In contrast to the conventional Viterbi algorithm (VA) which generates hard-outputs, an optimum soft-output algorithm (OSA) is derived under the constraint of fixed decision delay for detection of M-ary digital signals in the presence of intersymbol interference and additive white Gaussian noise. The OSA, a new type of the conventional symbol-by-symbol maximum a posteriori probability algorithm, requires only a forward recursion and the number of variables to be stored and recursively updated increases linearly, rather than exponentially, with the decision delay. Then, with little performance degradation, a suboptimum soft-output algorithm (SSA) is obtained by simplifying the OSA. The main computations in the SSA, as in the VA, are the add-compare-select operations. Simulation results of a convolutional-coded communication system are presented that demonstrate the superiority of the OSA and the SSA over the conventional VA when they are used as detectors. When the decision delay of the detectors equals the channel memory, a significant performance improvement is achieved with only a small increase in computational complexity.< >