State of the art narrowband noise cancellation techniques utilise the generalised eigenvalue decomposition (GEVD) for multichannel Wiener filtering which can be applied to independent frequency bins in order to achieve broadband processing. Here we investigate the extension of the GEVD to broadband, polynomial matrices, akin to strategies that have already been developed by McWhirter et. al on the polynomial matrix eigenvalue decomposition (PEVD)

Corr, Jamie

Moonen, Marc

Pestana, Jennifer

Redif, Soydan

Weiss, Stephan

Name not available

Investigation of a polynomial matrix generalised EVD for multi-channel Wiener filtering

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryCorr, Jamie and Pestana, Jennifer and Weiss, Stephan and Redif, Soydan and Moonen, Marc (2016) Investigation of a polynomial matrix generalised EVD for multi-channel Wiener filtering. In: 50th Asilomar Conference on Signals, Systems and Computers. IEEE. (In Press) , This version is available at http://strathprints.strath.ac.uk/57611/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  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 private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukExtended Summary for Asilomar’16 1Investigation of a Polynomial Matrix Generalised EVDfor Multi-Channel Wiener FilteringJamie Corr1,† Jennifer Pestana2, Stephan Weiss1, Soydan Redif3 and Marc Moonen41 Dept. Electronic & Electrical Engineering, University of Strathclyde, Glasgow Scotland2 Dept. Mathematics & Statistics, University of Strathclyde, Glasgow Scotland3 Electrical & Electronic Engineering Dept. European University of Lefke, Cyprus4 Dept. Electrical Engineering, KU Leuven, Leuven, BelgiumAbstract. State of the art narrowband noise cancellation techniques utilise the generalisedeigenvalue decomposition (GEVD) for multichannel Wiener filtering which can be applied toindependent frequency bins in order to achieve broadband processing. Here we investigatethe extension of the GEVD to broadband, polynomial matrices, akin to strategies that havealready been developed by McWhirter et. al on the polynomial matrix eigenvalue decomposition(PEVD).Extended SummaryMotivation. Recently the scalar generalised eigenvalue decomposition (GEVD) has beenutilised to enable low rank approximation in multichannel Wiener filters (MWF) [3,8]. At thestage where the signal only covariance matrix is estimated, the estimation process artificiallyincreases the rank of the matrix. To avoid using matrices with higher rank than required,the covariance matrix gets replaced by a low rank approximation [3]. Methods using both theconventional eigenvalue decomposition (EVD) and GEVD have been explored in [8] with resultsshowing the GEVD based approach performs better as it effectively selects the modes with thehighest SNR.For a broadband array vector x[n], where explicit lags τ need to be taken into accout, thespace-time matrix is defined as R[τ ] = E{x[n]xH[n − τ ]}. Its z-transform R(z) •—◦ R[τ ]forms a parahermitian polynomial matrix [9], for which a generalisation from the EVD to apolynomial EVD (PEVD) exist [5]. In this contribution, we extend the PEVD to a polynomialversion of the GEVD (PGEVD) in order to directly address the broadband MWF problem.Wiener filters have previously been formulated using polynomial matrix techniques [6] butinitially could not be solved due to an absence of tools. We here completement the polynomialWiener filter solution in [12] by a PGEVD approach.Generalised Eigenvalue Decomposition. The GEVD solves the problem of R1v = dR2v,v 6= 0, where v is an eigenvector of the pencil R1 − dR2 [2]. In matrix form,R1V = R2VD , (1)the diagonal matrix D contains the generalised eigenvalues and V holds the correspondinggeneralised eigenvectors. When the generalised eigenvectors are applied to R1 and R2 we getVHR1V = D , (2)VHR2V = I , (3)†POC: Jamie Corr, Dept. of EEE, University of Strathclyde, Glasgow G1 1XW, Scotland, UK. Tel. 0044-(0)141-548-2546 Fax 0044-(0)141-552-2487; e-mail jamie.corr@strath.ac.uk.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) Grantnumber EP/K014307/1 and the MOD University Defence Research Collaboration in Signal Processing.Corr et al.: Investigation of Polynomial Matrix GEVD 2where I is the identity. From the alternative form VHR−12 R1V = D to (2) and (3) it is evidentthat R−12 R1 is no longer Hermitian and therefore V not guaranteed to be unitary, even if —as in the case of MWF [3,8] — R1 and R2 are Hermitian covariance matrices.Amongst a number of options to calculate the GEVD or joint diagonalisation of Hermitianmatrices R1 and R2 [2] is a Cholesky-based approach. In a first step, a Cholesky decompositionR2 = LLH, with L lower left triangular, is determined. In a second step, an EVD is performedon the intermediate Hermitian matrix C = L−1R1L−H = YDYH, such that D contains thegeneralised eigenvalues and the generalised eigenvectors are the columns of V = L−HY.Extension of GEVD to Polynomial Matrices. Extending the GEVD in (1) to polynomialparahermitian matrices Ri(z), i = {1, 2}, leads to a joint diagonalisation problem akin to (2)and (3)V˜ (z)R1(z)V (z) = D(z) , V˜ (z)R2(z)V (z) = I , (4)where D(z) contains the polynomial generalised eigenvectors. The operator {˜·} denotes theparahermitian transpose, which includes both a Hermitian transpose and time reversal; thiscan be seen as the polynomial equivalent of the Hermitian transpose in the scalar case [9]. Thefactorisations in (4) can