Block-Skew-Circulant Matrices in Complex-Valued Signal Processing

Two main lines of approach can be identified in the recent literature on improper signals and widely linear operations. The augmented complex formulation based on the signal and its complex conjugate is considered as more insightful since it leads to convenient mathematical formulations for many considered problems. Moreover, it allows an easy distinction between proper and improper signals as well as between linear and widely linear operations. On the other hand, the composite real representation using the real and imaginary parts of the signal is closer to the actual implementation, and it allows to readily reuse results that have originally been derived for real-valued signals or proper complex signals. In this work, we aim at getting the best of both worlds by introducing mathematical tools that make the composite real representation more powerful and elegant. The proposed approach relies on a decomposition of real matrices into a block-skew-circulant and a block-Hankel-skew-circulant component. By means of various application examples from the field of signal processing for communications, we demonstrate the usefulness of the proposed framework.

[1]  Ting-Zhu Huang,et al.  The Inverses of Block Toeplitz Matrices , 2013 .

[2]  Tülay Adali,et al.  Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety , 2011, IEEE Transactions on Signal Processing.

[3]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[4]  Donatella Darsena,et al.  Universal Linear Precoding for NBI-Proof Widely Linear Equalization in MC Systems , 2008, EURASIP J. Wirel. Commun. Netw..

[5]  Pearson India,et al.  Applied Mathematical Methods , 1999 .

[6]  게르스타거볼프강,et al.  Method for cancelling interference during tdma transmission and/or fdma transmission , 2001 .

[7]  Eduard A. Jorswieck,et al.  Optimal Non-regenerative Relay Processing with Improper Signals , 2013, International Symposium on Wireless Communication Systems.

[8]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[9]  Visa Koivunen,et al.  Complex random vectors and ICA models: identifiability, uniqueness, and separability , 2005, IEEE Transactions on Information Theory.

[10]  Wolfgang Utschick,et al.  QoS Feasibility in MIMO Broadcast Channels With Widely Linear Transceivers , 2013, IEEE Signal Processing Letters.

[11]  L. Scharf,et al.  Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals , 2010 .

[12]  Mathini Sellathurai,et al.  Improved Linear Transmit Processing for Single-User and Multi-User MIMO Communications Systems , 2010, IEEE Transactions on Signal Processing.

[13]  Michael Joham,et al.  A Complete Description of the QoS Feasibility Region in the Vector Broadcast Channel , 2010, IEEE Transactions on Signal Processing.

[14]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[15]  Syed Ali Jafar,et al.  Interference Alignment With Asymmetric Complex Signaling—Settling the Høst-Madsen–Nosratinia Conjecture , 2009, IEEE Transactions on Information Theory.

[16]  Martin Haardt,et al.  Widely Linear Signal Processing for Two-Way Relaying with MIMO Amplify and Forward Relays , 2013, ISWCS.

[17]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[18]  Syed Ali Jafar,et al.  Interference Alignment and Degrees of Freedom of the $K$-User Interference Channel , 2008, IEEE Transactions on Information Theory.

[19]  Wolfgang Utschick,et al.  Performance gains due to improper signals in MIMO broadcast channels with widely linear transceivers , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[20]  Karl-Dirk Kammeyer,et al.  Weighted Sum Rate Maximization for MIMO-OFDM Systems with Linear and Dirty Paper Precoding , 2011 .

[21]  Wolfgang Utschick,et al.  Interference robustness for cellular MIMO networks , 2012, 2012 IEEE 13th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC).

[22]  Holger Boche,et al.  Solution of the multiuser downlink beamforming problem with individual SINR constraints , 2004, IEEE Transactions on Vehicular Technology.

[23]  Erry Gunawan,et al.  Optimized Transmission with Improper Gaussian Signaling in the K-User MISO Interference Channel , 2013, IEEE Transactions on Wireless Communications.

[24]  Michael Joham,et al.  Feasible rate region of the MIMO broadcast channel with linear transceivers , 2010, 2010 International ITG Workshop on Smart Antennas (WSA).

[25]  Martin Haardt,et al.  Widely linear adaptive beamforming algorithm based on the conjugate gradient method , 2011, 2011 International ITG Workshop on Smart Antennas.

[26]  Alfred Hanssen,et al.  A generalized likelihood ratio test for impropriety of complex signals , 2006, IEEE Signal Processing Letters.

[27]  Andrea J. Goldsmith,et al.  Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels , 2003, IEEE Trans. Inf. Theory.

[28]  Bernard C. Picinbono,et al.  Second-order complex random vectors and normal distributions , 1996, IEEE Trans. Signal Process..

