CSS, the purpose of this paper is to highlight the similarities by expressing the auto-focussing approach [5] in terms of polynomial matrix decomposit...

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2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

Broadband Angle of Arrival Estimation Methods in a Polynomial Matrix Decomposition Framework Stephan Weiss∗, Mohamed Alrmah∗ , Sangarapillai Lambotharan†, John G. McWhirter‡ , and Mostafa Kaveh§ ∗

†

Department of Electronic & Electrical Engineering, University of Strathclyde,Glasgow,Scotland, UK School of Electronic, Electrical and Systems Engineering, Loughborough University, Loughborough, Leistershire, UK ‡ School of Engineering, University of Cardiff, Cardiff, Wales, UK § Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, USA

Abstract—A large family of broadband angle of arrival estimation algorithms are based on the coherent signal subspace (CSS) method, whereby focussing matrices appropriately align covariance matrices across narrowband frequency bins. In this paper, we analyse an auto-focussing approach in the framework of polynomial covariance matrix decompositions, leading to comparisons to two recently proposed polynomial multiple signal classification (MUSIC) algorithms. The analysis is complemented with numerical simulations.

I. I NTRODUCTION The coherent signal subspace (CSS) technique was proposed in [1] as an effective method for the estimation of angles of arrival of spectrally overlapping broadband sources using narrowband direction-finding algorithms such as MUSIC. A fundamental feature of CSS is that it coherently combines narrowband covariance matrices at different frequency bins covering the band occupied by the sources. In its simplest form, CSS pre-steers the array data by focussing matrices such that the sources of interest appear in the vicinity of the array’s broadside, where array response vectors for all temporal frequencies approximately coincide. The focussing matrices can be constructed from approximate knowledge of the signals of interest’s directions of arrival (DoA). The long evolution of CSS algorithms since this inception, see e.g. [2], [3], includes a recently proposed auto-focussing approach [5]. In this method, focussing matrices are directly calculated from the array’s space-time covariance matrix without the requirement for explicit knowledge of approximate DoAs. Different from CSS, where the wideband approach is cleverly bypassed in favour of narrowband processing, an EVD algorithm for polynomial space-time covariance matrices [8] has recently led to a broadband MUSIC algorithm [7]. Applicable directly to broadband array data, these polynomial MUSIC algorithms exploit the idea of signal subspaces created by the polynomial EVD [8]. While this approach seems distinct from CSS, the purpose of this paper is to highlight the similarities by expressing the auto-focussing approach [5] in terms of polynomial matrix decompositions. Below, we define a broadband steering vector, the polynomial space-time covariance matrix and polynomial EVD in Sec. II, and review the auto-focussing broadband approach in Sec. III. The formulation of MUSIC based on the auto-

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focussing method of [5] is then related to the two polynomial MUSIC algorithms in [7] in Sec. IV. Two illustrative simulations are included in Sec. V, followed by conclusions in Sec. VI. II. S YSTEM M ODEL Based on the signal model for a broadband array described in Sec. II-A, Sec. II-B defines a polynomial space-time covariance matrix. A. Broadband Steering Vector An M -element array of omnidirectional sensors located at positions rm , m = 1 . . . M collects broadband data in a vector x[n] ∈ CM . For the lth far-field source sl [n], the array experiences a planar wavefront with normal kl . We are only interested in the relative delay between signals at the sensors, such that the contributions to x[n] in the absence of attenuation due to propagation is ∞ XX x[n] = al [ν]sl [n − ν] + v[n] (1) l

ν=0

with the broadband steering vector ai [n] δ[n − τl,0 ] 1 .. al [n] = √ . M δ[n − τl,M−1 ]

(2)

kH r

l m , whereby Ts is the and the normalised delays τl,m = cT s sampling period and c the propagation speed in the medium, such that kl /c is the lth source’s slowness vector. The vector v[n] adds spatially and temporally uncorrelated noise with covariance E v[n]vH [n] = σv2 I to the model in (1). Below, aϑ,ϕ [n] refers to a broadband steering vector determined by k with azimuth ϕ and elevation ϑ.

