This paper examines the followingM-level pulse-amplitude modulation (PAM) sequential communication system. Given that one of theMsignals is repetitively sent over an additive Gaussian channel during successive intervals of time, what is the optimum sequential procedure to follow at the receiver in order to pick the correct signal with a probability of wrong selection no greater than\epsilon?The optimum procedure is defined to be one that minimizes the expected number of transmissions (sample size) before a decision is reached. The paper extends the results of Hecht for the "simultaneous" test procedure (also obtained independently by the author). This paper shows the following. 1) The maximum a posteriori (MAP) test procedure is shown to be comparable in performance with the simultaneous test. 2) MAP test with a threshold 1 - ε does not yield error probabilities equal to ε ifM \geq 3. 3) Lower bounds on the expected sample sizes for any test procedure for theM-level PAM are obtained with the results from Hoeffding and Simons. 4) Both the simultaneous and the MAP tests are shown to achieve the lower bound when\epsilon \rightarrow 0. 5) Approximations used by Hecht for the simultaneous test are made rigorous and slightly tighter expressions of error probability than the ones given by Hecht are obtained for some cases. 6) Energy savings of both the MAP and the simultaneous tests over the nonsequential scheme are presented for different values ofM.

This paper examines aM-ary sequential orthogonal (OM) communications system. A similarM-ary sequential pulse-amplitude modulation (PAM) problem was previously considered by Hecht and Schwartz [10], and by the author. We examine the following problem: given that one of them signals is repetitively sent over an additive Gaussian channel successive intervals of time, determine the optimum sequential procedure to follow at the receiver to pick the correct signal with a probability of wrong selection no greater than\varepsilon. The optimum procedure is defined to be one that minimizes the expected number of transmissions (sample size) before a decision is reached. This paper extends the works of Viterbi [1] and Kramer [2] who proposed two ad hoc test procedures for this OM problem. From the results of Simons [3] and Hoeffding [4], a lower bound on the expected sample size is found for any sequential test procedure for the proposed OM system. Both the maximum a posteriori probability (MAP) and the simultaneous tests are shown to achieve this lower bound in the limit as\epsilon \rightarrow 0. Viterbi [1] computed the expected sample size for the MAP test procedure whenM = 2. For the case whereM \geq 3, this paper derives approximations of the expected sample sizes for both the MAP and the simultaneous test procedures when\varepsilon \rightarrow 0. Computer simulation shows that the derived expected sample size gives accurate estimation of the actual sample size whenM \epsilon < 0.1. These sequential procedures offer 1 to 2 dB more power saving than the test procedures of Viterbi [1] and Kramer [2] if\epsilon > 10^{-7}.