be shown to exist if the PEVDs of Ri(z) exist [4], and if R2(z) isinvertible. Due to non-uniqueness of the PEVD [1], the generalised polynomial eigenvectors inV (z) can at the very least be arbitrarily delayed w.r.t. each other, leading to a variability intheir polynomial degree.For the computation of V (z) and D(z), a two-step Cholesky-style approach can be per-formed. The first step uses the PEVD R2(z) = Q˜2(z)D2(z)Q2(z) and the spectral fac-torisation [10] of D2(z) into its minimum and maximum phase components, D(+)2 (z) andD(−)2 (z) = D˜(+)2 (z). Therefore the factor L(z) = Q˜2(z)D(+)2 (z) is now not lower left trian-gular but easily inverted [11], such that L−1(z) =(D(+)2 (z))−1Q2(z). For the second step,the PEVD C(z) = L−1(z)R1(z)L˜−1(z) = Q(z)D(z)Q˜(z) is calculated. This leads to thepolynomial generalised eigenvalues on the diagonal of D(z) and the polynomial generalisedeigenvectors in V (z) = L˜−1(z)Q(z), noting a possible order reduction due to ambiguity [1].Results. As an example of a PGEVD, the source model described in [7] generates two para-hermitian matrices R1(z) and R2(z) as shown in Fig. 1. The sequential matrix diagonalisation(SMD) [7] PEVD algorithm is used to decompose R2(z) and to calculate the intermediatematrix C(z). The determined V (z) is then applied to both R1(z) and R2(z), leading to thejointly diagonalised systems in Fig. 2, closely approximating (4).Conclusions and Full Paper. This summary has introduced the ideas and motivation be-hind a polynomial matrix generalised eigenvalue decomposition and some initial results fromthe Cholesky type method based on the polynomial EVD.The final paper will provide details on the approach and numerical results for joint diagonal-isation over an ensemble of differently conditioned parahermitian matrices. These results willbe applied to broadband multichannel Wiener filtering, with a comparison to the narrowbandGEVD applied in independent frequency bins. We also aim to generalise advantages in theGEVD to the polynomial case if R1(z) has low rank [3], which can be expected to providesimplifications over the polynomial Wiener approach outlined but not solved in [6].Corr et al.: Investigation of Polynomial Matrix GEVD 3−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4lag τ|R1[τ]|(a)−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4−5 0 500.20.4lag τ|R2[τ]|(b)Fig. 1: Space-time covariance matrix (a) R1[τ ] ◦—• R1(z) and (b) R2[τ ] ◦—• R2(z), onlyshowing the absolute value of coefficients.−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246−10 0 100246lag τ|VH[−τ]∗R1[τ]∗V[τ]|(a)−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51−5 0 500.51lag τ|VH[−τ]∗R2[τ]∗V[τ]|(b)Fig. 2: Results of joint diagonalisation by generalised modal matrix V (z) •—◦ V[τ ] for (a)R1[τ ] ◦—• R1(z) and (b) R2[τ ] ◦—• R2(z), only showing the absolute value of coefficients.References:[1] J. Corr, K. Thompson, S. Weiss, I. Proudler, and J. McWhirter. Row-shift corrected trun-cation of paraunitary matrices for PEVD algorithms. In 23rd European Signal ProcessingConference, pages 849–853, Nice, France, August/September 2015.[2] G. H. Golub and C. F. Van Loan. Matrix Computations. John Hopkins University Press,Baltimore, Maryland, 3rd edition, 1996.[3] A. Hassani, A. Bertrand, and M. Moonen. Gevd-based low-rank approximation for dis-tributed adaptive node-specific signal estimation in wireless sensor networks. IEEE Trans-actions on Signal Processing, 64(10):2557–2572, May 2016.[4] S. Icart and P. Comon. Some properties of Laurent polynomial matrices. In 9th IMAConference on Mathematics in Signal Processing, Birmingham, UK, December 2012.Corr et al.: Investigation of Polynomial Matrix GEVD 4[5] J. G. McWhirter, P. D. Baxter, T. Cooper, S. Redif, and J. Foster. An EVD Algo-rithm for Para-Hermitian Polynomial Matrices. IEEE Transactions on Signal Processing,55(5):2158–2169, May 2007.[6] A. Mertins. MMSE Design of Redundant FIR Precoders for Arbitrary Channel Lengths.IEEE Transactions on Signal Processing, 51(9):2402–2409, September 2003.[7] S. Redif, S. Weiss, and J. McWhirter. Sequential matrix diagonalization algorithms forpolynomial EVD of parahermitian matrices. IEEE Transactions on Signal Processing,63(1):81–89, January 2015.[8] R. Serizel, M. Moonen, B. V. Dijk, and J. Wouters. Low-rank approximation based multi-channel wiener filter algorithms for noise reduction with application in cochlear implants.IEEE/ACM Transactions on Audio, Speech, and Language Processing, 22(4):785–799,April 2014.[9] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice Hall, Englewood Cliffs,1993.[10] Z. Wang, J. G. McWhirter, and S. Weiss. Multichannel spectral factorization algorithm us-ing polynomial matrix eigenvalue decomposition. In 49th Asilomar Conference on Signals,Systems and Computers, Pacific Grove, CA, November 2015.[11] S. Weiss, A. Millar, and R. W. Stewart. Inversion of parahermitian matrices. In 18thEuropean Signal Processing Conference, pages 447–451, Aalborg, Denmark, August 2010.[12] S. Weiss, C. H. Ta, and C. Liu. A wiener filter approach to the design of filter bank basedsingle-carrier precoding and equalisation. In IEEE International Symposium on PowerLine Communications and Its Applications, pages 493–498, Pisa, Italy, March 26-28 2007.

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Investigation of a polynomial matrix generalised EVD for multi-channel Wiener filtering

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