B. Space-Time Covariance Matrix and Polynomial EVD Collecting an M -element broadband array data vector x[n] ∈ CM , its space-time covariance matrix is given by R[τ ] = E x[n]xH [n − τ ] , (3) which forms a transform pair with the cross power spectral density (CSD) matrix R(z) •—◦ R[τ ], X R(z) = R[τ ]z −τ . (4)

109

τ

2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

˜ The CSD matrix is parahermitian, i.e. R(z) = R(z) = H ∗ R (1/z ). Based on (1) and with al [n] ◦—• al (z), it can also be expressed as X R(z) = al (z)Sl (z)˜ al (z) + σv2 I , (5) l

with Sl (z)|z=ejΩ the power spectral density (PSD) of the lth source signal, sl [n]. A polynomial EVD [8] decouples the parahermitian R(z) by means of a paraunitary Q(z), ˜ Γ(z) = Q(z)R(z)Q(z) ,

(6)

such that Γ(z) = diag{Γ1 (z) Γ1 (z) . . . ΓM (z)} is diagonalised and spectrally majorised with PSDs Γi+1 (ejΩ ) ≥ Γi (ejΩ ) ∀Ω, i = 1 . . . (M − 1) , with Γi (ejΩ ) = Γi (z)|z=ejΩ . Below, we use this decomposition framework to express the CSS approach. III. C OHERENT C OVARIANCE AND AUTO -F OCUSSING M ATRICES

B. Auto-Focusssing Approach via CSD Matrix and PEVD With the modal matrix Q0 obtained at the reference frequency via EVD of (7), the focussing matrix can be formulated as a paraunitary matrix T(z)|z=ejΩ = Q0 QH (ejΩ ). Replacing the summation over frequency bins in (8) by the integration over the Fourier transform (i.e. K → ∞) leads to I n o 1 ˜ Rcoh ≈ T(z)R(z)T(z) dΩ (12) 2π z=ejΩ I n o 1 ˜ Q(z)R(z)Q(z) = Q0 dΩ QH 0 . (13) 2π z=ejΩ Since the paraunitary matrix Q(z) diagonalises R(z), the argument under the integral is the polynomial EVD of (6), resulting in a diagonal matrix of power spectral densities, H 1 jΩ Γ(e )dΩ = Γ[0], where Γ[0] is the evaluation of 2π Γ[τ ] ◦—• Γ(z) for zero lag. Therefore 2 σ1 H .. Rcoh ≈ Q0 Γ[0]QH (14) Q0 . 0 = Q0 2 σM

Based on a K-point DFT of the space-time covariance matrix, K−1 X R(ejΩk ) = R[τ ]e−jΩk τ (7)

represents the coherent covariance matrix in terms of the polynomial EVD of the CSD matrix. Given that the DFT in (7) is a sufficiently accurate representation of the Fourier transform formulation in (12), then (11) and (14) are equivalent with Λ = Γ[0]. Further, the PEVD of the CSD matrix provides a paraunitary Q(z) that leads to an ˜ auto-focussing matrix Q0 Q(z) that is continuous in frequency.

with Ωk = 2π K k, k = 0 . . . K − 1, the CSS method is based on a covariance matrix

IV. B ROADBAND A NGLE OF A RRIVAL E STIMATION

A. Coherent Signal Subspace Method

τ =0

Rcoh

K−1 1 X = T(ejΩk )R(ejΩk )TH (ejΩk ) , K

A. MUSIC Based on Auto-Focussing Approach (8)

k=0

obtained by coherently combining across frequency bins through unitary and frequency-dependent focussing matrices T(ejΩ ). Following the “auto-focussing” approach of [5], for a reference frequency Ω0 , an EVD of the appropriate frequency-bin covariance matrix R(ejΩ0 ) yields jΩ0 )Q0 Λo = Q H 0 R(e

.

