Sequential M-ary PAM System

This paper examines the following
M
-level pulse-amplitude modulation (PAM) sequential communication system. Given that one of the
M
signals 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 if
M \geq 3
. 3) Lower bounds on the expected sample sizes for any test procedure for the
M
-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 of
M
.