  Motivated by recent high bandwidth communication systems, Inter-Symbol
Interference (ISI) channels with 1-bit quantized output are considered under an
average-power-constrained continuous input. While the exact capacity is
difficult to characterize, an approximation that matches with the exact channel
output up to a probability of error is provided. The approximation does not
have additive noise, but constrains the channel output (without noise) to be
above a threshold in absolute value. The capacity under the approximation is
computed using methods involving standard Gibbs distributions. Markovian
achievable schemes approaching the approximate capacity are provided. The
methods used over the approximate ISI channel result in ideas for practical
coding schemes for ISI channels with 1-bit output quantization.
