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A discrete (binary-input 2q-ary output) communication channel with memory is introduced to judiciously capture both the statistical memory and the soft-decision information of a time-correlated discrete fading channel (DFC) used with antipodal signaling and soft output quantization of resolution q. The discrete channel, which can be explicitly described via its binary input process and a 2q-ary noise process, is shown to be symmetric, thus admitting a simple expression for its capacity when its noise is stationary ergodic. It is observed that considerable capacity gains can be achieved due to the channel's memory and the use of as few as 2 bits for soft-decision over interleaving the channel (to render it memoryless) and hard-decision demodulation (q=1). The 2q-ary noise process is next modeled via a queue-based (QB) ball-sampling mechanism to produce a mathematically tractable stationary ergodic Markovian noise source. The DFC is fitted by the QB noise model via an iterative procedure that minimizes the Kullback-Leibler divergence rate between the DFC and QB noise sources. Modeling results, measured in terms of channel noise correlation function and capacity reveal a good agreement between the two channels for a broad range of fading conditions.