[29]  Christoph W. Ueberhuber,et al.  Spectral decomposition of real circulant matrices , 2003 .

[30]  Christian Lameiro,et al.  Degrees-of-freedom for the 4-user SISO interference channel with improper signaling , 2013, 2013 IEEE International Conference on Communications (ICC).

[31]  Patrik Wahlberg,et al.  Spectral Relations for Multidimensional Complex Improper Stationary and (Almost) Cyclostationary Processes , 2008, IEEE Transactions on Information Theory.

[32]  D. Mandic,et al.  Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models , 2009 .

[33]  Eduard A. Jorswieck,et al.  Improper Gaussian signaling on the two-user SISO interference channel , 2011, 2011 8th International Symposium on Wireless Communication Systems.

[34]  Marco Lops,et al.  Widely linear reception strategies for layered space-time wireless communications , 2006, IEEE Transactions on Signal Processing.

[35]  Harry Leib,et al.  Maximizing SNR in improper complex noise and applications to CDMA , 1997, IEEE Communications Letters.

[36]  Danilo P. Mandic,et al.  Complex Valued Nonlinear Adaptive Filters , 2009 .

[37]  Thomas M. Cover,et al.  Elements of information theory (2. ed.) , 2006 .

[38]  Wolfgang Utschick,et al.  Optimality of proper signaling in Gaussian MIMO broadcast channels with shaping constraints , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[39]  Luigi Paura,et al.  Widely linear MMSE transceiver for real-valued sequences over MIMO channel , 2006, 2006 14th European Signal Processing Conference.

[40]  Louis L. Scharf,et al.  Second-order analysis of improper complex random vectors and processes , 2003, IEEE Trans. Signal Process..

[41]  Erry Gunawan,et al.  MISO interference channel with improper Gaussian signaling , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[42]  Mikko Valkama,et al.  RF-Aware Widely-Linear MMSE Beamforming , 2013, ISWCS.

[43]  Donatella Darsena,et al.  Widely-linear precoders and decoders for MIMO channels , 2013, ISWCS.

[44]  Wolfgang Utschick,et al.  On Optimal Gaussian Signaling in MIMO Relay Channels With Partial Decode-and-Forward , 2014, IEEE Transactions on Signal Processing.

[45]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[46]  M. Joham,et al.  A combinatorial approach to maximizing the sum rate in the MIMO BC with linear precoding , 2008, 2008 42nd Asilomar Conference on Signals, Systems and Computers.

[47]  Erry Gunawan,et al.  Ieee Transactions on Signal Processing, Accepted 1 Transmit Optimization with Improper Gaussian Signaling for Interference Channels , 2022 .

[48]  G. Tee.  Eigenvectors of block circulant and alternating circulant matrices , 2005 .

[49]  Léo Ducas,et al.  Faster Gaussian Lattice Sampling Using Lazy Floating-Point Arithmetic , 2012, ASIACRYPT.

[50]  Pascal Chevalier,et al.  Widely linear estimation with complex data , 1995, IEEE Trans. Signal Process..

[51]  Robert Schober,et al.  Receivers with widely linear processing for frequency-selective channels , 2003, IEEE Trans. Commun..

[52]  Shlomo Shamai,et al.  The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel , 2006, IEEE Transactions on Information Theory.

[53]  B. Friedman.  Eigenvalues of Composite Matrices , 1961, Mathematical Proceedings of the Cambridge Philosophical Society.

[54]  Faezeh Toutounian,et al.  Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems , 2012, Adv. Numer. Anal..

[55]  Nicolas C. Menicucci,et al.  The optical frequency comb as a one-way quantum computer , 2008, 0811.2799.

[56]  Wolfgang Utschick,et al.  Minimax Duality for MIMO Interference Networks , 2016, Inf..

[57]  Konstantinos N. Plataniotis,et al.  An enhanced widely linear CDMA receiver with OQPSK modulation , 2006, IEEE Transactions on Communications.

[58]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[59]  James L. Massey,et al.  Proper complex random processes with applications to information theory , 1993, IEEE Trans. Inf. Theory.

[60]  Mathini Sellathurai,et al.  Iterative Receiver Design for MIMO Systems with Improper Signal Constellations , 2009, 2009 IEEE International Conference on Communications.

[61]  Giacinto Gelli,et al.  FIR Zero-Forcing Multiuser Detection and Code Designs for Downlink MC-CDMA , 2007, IEEE Transactions on Signal Processing.

[62]  Donatella Darsena,et al.  Widely linear equalization and blind channel identification for interference-contaminated multicarrier systems , 2005, IEEE Transactions on Signal Processing.

[63]  On the Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with Fibonacci and Lucas Numbers , 2013 .