(9)

Based on the coherent covariance matrix, the standard MUSIC algorithm can be applied by probing the noise-only subspace of Rcoh with a set of steering vectors at the reference frequency Ω0 . If the eigenvalues Rcoh reveal R linearly independent sources, then the last M − R columns of Q0 = Q0,s Q⊥ 0,s M×(M−R) contained in Q⊥ span the noise-only subspace 0,s ∈ C of the coherent covariance matrix. Scanning for azimuth and elevation angles, aϕ,ϑ (z) can be evaluated at the reference frequency Ω0 , leading to the MUSIC spectrum jΩ0 −2 SAF (ϕ, ϑ) = kQ⊥ )k2 0,s aϕ,ϑ (e 1 = ⊥,H ⊥ H jΩ 0 aϕ,ϑ (e )Q0,s Q0,s aϕ,ϑ (ejΩ0 )

Together with the modal matrix QH (ejΩk ) extracted for frequency bin k, k = 0 . . . (K − 1), the auto-focussing matrix is constructed as T(ejΩk ) = Q0 QH (ejΩk ) .

(10)

Therefore, the coherent covariance matrix in (8) can be diagonalised by Q0 to provide Λcoh = QH 0 Rcoh Q0 = diag{λ1 λ2 . . . λM }

,

with λm , m = 1 . . . M the eigenvalues of Rcoh in (8).

(11)

. (15)

B. Polynomial Spatio-Spectral MUSIC For the polynomial MUSIC algorithm, a spatial and a spatiospectral version have been suggested in [7]. Both versions require the identification of the polynomial noise-only subspace given R sources identified from Γ(z), Q(z) = Qs (z)Q⊥ , (16) s (z)

110

SAF (ϕ, ϑ) = SPSS (ϕ, ϑ, ejΩ )|Ω=Ω0

.

(18)

Therefore, the auto-focussing approach to coherent signal subspace MUSIC estimation is equivalent to evaluating the polynomial spatio-spectral MUSIC spectrum at the reference frequency Ω0 . To obtain the same spatio-spectral characterisation of the array data as provided by PSS-MUSIC with the auto-focussing approach, a sequence of different modal matrices Q0 at different reference frequencies Ω0 could be calculated, for all of which (15) is evaluated. C. Polynomial Spatial MUSIC The polynomial spatial (PS) MUSIC estimate [7] integrates the PSD in the denominator of (17), providing a power term I 1 ˜ ⊥ (z)Q⊥ (z)aϕ,ϑ (z) jΩ dΩ . (19) ˜ϕ,ϑ (z)Q a γ= s s z=e 2π The PS-MUSIC spectrum is given by the reciprocal of (19), 1 SPS (ϕ, ϑ) = . (20) γ If the integral in (19) is approximated by a sum over discrete frequency bins, i.e. γ≈

1 K

K−1 X

jΩk jΩk aH )Q⊥,H (eΩk )Q⊥ )aϕ,ϑ (ejΩk ), ϕ,ϑ (e s s (e

k=0

(a) 0

−50 1 0.5

Ω/π 0

Sdiff (ϑ, ejΩ )/[dB]

M×R where Q⊥ (z). The polynomial spatio-spectral (PSS) s ∈C MUSIC is based on inverting a PSD-type function, 1 jΩ . SPSS (ϕ, ϑ, ejΩ ) = ⊥ ⊥ ˜ ˜ϕ,ϑ (z)Qs (z)Qs (z)aϕ,ϑ (z) z=e a (17) Provided that the estimation of the number of linearly independent sources, R, is the same from (11) for auto-focussing (AF) and from (14) of the polynomial approach, then with Q0 being the evaluation of the paraunitary Q(z) at the reference frequency Ω0 , i.e. Q0 = Q(z)|z=ejΩ0 , it follows that

SPSS (ϑ, ejΩ )/[dB]

2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

−60

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0

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ϑ/◦

(b) −20 −40 −60 −80 −100 −120 1 0.5

Ω/π 0

−60

−40

40

ϑ/◦

Fig. 1. (a) PSS-MUSIC spectrum and (b) difference to AF-MUSIC for a single source at ϑ = 90◦ .

B. Idealistic Example with Exact PEVD As a simple toy problem, a single source emits an uncorrelated Gaussian signal. In an otherwise noise-free scenarion, this signal is received by an M = 4 element, spatially and temporally critically sampled array from endfire position, such that the broadband steering vector of the source is 1 a1 (z) = √ [1 z −1 M providing a space-time covariance 1 z1 z −1 1 R1 (z) = .. . z −M+1 . . .

