)with independent and identically distributed white noise v[n],such that E{v[n]vH[n− τ ]}= δ[τ ]σ2vI.3. NARROWBANDMUSIC3.1. Narrowband Covariance MatrixFor narrowband signals with frequency Ω, only correlationsfor lag zero need to be considered in the covariance matrixR = E{x[n]xH[n]}∈ CM×M , where E{·} is the expecta-tion operator. This covariance matrix entirely describes thedata as modelled in the narrowband scenario (6), since inthe case of L independent source signals with power σ2l , l ∈(1, L),R =L∑l=1σ2l aΩ,ϑlaHΩ,ϑl+ σ2vI . (7)The maximum rank of R, rank{R} = M is achieved in thecase of linear independence of all steering vectors.For data acquired over a data window of N samples, thedata matrixXn = [x[n−N + 1] . . .x[n− 1] x[n]] (8)can be utilised to estimate the covariance matrix as Rˆn =1NXnXHn . Below, we assume an appropriate estimation pro-cedure and for convenient continue to use R for the analysis.3.2. Narrowband MUSIC AlgorithmDirect eigenanalysis of the covariance matrixR can only leadto the correct angles of arrivals for sources, if all steering vec-tors in (7) are orthogonal. Otherwise, the eigenvalue decom-position (EVD)R = [QsQn][Λs 00 Λn] [QHsQHn](9)is likely to extract the steering vector of only the strongestsource correctly, but otherwise contain orthonormalised basisvectors of the signal subspace in Qs.The idea of the MUSIC algorithm is to scan the noise-only subspace Qn, which is spanned by eigenvectors corre-sponding to eigenvalues close to the noise floor, Λn ≈ σ2vI.The steering vectors of sources that contribute to R will de-fine the signal-plus-noise subspace Qs and therefore lie inthe nullspace of its complement Qn. Therefore, the vectorQHnaΩ,ϑ has to be close to the origin for aΩ,ϑ to be a steer-ing vector of a contributing source. Therefore, in the MUSICalgorithm the inverse of the squared Euclidean norm of thisvector, as proposed by [4].PMU(ϑ) =1aHΩ,ϑQnQHnaΩ,ϑ, (10)is calculated as the MUSIC spectrum PMU(ϑ).4. BROADBAND DOA ESTIMATION4.1. Coherent Signal Subspace MethodThe CSSM approach [6] calculates covariance matrices in anumber of frequency bins, which are then combined such thattheir signal subspaces allign into one single correlation matrixto which narrowband high resolution DoA techniques such asMUSIC can be applied. The coherence across different fre-quency bins is created by a frequency-dependent and unitaryfocussing matrix T(ejΩ), such thatRcoh =N−1∑i=nαnT(ejΩn )R(ejΩn)TH(ejΩ) , (11)where αn a weighting for maximum ratio combination ofits coherently rotated contributions. In [6] and subsequentderivative works, the focussing matrix T(ejΩ) is estimatedbased on a set of steering vectors. A poor estimate of the an-gle of arrival may also lead to poor results of this approach.However, a recent auto-focussing method in [7] allows com-putation based on the EVDs of R(ejΩn), n ∈ (0, N − 1), indifferent frequency bins.4.2. Parameterised Spatial Correlation MatrixThe idea of the broadband DoA estimation method in [2, 5]is based on testing the zero-lag coherence of a spatial corre-lation matrix calculated from appropriately pre-steered arraydata. Knowing the array configuration, a broadband steeringvector similar to (2) can be defined for a specific DoA. As-suming a linear array which only resolves a single angle ϑ,the covariance matrix of the pre-steered data is given byRϑ = E{yϑ[n]yHϑ [n]} (12)yϑ[n] =x[n− τ0(ϑ)]...x[n− τM−1(ϑ)] = Γϑ[n] ∗ x[n] (13)with the delay τm(ϑ) calculated akin to (3) and the diagonalpre-steering systemΓϑ[n] = diag{δ[n− τ0(ϑ)] . . . δ[n− τM−1(ϑ)]} . (14)The proposed method then evaluates the maximum eigen-value of Rϑ in (12) for a range of angles ϑ, with the bestmatch indicated by ϑopt = argmaxϑ{maxi λi(Rϑ)}, whereλi(Rϑ) indicates the ith eigenvalue of Rϑ.4.3. Space-Time Covariance Matrix and PolynomialEigenvalue DecompositionTo generalise (10) to the broadband case, we first define apolynomial space-time covariance matrix. This matrix can bedecomposed by McWhirter’s