On symbol error probability bounds for ISI-like channels

Some issues with Forney’s upper and lower bounds for the symbol error probability in systems with memory (e.g., intersymbol interference channels) have been pointed out in the literature. We expound on these issues. For the upper bound, we show that, although the most commonly cited proofs are not logically consistent, the bound is true for more general conditions. The reasoning leading to the lower bound is shown to be flawed and, in general, to lead to invalid lower bounds. We suggest a lower bound based on Mazo’s bound as an alternative.