AnM-ary decision feedback communication system with the MAP strategy is investigated. The performance of this system is analytically evaluated as follows. First, an approximate lower performance bound is found by means of the Cauchy inequality. Then an approximate lower performance bound is found by means of the Jensen inequality. Finally, the bounds are compared and found to coincide, yielding an approximate estimate of the system performance. The same estimate may also be obtained by means of mean path approximation. Thus, the mean path approximation tends to underestimate the system average transmission time. For verification purposes, the results are compared with those obtained from computer simulation and are found to be in good agreement. Furthermore, the feedback transmission is compared with one-way transmission. It is shown that feedback systems are better than one-way systems in terms of the signal/noise ratio required to achieve a prescribed error probability. The estimated power advantage of feedback systems is between 5.4 dB and 3 dB, depending on the number of possible signalsM; the power advantage decreases asMincreases. The results obtained forM→ ∞ imply that the feedback increases the capacity of an infinite-bandwidth Gaussian channel. However, the last result was not proved with the degree of rigour required in mathematics; rather, we have used approximate performance bounds and computer simulation data in our analysis.