Joanne A. Foster

An EVD Algorithm for Para-Hermitian Polynomial Matrices

An algorithm for computing the eigenvalue decomposition of a para-Hermitian polynomial matrix is described. This amounts to diagonalizing the polynomial matrix by means of a paraunitary similarity transformation. The algorithm makes use of elementary paraunitary transformations and constitutes a generalization of the classical Jacobi algorithm for conventional Hermitian matrix diagonalization. A proof of convergence is presented. The application to signal processing is highlighted in terms of strong decorrelation and multichannel data compaction. Some simulated results are presented to demonstrate the capability of the algorithm

An Algorithm for Calculating the QR and Singular Value Decompositions of Polynomial Matrices

In this paper, a new algorithm for calculating the QR decomposition (QRD) of a polynomial matrix is introduced. This algorithm amounts to transforming a polynomial matrix to upper triangular form by application of a series of paraunitary matrices such as elementary delay and rotation matrices. It is shown that this algorithm can also be used to formulate the singular value decomposition (SVD) of a polynomial matrix, which essentially amounts to diagonalizing a polynomial matrix again by application of a series of paraunitary matrices. Example matrices are used to demonstrate both types of decomposition. Mathematical proofs of convergence of both decompositions are also outlined. Finally, a possible application of such decompositions in multichannel signal processing is discussed.

Polynomial Matrix QR Decomposition and Iterative Decoding of Frequency Selective MIMO Channels

For a frequency flat multi-input multi-output (MIMO) system the QR decomposition can be applied to reduce the MIMO channel equalization problem to a set of decision feedback based single channel problems. Using a novel technique for polynomial matrix QR decomposition (PMQRD) based on Givens rotations, we show the PMQRD can do likewise for a frequency selective MIMO system. Two types of transmitter design, based on Horizontal and Vertical Bell Laboratories Layered Space Time (H-BLAST, V-BLAST) encoding have been implemented. Receiver processing utilizes Turbo equalization to exploit multipath delay spread and to facilitate multi-stream data feedback. n nAverage bit error rate simulations show a considerable improvement over a benchmark orthogonal frequency division multiplexing (OFDM) technique. The proposed scheme thereby has potential applicability in MIMO communication applications, particularly for a TDMA system with frequency selective channels.

A novel algorithm for calculating the QR decomposition of a polynomial matrix

A novel algorithm for calculating the QR decomposition (QRD) of polynomial matrix is proposed. The algorithm operates by applying a series of polynomial Givens rotations to transform a polynomial matrix into an upper-triangular polynomial matrix and, therefore, amounts to a generalisation of the conventional Givens method for formulating the QRD of a scalar matrix. A simple example is given to demonstrate the algorithm, but also illustrates two clear advantages of this algorithm when compared to an existing method for formulating the decomposition. Firstly, it does not demonstrate the same unstable behaviour that is sometimes observed with the existing algorithm and secondly, it typically requires less iterations to converge. The potential application of the decomposition is highlighted in terms of broadband multi-input multi-output (MIMO) channel equalisation.

An Algorithm for Computing the QR Decomposition of a Polynomial Matrix

This paper introduces an algorithm for computing a QR decomposition of a polynomial matrix. The algorithm proceeds to perform the decomposition by following the same strategy in eliminating entries of the matrix as is used in the Givens method for a QR decomposition of a scalar matrix. However scalar Givens rotation matrices can no longer be applied. Instead, a polynomial Givens rotation is introduced, enabling the QR decomposition of a polynomial matrix. Convergence of the algorithm is discussed and through simulations the capability of the algorithm is assessed.

A Polynomial Matrix QR Decomposition with Application to MIMO Channel Equalisation

An algorithm for computing the QR decomposition of a polynomial matrix is introduced. The algorithm proceeds to perform the decomposition by following the same strategy in eliminating entries of the matrix as is used in the Givens method for a QR decomposition of a scalar matrix, however polynomial Givens rotations are now required. A possible application of the decomposition is in MIMO communications, where it is often required to reconstruct data sequences that have been distorted due to the effects of co-channel interference and multipath propagation, leading to intersymbol interference. If the channel matrix for the system is known, its QR decomposition can be calculated and used to transform the MIMO channel equalisation problem into a set of single channel problems, which can then be solved using a maximum likelihood sequence estimator. Some simulated average bit error rate results are presented to support the potential application to MIMO channel equalisation.

Polynomial matrix QR decomposition for the decoding of frequency selective multiple-input multiple-output communication channels

This study proposes a new technique for communicating over multiple-input multiple-output (MIMO) frequency selective channels. This approach operates by calculating the QR decomposition of the polynomial channel matrix at the receiver on the basis of channel state information, which in this work is assumed to be perfectly known. This then enables the frequency selective MIMO system to be transformed into a set of frequency selective single-input single-output systems without altering the statistical properties of the receiver noise, which can then be individually equalised. A like-for-like comparison with the orthogonal frequency division multiplexing scheme, which is typically used to communicate over channels of this form, is provided. The polynomial matrix system is shown to achieve improved performance in terms of average bit error rate results, as a consequence of time-domain symbol decoding.

A Polynomial QR Decomposition Based Turbo Equalization Technique for Frequency Selective MIMO Channels

In the case of a frequency flat multiple-input multiple-output (MIMO) system, QR decomposition can be applied to reduce the MIMO channel equalization problem to a set of decision feedback based single channel equalization problems. Using a novel technique for polynomial matrix QR decomposition (PMQRD) based on Givens rotations, we extend this work to frequency selective MIMO systems. A transmitter design based on Diagonal Bell Laboratories Layered Space Time (D-BLAST) encoding has been implemented. Turbo equalization is utilized at the receiver to overcome the multipath delay spread and to facilitate multi-stream data feedback. The effect of channel estimation error on system performance has also been considered to demonstrate the robustness of the proposed PMQRD scheme. Average bit error rate simulations show a considerable improvement over a benchmark orthogonal frequency division multiplexing (OFDM) technique. The proposed scheme thereby has potential applicability in MIMO communication applications, particularly for TDMA systems with frequency selective channels.

A study of the effect of uncertainties when calculating the singular value decomposition of a polynomial matrix

An algorithm has been recently proposed by the authors for calculating a polynomial matrix singular value decomposition (SVD) based upon polynomial matrix QR decomposition. In this work we examine how this method compares to a previously proposed method of formulating this decomposition. In particular, the performance of the two methods is examined when each is used as part of a broadband multiple-input multiple-output (MIMO) communication system by means of average bit error rate simulations. These results confirm a clear advantage of using the new polynomial matrix SVD method over the existing technique. This paper also discusses the possible errors that are encountered when formulating the SVD of a polynomial matrix and investigates how these errors affect the error rate performance of both SVD methods within the proposed application.