 for example, in Figure 1.45. The equa- tions describing the input x(t) and output y(t) under these conditions are given by (1.58)-(1.60). Furthermore, if y(t) is sampled at a rate 1/B, then the samples Yk have a probability density function Po(yk) given by (1.63) or (yk) given by (1.61), where the subscript on p(yk) now corresponds to that on the particular hypothesis being tested. Define the log-likelihood function (1.250) Then the sequential probability ratio test [27] for the non-coherent detec- tion of a sine wave signal in Gaussian noise sequentially compares the run- ning sum (1.251) to two thresholds n'1 and n'2(2'2 n'1), where as before it is convenient to talk about the normalized samples yKyk/2o2yk/NoB.The compari- son is made for each i = 1,2, until the inequalities Vi < n'1 are not satisfied. Then, if it is decided that the signal is not present (dis- missal), and if nisuit is decided that the signal is present (alarm). Figure SAMPLE AT RATE I/B RECEIVED SIGNAL SQUARE-LAW N(1)% ENVELOPE Int(1) (BANDWIDTH=B) DETECTOR NOISE A2/202 nzevieni INCREMENT i BY I LOCAL CODE PHASE VISM2 DOUBLE PN CODE THRESHOLD GENERATOR UPDATE COMPARISON vizni SIGNAL PRESENT Figure 1.45. Block diagram of a sequential detection PN acquisition system. Pseudonoise Code Acquisition in Direct-Sequence Receivers 1.45 is a block diagram of the sequential detector with test statistic as in (1.251). In some discussions of sequential detection, an envelope detector alone is postulated instead of a square-law envelope detector. In this case, the sam- ples of the envelope detector output would have probability density func- tions otherwise otherwise (1.252) and log-likelihood function (1.253) Comparing (1.253) with (1.250), we note that the block diagram of Figure 1.45 can be made to apply to the sequential detection system with an enve- lope detector by simply removing the square root device preceding the mul- tiplication by A/o2. This result is intuitively obvious since the tandem combination of square-law and square-root devices follo