Infinite Length Results for Channel Shortening Equalizers

Time-domain equalization is crucial in reducing state dimension in maximum likelihood sequence estimation, and inter-carrier and inter-symbol interference in 802.11a and ADSL multicarrier systems. A time-domain equalizer, or TEQ, which is a finite impulse response (FIR) filter, placed in cascade with the channel produces an effective impulse response of ν + 1 samples that is shorter than the channel impulse response. This paper analyzes the two families of TEQ design methods amenable to cost-effective real-time implementation: minimum mean squared error (MMSE) and maximum shortening SNR (MSSNR) methods. For infinite length TEQs, we prove that MMSE target impulse responses are symmetric and have all ν zeros on the unit circle, and MSSNR TEQs have ν of their zeros on the unit circle. Consequently, finite-length MMSE and MSSNR TEQs will eventually yield increasing bit error rates (for broadcast systems) or decreasing bit rates (for point-to-point systems that allow bit allocation) with increasing filter length.