On the equivalence of the simultaneous and separate MMSE optimizations of a DFE FFF and FBF

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Abstract
The equivalence of the simultaneous and separate minimum-mean-square-error (MMSE) optimizations of the feed- forward filter (FFF) and feedback filter (FBF) of a finite-length decision feedback equalizer (DFE) is established. HERE IS currently great interest in the use of decision feedback equalization (DFE) to mitigate the effects of intersymbol interference (ISI) on wireless multipath fading channels (1), (2). The coefficients of a DFE feedforward filter (FFF) and feedback filter (FBF) are usually adjusted based on the minimum-mean-square-error (MMSE) criterion (3). The equalizer coefficients can be calculated by recursive adaptation or by direct computation based on a channel estimate. The latter approach was used in a recent patent (2) that considered a DFE for use with the digital cellular standard IS-54. In the case of direct computation, two different approaches for computing the tap coefficients are frequently used, i.e., simultaneous optimization and separate optimization. In the simultaneous MMSE optimization, the DFE FFF and FBF co- efficients can be calculated by application of the orthogonality principle (OP) to an equalizer contents vector whose length spans the FFF and the FBF. The OP states that the error FBF will perfectly cancel the postcursor ISI within its span. Furthermore, our development shows that the separate solution to the MMSE FFF in an often cited reference (6, eq. 37) is (perhaps typographically) incorrect. The simultaneous optimization involves the inversion of a partitionable matrix whose dimension is given by the number of FFF and FBF taps. In the separate optimization the FFF taps are found by inverting a matrix whose dimension is given by the number of FFF taps, and the FBF taps are calculated without matrix inversion. Due to the absence of special structure in the separate optimization autocorrelation matrix, the complexity of the matrix inversion would be cubic in the number of FFF taps. This is to be contrasted with the Toeplitz autocorrelation matrixes that arise in linear equalizers, and which can be inverted with complexity quadratic in the number of taps. The issue of the computational complexity of calculating the MMSE DFE taps based on a channel estimate is treated thoroughly in a recent paper (10) (published after the submission of this letter) where it is shown that, by using a Cholesky factorization, the computational complexity of the MMSE DFE can be made quadratic in the number of FFF and FBF taps.

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