d in [27], the latter term in (1.259) is further approximated by its expected value when no signal is present. Thus, from (1.76) (1.260) = -(1+2)] (1.261) Pseudonoise Code Acquisition in Direct-Sequence Receivers Normalizing Vi by NoBly and recalling the relation between Yk and y gives the sequential test statistic corresponding to the optimum detector for small SNR, namely, (1.262) where from (1.261) the bias b is given by (1.263) A block diagram of the small SNR sequential detector is illustrated in Figure 1.46. Identifying (1.262) with Z of (1.64) when NB = 1, one observes that the optimum bias of (1.263) lies halfway between the means of the square-law detector output under the hypotheses H and H. Equivalently, for this choice of bias, the integrator output in the presence of the bias voltage will tend to increase linearly when signal plus noise are present at the same rate that it tends to decrease linearly when noise only is present (see Figure 1.44b). Before proceeding to a discussion of performance, we hasten to point out that for a two-hypothesis test of a sampled random variable, the sequential probability-ratio test is optimum in the sense that, for given PFA and PD, it requires the minimum average number of samples to produce a decision. For our application, namely, rapid acquisition of a PN sequence in noise, mini- mizing the average number of samples required is equivalent to minimizing the average dwell time per code phase cell being searched. Although the sequential probability-ratio test was discovered by Wald in the early 1940's while working on radar detection, an accurate characteri- zation of its performance did not come until many years later. In particular, the integral equation approach of Albert [28]31 was applied by Kendall [29] in 1965 to give exact solutions for the false alarm probability and average test duration of the biased square-law sequential detector in the absence of signal. These results were unique in that they were valid without any restric- tions on either the pre-detecti