wing an envelope detector has no effect on its output. Wald showed [10] that the thresholds n'1 and n'2 are approximately related to the probabilities of false alarm PFA and detection PD by (1.254) (1.255) exp(ni) - exp(n2) Furthermore, the average number of samples N required to dismiss when signal is not present was shown to be approximately given by the expres- E{Ak} E{Ak} (1.256) PN Synchronization Using Sequential Detection where from (1.73) and (1.251) (1.257) Evaluation of (1.256) is difficult because of the requirement to accurately compute the expectation in (1.257) for each value of Y of interest. Since the integral cannot be obtained in closed form, one can either evaluate it using numerical integration or first expand the In I function in a power series and compute as many moments of yk as necessary to achieve the desired degree of accuracy. In the case of the latter, approximation of In Io(x) by only the first few terms in the expansion restricts its validity to small values of X or, equivalently, small values of pre-detection SNR. Another approach based on the small SNR assumption is to make a sim- ilar expansion of In Io(x) directly in the sequential test statistic of (1.251) and keep only the first few terms. Doing this has the effect of simplifying the detector implementation, which, except when digital processing allows use of a look-up table for In Io(x), is ordinarily difficult. In the next section, we discuss a small SNR sequential test and the corresponding detector perfor- mance corresponding to a two-term approximation of the In I function. 1.7.2 The Biased Square-Law Sequential Detector Although (1.251) defines a sequential test for the optimum detector, the important case of small pre-detection SNR allows one to simplify this equa- tion into a similar test for an easier to implement (and analyze) detector as follows. When y is small, the In I function can be approximated by the first two terms of its power series, namely, (1.258) Applying (1.258) to (1.251) then gives (1.259) As suggeste