When are the MLSD respectively the matched filter receiver optimal with respect to the BER?

We reconsider the minimum/optimal bit-error probability receiver (OBER) for intersymbol interference channels with Gaussian noise and the reception of finite blocks of bits. We view the OBER as a function with two inputs: the received sequence and an expected signal-to-noise ratio; and one output: the estimated block of bits. Assuming that all sequences are equally probable to be transmitted we prove two results about the behaviour of the OBER. We show that the OBER coincides with the maximum likelihood sequence detector (MLSD) when designed for high signal-to-noise ratios and that it collapses to a matched filter followed by a hard-limiting device for low expected signal-to-noise ratios