polynomial EVD [3], followedby an appropriate selection of a broadband steering vector toprobe its noise-only subspace.Different from the narrowband case, in a broadband sc-neario time signal wavefronts travelling across the array atfinite speed must be characterised by time delays rather thanjust phase shifts. This motivates the definition of a polyno-mial space-time covariance matrix R(z) •—◦ R[τ ],R[τ ] = E{x[n]xH[n− τ ]},which includes a time delay in form of the lag value τ . Thispower spectral matrix can be decomposed by an iterative al-gorithm [3] to yield a polynomial EVDR(z) = Q(z)Λ(z)Q˜(z) =M−1∑m=0λm(z)qm(z)q˜m(z)with paraunitary Q(z), i.e. Q(z)Q˜(z) = I. The diagonalmatrix Λ(z) contains the polynomial eigenvalues λm(z).Thresholding the latter reveals the number of independentbroadband sources contributing to R(z), and permits a dis-tinction between signal-plus-noise and noise only subspaces,R(z) = [Qs(z)Qn(z)][Λs(z) 00 Λn(z)] [QHs (z)QHn (z)](15)similar to the narrowband EVD in (9). To probe the nullspaceof Q˜n(z),Q˜n(z) =q˜L(z)...q˜M−1(z) (16)a broadband steering vector is required instead of the narrow-band one in (5).4.3.1. Broadband Steering VectorTo accurately reflect the time delays required to describe (4),a polynomial vector containing fractional delay transfer func-tions is proposed here. One possibility to implement thesefractional delays is by means of an appropriately sampled sincfunction, such thatal[n] = sinc(nTs −∆τl) . (17)WithAl(z) •—◦ al[n], a broadband steering vector can be de-fined asaϑ(z) =A0(z)...AM−1(z) . (18)The parameter ϑ on the l.h.s. of (18) indicates the dependencyof ∆τl on the angle of arrival. This vector is equivalent to themain diagonal of the parameterised spatial correlation matrixapproach in (14).4.3.2. Polynomial MUSIC AlgorithmBased on the concept of the narrowband MUSIC algorithm,the generalised quantityΓϑ(z) = a˜ϑ(z)Qn(z)Q˜n(z)aϑ(z)is no longer a norm measuring the vicinity of aϑ(z) to thenullspace of Q˜n(z), but a power spectral density. This hasmotivated two versions of the a polynomial MUSIC (P-MUSIC) algorithm [1] outlined below.Spatial P-MUSIC. The energy contained in the signal vectorQ˜n(z)aϑ(z) is related to the zero lag term γϑ[0] of the auto-correlation-type sequence γϑ[τ ] ◦—• Γϑ(z). This measure isonly dependent on the angle of arrival ϑ, and collects all en-ergy across the spectrum. Instead of searching for the steeringvectors providing minimum energy, the reciprocalPSP−MU (ϑ) =1γϑ[0]. (19)is maximised by the angle of arrival ϑ of signal sources.Spatio-Spectral P-MUSIC. With (4.3.2) describing a powerspectral density, spectral clues can be exploited in addition tothe spatial information extracted by (19). Therefore in addi-tion to spatial localisation of sources,PSSP−MU (ϑ,Ω) =(∞∑τ=−∞γϑ[τ ]e−jΩτ)−1(20)can determine over which frequency range sources in thedirection defined by the steering vector aϑ(z) are active.SSP-MUSIC was introduced in [1], but will be omitted fromthe comparison below, since the benchmark method onlyretrieves DoA information.5. COMPARISONThe broadband steering vector for a linear uniform array sen-sors separated by distances d = cfstakes on simple forms−90 −80 −70 −60 −50 −40 −30 −20 −10 000.10.20.30.40.50.60.70.80.91theta / [o]norm. spectrum  paramet. cov. mat.coherent signal subspaceP−MUSIC (est.)P−MUSIC (ideal)Fig. 1. Direction of arrival estimation for a single source lo-cated at end fire (ϑ = −90◦) using different approaches.−80 −60 −40 −20 0 20 40 60 80−50−45−40−35−30−25−20−15−10−505theta / [o]spectrum /[dB]  paramet. cov. mat.P−MUSIC (estimated)coherent signal subspaceFig. 