On Sequential M-ary Orthogonal Modulation

This paper examines a
M
-ary sequential orthogonal (OM) communications system. A similar
M
-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 when
M = 2
. For the case where
M \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 when
M \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}
.