d the average test duration (average sample number) M1(xo) satisfies the integral equation (1.266) For most cases of interest, these integral equations are difficult if not impossible to solve. However, for the non-coherent sequential detection of a sine wave in Gaussian noise using a biased square-law detector, Kendall [29] was able to obtain exact solutions. In particular, the sequence {xi} now corresponds to {Z}} of (1.262), d1 is the dismissal decision, and d2 is the alarm decision. Since from (1.262), Z = Zi-1 Y = Zi-1 - b, then, using (1.63), dF(Z|Zi-1) otherwise (1.267) Also, the set of decision probabilities Ti(Zk), = 0,1,2 is stationary (i.e., inde- pendent of k) and given by otherwise (1.268) PN Synchronization Using Sequential Detection Finally, letting Z = 0, then, for n2<<<<< exp(-n2/y)G(-Dn2;Db') (1.269) where we have introduced the normalizations (1.270) (1.271) Also, the function G(x;c) is defined by (1.272) where N is an integer chosen to satisfy the inequalities (1.273) By similar methods, Kendall [29] obtains a solution to (1.266) for the aver- age sample number N which is given by Na M1(0) = - n'2+b'); Db']} (1.274) where P2(0) is given by (1.269) and (1.275) with N still obtained from (1.273). The average test duration Td, for the par- ticular noise only cell under investigation, is simply (1.276) since for the real system the Z represent samples taken at a rate 1/B. Before presenting numerical illustrations of these results, we point out that with suitable approximations they can be shown to agree with Wald's results [10]. In particular, since Wald's results are approximate in that they neglect the "excess over the bounds," i.e., at the end of the test we have either Z VI N2 or N1 < Z, not simply Z = N2 or Z = N1, then if the values of the thresh- olds are such that this effect is negligible, Albert's results simplify to those of Wald. Also, the normalized threshold b' of (1.270) is not required to Pseudonoise Code Acquisition in Direct-Sequence Receivers correspond to the optimum bias of (1