2. Direction of arrival estimation for a scenario withtwo independent sources of equal strength located at broad-side and end fire positions.for the broadside and end-fire directions, where for M = 4sensorsaϑ=0◦(z) = [1 1 1 1]T (21)aϑ=−90◦(z) = [1 z−1 z−2 z−3]T (22)results, with the covariance matrix R(z) = aϑ(z)a˜ϑ(z) foruncorrelated unit-variance sources.Example 1. For a single source at end-fire position, the sim-ulation results normalised to a spectrum peak value of oneare shown in Fig. 1, whereby the PCV approach fares worst.Close to end-fire position, the CSS provide slightly higher res-olution than P-MUSIC based on an estimate covariance ma-trix and iterated decomposition. Knowing the covariance ma-trix and its true noise subspace in the PEVD sense howevercan achieve a much higher selectivity, with values quickly de-scending towards -300dB away from end-fire.Example 2. Assuming two independent sources at broad-side and end-fire position, the space-time covariance matrixis given by R3(z) = R1(z) +R2(z). The Spatial P-MUSICalgorithm identifies two large polynomial eigenvalues, andfrom the noise-only subspace derives the result in Fig. 2. TheDoAs and relative strengths of the sources are correctly ex-tracted. CSS provide higher resolution towards broadside,while the end-fire source is not detected equally. The PSCapproach fails to identify more than one source, and is onlyable to provide a very poor estimate for the broadside source.As for example 1, the spectra in Fig. 2 are normalised suchthat the maximum is unity.The simulations in Example 1 suggest that the proposedP-MUSIC algorithm can outperform the other methods if anidea polynomial eigenvalue decomposition is available. Inpractical cases where this matrix is estimated and the de-composition iteratively approximated, the performance ofP-MUSIC is often inferior to the coherent signal subspaceapproach [6, 7], such as in the accuracy of detecting thebroadside source in Example 2. In contrast, the more recentmethod in [2, 5] appears weaker and is unable to resolve morethan one source.6. CONCLUSIONSThis paper has compared three linear algebraic broadband di-rection of arrival estimation techniques, wherebya recentlyproposed polynomial matrix decomposition approach to ex-tend the MUSIC algorithm to the broadband case has been re-viewed. Simulations indicate that the performance for the op-timum decomposition works very well and outperforms otheralgorithms. The performance degrades when based on esti-mated values and an iterative approximate decomposition andmay therefore be enhanced by the develpment of a more ded-icated decomposition algorithm.7. REFERENCES[1] M. Alrmah, S. Weiss, and S. Lambotharan. An extensionof the music algorithm to broadband scenarios using polyno-mial eigenvalue decomposition. In EUSIPCO, pp. 629–633,Aug. 2011.[2] J. Dmochowski, J. Benesty, and S. Affes. Direction of arrivalestimation using the parameterized spatial correlation matrix.IEEE Trans. Audio, Speech, Lang. Proc., 15(4):1327–1339,May 2007.[3] J. G. McWhirter, P. D. Baxter, T. Cooper, S. Redif, and J. Fos-ter. An EVD Algorithm for Para-Hermitian Polynomial Ma-trices. IEEE Trans. Sig. Proc., 55(5):2158–2169, May 2007.[4] R. O. Schmidt. Multiple emitter location and signal pa-rameter estimation. IEEE Trans. Ant. Prop., 34(3):276–280,Mar. 1986.[5] M. Souden, J. Benesty, and S. Affes. Broadband source local-ization from an eigenanalysis perspective. IEEE Trans. Audio,Speech, and Lang. Proc., 18(6):1575–1587, Aug. 2010.[6] H. Hung and M. Kaveh. Focussing matrices for coherentsignal-subspace processing. IEEE Trans. ASSP 36(8):1272–1281, Aug. 1988.[7] P. Pal and P. P. Vaidyanathan. A novel autofocusing approachfor estimating directions-of-arrival of wideband signals. In43rd Asilomar Conf. SSC, pp:1663–1667, Pacific Grove, CA,Nov. 2009.

A novel autofocusing approach for estimating directions-of-arrival of wideband signals.

An EVD Algorithm for Para-Hermitian Polynomial Matrices.

An extension of the music algorithm to broadband scenarios using polynomial eigenvalue decomposition.

Broadband source localization from an eigenanalysis perspective.

Direction of arrival estimation using the parameterized spatial correlation matrix.

Comparative study for broadband direction of arrival estimation techniques

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