. . . z −M+1 ]T

(22)

matrix z M−1 .. . .. .. . . ... 1

...

.

(23)

Because R1 (z) is rank one, a manifold of diagonalising decompositions exists, with one possibility Q(z) = diag 1 z −1 . . . z −M+1 TDFT , (24)

(21) then (21) is the summation over the denominator terms of (15) for all possible reference frequencies Ωk with Ωk = 2π K k, k = 0 . . . (K − 1). The paraunitary matrix Q(z) that feeds into (21) has been demonstrated in (14) to cohere the spatiotemporal covariance matrixin the auto-focussing sense.

where TDFT is a normalised, unitary M -point DFT matrix. For PSS-MUSIC in (17), the spectrum in Fig. 1(a) emerges, identifying the DoA of the end-fire source. In line with broadband arrays, at lower frequencies the fixed aperture degrades the spatial resolution, with no ability to discern sources at DC.

V. N UMERICAL S IMULATIONS A. Implementational Aspects It has been shown in [9] that the polynomial EVD in (6) fulfilling spectral majorisation can be approximated very closely by FIR paraunitary matrices even if an exact decomposition by FIR filter banks does not exist. Therefore, here we rely on the second order sequential best rotation (SBR2) algorithm [8], which iteratively approaches the decomposition in (6), and has been proven to converge, whereby the number of iterations will determine the accuracy with which diagonalisation and spectral majorisation are approximated. Broadband steering vectors are based on fractional delay filters constructed from truncated sinc functions, which can be substantially improved by applying a tapered window [10].

For auto-focussing, at a given reference frequency Ω0 , it can be shown that Rcoh,Ω0 = R(z)|z=ejΩ0 and Λcoh,Ω0 = jΩ0 diag{1, 0 · · · 0}. Using the nullspace Q⊥ ) derived from s (e the EVD of Rcoh,Ω0 , the MUSIC spectrum is evaluated for a range of K = 64 reference frequencies Ω0 . This leads to a spectrum very closely related to PSS-MUSIC, with the difference, Sdiff (ϑ, ejΩ ) = |SPSS (ϑ, ejΩ ) − SAF (ϑ, Ω)|, plotted in Fig.1(b). The error reaches a maximum of 10dB where the PSS-MUSIC spectrum is numerically most sensitive, i.e. towards the source at ϑ = 90◦ , and for DC, Ω = 0, which can be attributed to the inaccuracies in implementing broadband steering vector. Note the trivial broadband steering 1 vector towards broadside ϑ = 0◦ , a0◦ (z) = M [1 1 . . . ]T , for which the error in Fig. 1(b) is neglegible.

111

AF-MUSIC (Ω0 = π/2) AF-MUSIC (integrated) PS-MUSIC (SBR2) PS-MUSIC (ideal)

0.6 0.4 0.2 0 87

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(a) 40

1 0.8

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0

0.4 −80

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−20

ϑ/◦

Fig. 2. Comparison between AF-MUSIC (i) at Ω0 = π2 , (ii) integrated according to (21), and PS-MUSIC (iii) with estimated and (iv) ideal PEVD.

Due to the trivial space-time covariance matrix, SBR2 converges instantly to the exact PEVD, yielding exact results for PS-MUSIC in Fig. 2. AF-MUSIC is shown both for a single reference frequency, and integrated over a range of reference frequencies, with results somewhat degraded compared to PSMUSIC.

0.2 0

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Ω/π

0

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ϑ/◦

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SAF (ϑ, ejΩ )/[dB]

normalised spectrum

1 0.8

SPSS (ϑ, ejΩ )/[dB]

2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

(b) 40

1 0.8

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0

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0.2 0

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ϑ/◦ Fig. 3.

(a) PSS- and (b) AF-MUSIC spectra for 3 overlapping sources.

