On the generalized symmetric cutoff rate for finite-state channels

Finite-state discrete-time channels with equiprobable M-ary inputs are considered. The generating function bound, which is ubiquitously applied to upper-bound the error probability of uncoded signaling over these channels, is used here to lower-bound the corresponding generalized symmetric cutoff rate, which lower-bounds the practically achievable rates and error exponents in these channels with symmetric signaling. The bound accounts for general additive metrics. For the special case of the optimal maximum-likelihood metric, the corresponding bound is shown to be asymptotically tight in the region where the symmetric cutoff rate approaches its maximum value of log₂ M (bits-per-channel use). For a finite-state additive white Gaussian noise (AWGN) channel this feature is used to relate the minimum Euclidean distance of an uncoded symmetric system to the corresponding symmetric cutoff rate. The results are demonstrated for AWGN channels corrupted by linear and nonlinear intersymbol interference. They are also used to assess the efficiency of concatenated coding over the 4-ary AWGN channel where the finite-state mechanism is defined by simple rate 1/2, four-state Ungerboeck 4-AM trellis code