A simple expression for the matrix gradient of a diagonal element of R in QR decomposition – For use in MIMO Communications and Signal Processing

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Authors
  1. Yasotharan, A.
Corporate Authors
Defence R&D Canada - Ottawa, Ottawa ONT (CAN)
Abstract
The QR matrix decomposition (QRD), or factorization, has many applications. In matrix computations, it is used to solve linear equations and least squares problems. In signal processing, it is used for adaptive filtering, adaptive beamforming/ interference nulling, and direction finding. In communications, it is used for adaptive equalization and transceiver design for multiple-input multiple-output (MIMO) channels. When QRD is used in signal processing and communications, it is of interest to know the effects of noise. This work is a first step towards that goal. Given matrix A with full column-rank M, we consider the unique decomposition A=QR where Q is a matrix with M orthonormal columns and R is an M x M upper triangular matrix with real positive diagonal elements r1, r2,….,rM. Treating ri as a function of the elements of A, a simple expression is derived for its matrix gradient with respect to A. Future work will aim to derive expressions for the gradients of all elements of Q and R, and use these expressions to evaluate the effect of noise perturbation in A. The present result is useful in optimizing certain MIMO decision-feedback communication systems.

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Report Number
DRDC-OTTAWA-TM-2010-247 — Technical Memorandum
Date of publication
01 Dec 2010
Number of Pages
38
DSTKIM No
CA035089
CANDIS No
534840
Format(s):
Electronic Document(PDF)

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