VI. C ONCLUSIONS Polynomial MUSIC algorithms have been compared to a recently proposed auto-focussing (AF) approach, which is claimed to be in the line of coherent signal subspace methods [5]. With the AF approach expressed in the framework of polynomial space-time covariance matrices and their polynomial eigenvalue decomposition, and under the assumption the DFT sufficiently well approximating the Fourier transform, the polynomial spatio-spectral MUSIC algorithm has been shown to equate to the AF approach when evaluated at the reference frequency, while the polynomial spatial MUSIC algorithm has been shown to relate to a summation of AF terms for a set of reference frequencies. Numerical simulations have indicated that the polynomial MUSIC methods perform similar to the AF-approach where the exact PEVD is known or easily determined. For more realistic scenarios, restricting AF to sensible fractional bandwidths will provide superior resolution over polynomial MUSIC; however, the latter does not rely on a-priori spectral information and can be calculated over the entire bandwidth with appealing results.

normalised spectrum / [dB]

0

C. Realistic Scenario For a more realistic scenario, we consider an M = 8 element array illuminated by a mixture of three mutually uncorrelated Gaussian sources of equal power, 3π ◦ • ϑ1 = −30 , active over range Ω ∈ [ 8 ; π], π ◦ • ϑ2 = 40 , active over range Ω ∈ [ 2 ; π], and 2π 7π ◦ • ϑ3 = 20 , active over range Ω ∈ [ 8 ; 8 ], mixed with uncorrelated Gaussian noise 30dB below the three directional signals. Fig. 3 shows the PSS- and CSS-MUSIC spectra, in the latter case evaluated for a set of reference frequencies over small fractional bandwidths. In spectral ranges where all sources are active, AF-MUSIC provides superior resolution, also indicated by a snapshot for Ω = 3π 4 in Fig. 4; outside the overlap region, the performance is degraded. PSMUSIC in Fig. 4 provides a lower resolution than autofocussing, but is calculated over the entire spectrum, hence not requiring any prior spectral knowledge.

sources AF−MUSIC PS−MUSIC

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Fig. 4.

PS- and AF-MUSIC (measured at Ω0 =

3 π) 4

VII. ACKNOWLEDGEMENT The work of S. Weiss, S. Lambotharan and J.G. McWhirter has been supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) and the Defence Science and Technoloy Laboratory (Dstl) under contract EP/K014307/1. R EFERENCES [1] H. Wang and M. Kaveh, “Coherent signal-subspace processing for the detection and estimation of angles of arrival of multiple wide-band sources,” IEEE Trans ASSP, 33(4):823–831, Aug. 1985. [2] H. Wang and M. Kaveh, S “On the performance of signal-subspace processing–part ii: Coherent wide-band systems,” IEEE Trans ASSP, 35(11):1583–1591, Nov. 1987. [3] H. Hung and M. Kaveh, “Focussing matrices for coherent signalsubspace processing,” IEEE Trans ASSP, 36(8):1272–1281, Aug. 1988. [4] M.A. Doron and A.J. Weiss, “On focusing matrices for wide-band array processing,” IEEE Trans SP, 40(6):1295–1302, June 1992. [5] P. Pal and P.P. Vaidyanathan, “A novel autofocusing approach for estimating directions-of-arrival of wideband signals,” in 43rd Asilomar Conf. Signals, Systems & Computers, Pacific Grove, CA, pp. 1663–1667, Nov. 2009. [6] M.A. Doron, E. Doron, and A.J. Weiss, “Coherent wide-band processing for arbitrary array geometry,” IEEE Trans SP, 41(1):414, Jan. 1993. [7] M. Alrmah, S. Weiss, and S. Lambotharan, “An extension of the music algorithm to broadband scenarios using polynomial eigenvalue decomposition,” in 19th European Signal Processing Conf., Barcelona, Spain, pp. 629–633, Aug. 2011. [8] J.G. McWhirter, P.D. Baxter, T. Cooper, S. Redif, and J. Foster, “An EVD Algorithm for Para-Hermitian Polynomial Matrices,” IEEE Trans SP, 55(5):2158–2169, May 2007. [9] S. Icart and P. Comon, “Some properties of Laurent polynomial matrices,” in 9th IMA Conf. Mathematics in Signal Processing, Birmingham, UK, Dec. 2012. [10] J. Selva, “An efficient structure for the design of variable fractional delay filters based on the windowing method,” IEEE Trans SP, 56(8):3770– 3775, Aug. 2008.

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