lute-value envelope detector (Figure 1.46 with an absolute value instead of a square-law envelope detector). If, for the square-law envelope detector of case (2), the detector output sam- ples Yk are characterized as in (1.60), namely, (1.283) where then for the envelope detector of case (3) we would have (1.284) Similarly, the absolute-value detector model of case (4) would imply (1.285) which simulates inphase and quadrature sampling with a large number of quantization levels and the sign bit discarded. From our previous discussions, it is clear that for all of the above imple- mentations, the sequential test is governed by the choice of three variables, namely, the upper and lower thresholds and the bias. Typically, the upper threshold and bias determine the false alarm probability, PFA, whereas the lower threshold and bias determine the average number of samples to dis- miss Nd. Simulation results [30] performed on all four detector types reveal that the false alarm probability is governed by the relation (1.286) where f is a false-alarm rate parameter whose variation with detector type and bias is illustrated in Figure 1.51 and Tu denotes the upper test thresh- old. The table below relates, for each of the four above cases, the bias and T to the analogous parameters previously introduced in this section. Detector Type Ideal yb/2o b' yn1/202 Square-law b/202 = b'ly n1/202 = ni/r Envelope b/V2o n1/V20 Absolute-value b/V2o n1/V2o From the results in Figure 1.51, we observe that, for the ideal detector, f = 1 independent of the bias point, whereas the other detector types have a linear dependence on f on the logarithm of the bias point. This dependence of cases (2), (3), and (4) on bias point could no doubt be removed by a gain normalization of the detector output samples analogous to the A/o2 multiplication for the ideal detector (see Figure 1.45). However, the primary purpose of the comparison being to discover the effects of simple imple- PN Synchronization Using Sequential Detection ENVELOPE DETECTOR PFA=0.5