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

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. 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 a TDMA system with frequency selective channels

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. 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 a TDMA system with frequency selective channels

Broadband MIMO Beamforming for Frequency Selective Channels using the Sequential Best Rotation Algorithm

For a narrowband multi-input multi-output (MIMO) system the singular value decomposition has the ability to provide multiple spatial channels for data transmission. We extend this work to obtain spatial diversity techniques for frequency selective MIMO systems using a polynomial matrix decomposition known as the sequential best rotation using second order statistics (SBR2) method. This algorithm diagonalizes a MIMO frequency selective channel yielding various spatial modes for data transmission. We evaluate the diversity performance of the dominant channel provided by the SBR2 based broadband decomposition and compare it with a transmit antenna selection method (TAS) and a MIMO orthogonal frequency-division multiplexing (OFDM) singular value decomposition (SVD) based approach. Simulation results show SBR2 significantly outperforms the average bit error rate (BER) of TAS, making it very suitable for time division multiple access (TDMA) and code division multiple access (CDMA) systems. SBR2 and MIMO-OFDM systems are shown to have identical BER performance, confirming the efficiency of the proposed low delay spatial-temporal scheme

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