Exact distribution functions of test length for sequential processors with discrete input data

In studies of sequential detection of radar signals, the parameter of primary interest is the length of the sequential test, denoted by
n
. Since this test length is a random variable, moments and/or probability distribution functions of
n
are desirable. A procedure is described in this communication for obtaining exact probability distribution functions
P(n)
and exact average values of
n
,
E(n)
, when the input to the sequential processor is discrete radar data (radar data in quantized form). This procedure is based upon the representation of the sequential test as a Markov process. The results are quite general in that they apply to multilevel quantization of the data. However, the procedure appears especially attractive when the number of levels is small as is usually the case when dealing with discrete radar data. The procedure for determining exact distribution functions and average values of
n
presented herein is compared with the Wald-Girshick approach for obtaining
P(n)
and
E(n)
, and the superiority of the former approach in computational convenience is indicated.