On the Periodicity of Singular Vectors and the Holomorphic Block-Circulant SVD on the Unit Circumference (2312.00707v1)
Abstract: We investigate the singular value decomposition of a rectangular matrix that is analytic on the complex unit circumference, which occurs, e.g., with the matrix of transfer functions representing a broadband multiple-input multiple-output channel. Our analysis is based on the Puiseux series expansion of the eigenvalue decomposition of analytic para-Hermitian matrices on the complex unit circumference. We study the case in which the rectangular matrix does not admit a full analytic singular value factorization, either due to partly multiplexed systems or to sign ambiguity. We show how to find an SVD factorization in the ring of Puiseux series where each singular value and the associated singular vectors present the same period and multiplexing structure, and we prove that it is always possible to find an analytic pseudo-circulant factorization, meaning that any arbitrary arrangements of multiplexed systems can be converted into a parallel form. In particular, one can show that the sign ambiguity can be overcome by allowing non-real holomorphic singular values.
- A. Ahrens, A. Sandmann, S. Lochmann, Z. Wang, “Decomposition of optical mimo systems using polynomial matrix factorization.” in IET ISP, London, UK, (2015).
- W. Al-Hanafy, A.P. Millar, C.H. Ta, S. Weiss, “Broadband SVD and non-linear precoding applied to broadband MIMO channels.” in Asilomar SSC, Pacific Grove, CA, 2053–2057 (2008).
- J. Corr, K. Thompson, S. Weiss, I. Proudler, J. McWhirter, “Row- shift corrected truncation of paraunitary matrices for PEVD algorithms.” in EUSIPCO, Nice, France, 849–853 (2015).
- D. Hassan, S. Redif, J. G. McWhirter, S. Lambotharan, “Polynomial GSVD beamforming for two-user frequency-selective MIMO channels.” IEEE TSP, 69, 948–959 (2021).
- W. Kellermann, H. Buchner, “Wideband algorithms versus narrow-band algorithms for adaptive filtering in the DFT domain.” in Asilomar SSC, 2:1278–1282 (2003).
- J.G. McWhirter, P.D. Baxter, T. Cooper, S. Redif, J. Foster, “An EVD algorithm for para-Hermitian polynomial matrices.” IEEE TSP, 55(5), 2158–2169 (2007).
- X. Mestre, D. Gregoratti, “A parallel processing approach to filter-bank multicarrier MIMO transmission under strong frequency selectivity.” in IEEE ICASSP, 8078–8082 (2014).
- B. De Moor, S. Boyd, “Analytic properties of singular values and vectors.” KU Leuven, Tech. Rep., 1989.
- N. Moret, A. Tonello, S. Weiss “MIMO Precoding for Filter Bank Modulation Systems Based on PSVD.” VTC Spring (2011).
- V.W. Neo, P.A. Naylor, “Second order sequential best rotation algorithm with householder reduction for polynomial matrix eigenvalue decomposition.” in ICASSP, Brighton, UK, 8043–8047 (2019).
- A. I. Pérez-Neira, M. Caus, R. Zakaria, D. L. Ruyet, E. Kofidis, M. Haardt, X. Mestre, Y. Cheng, “MIMO signal processing in offset- QAM based filter bank multicarrier systems.” IEEE TSP, 64(21), 5733– 5762 (2016).
- S. Redif, J. McWhirter, S. Weiss, “Design of FIR paraunitary filter banks for subband coding using a polynomial eigenvalue decomposition,” IEEE TSP, 59(11), 5253–5264 (2011).
- S. Redif, S. Weiss, J. McWhirter, “Sequential matrix diagonalization algorithms for polynomial EVD of parahermitian matrices.” IEEE TSP, 63(1), 81–89 (2015).
- G.W. Stewart, “On the early history of the singular value decomposition.” SIAM Review, 35(4), 551–566 (1993).
- C.H. Ta, S. Weiss, “A jointly optimal precoder and block decision feedback equaliser design with low redundancy.” in EUSIPCO, Poznan, Poland, 489–492 (2007).
- M. Tohidian, H. Amindavar, A.M. Reza, “A DFT-based approximate eigenvalue and singular value decomposition of polynomial matrices.” EURASIP J. Adv. SP, 2013(1), 1–16 (2013).
- M. Vu, A. Paulraj, “MIMO wireless linear precoding.” IEEE SP Mag., 24(5), 86–105 (2007).
- S. Weiss, N. Moret, A. Millar, A. Tonello, R. Stewart, “Initial results on an MMSE precoding and equalisation approach to MIMO PLC channels.” in IEEE ISPLC, 146–152 (2011).
- S. Weiss, J. Pestana, I. K. Proudler, F. K. Coutts, "Corrections to “On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix”." IEEE Trans. Signal Process. 66(23), 6325–6327 (2018).
- S. Weiss, I.K. Proudler, “Comparing Efficient Broadband Beamforming Architectures and Their Performance Trade-Offs.” in DSP, Santorini, Greece, 417–422, (2002).
- H. Zamiri-Jafarian, M. Rajabzadeh, “A polynomial matrix SVD approach for time domain broadband beamforming in MIMO-OFDM systems.” in VTC Spring, 802–806, (2008).