On the convergence of the generalized DFE to the MMSE-DFE

This paper shows that under certain conditions the generalized decision feedback equalizer (GDFE) asymptotically converges to the infinite length minimum-mean-square-error decision feedback equalizer (MMSE-DFE), in the sense that both structures perform essentially the same signal processing operations at both the transmitter and the receiver. The GDFE (developed by Cioffi and Forney (1997)) is a multiple-input-multiple-output (MIMO) generalization of the well-known DFE structure, whose special cases include various transmission schemes such as discrete multi-tone (DMT). The proofs are based on a fundamental equivalence theorem holding between the Cholesky factorization of a doubly infinite Toeplitz covariance matrix and the spectral factorization of a stationary sequence. The results extend to the case where the optimum spectrum consists of several disjoint bands (which implies the operation of multiple corresponding independent MMSE-DFEs). The convergence of GDFE to the potentially many MMSE-DFEs requires performing a "generalized resampling" operation, which is equivalent to partitioning the available bandwidth into disjoint bands and eliminating the unused regions