M-ary Sequential Detection for Amplitude-Modulated Signals in One and Two Dimensions

M
-ary sequential tests are proposed and analyzed for amplitude-modulated signals that can be represented in a one-or two-dimensional signal space. Coherent detection in Gaussian noise has been assumed. Expressions for the probability of error and average sample size are found as a function of the threshold constants. The probability of error as a function of the average sample size for the sequential test is then compared to the optimum fixed sample test. For error rates in the range of 10
^-6
, the average sample size of the sequential tests is about 2.5 times (4 dB) smaller than the sample size of the corresponding fixed sample tests. For comparison purposes the length of the binary sequential test will be 3.2 times (5.2 dB) smaller than the length of the corresponding binary fixed sample test. The sequential likelihood ratio test for amplitude-modulated signals in one dimension leads to a fixed sample test of a certain length to determine the two most likely signals and then a binary sequential test between these two signals to determine the most likely one. The binary sequential portion of the test incorporates the data obtained from the fixed portion of the test. For two-dimensional amplitude-modulated signals, the sequential test consists of a fixed sample test through a certain length and then two concurrent binary sequential tests on the two most likely amplitudes of each dimension.