he single dwell time acquisition system. While, in principle, the problems are quite similar, we shall soon see that many of the simplifying assumptions made in the previous analysis are not valid here for the matched filter system. Nevertheless, having gone through the details of such an analysis in Section 1.2.4, our presentation here will be brief with emphasis placed on the differences between the two. First, unlike (1.58), the signal component of the inputs to the square-law detectors in Figure 1.25b is treated as a random variable reflecting the par- tial correlation effect produced by correlating the input code with the stored reference code over a finite time interval which is much less than the code period. In particular, if V2Ac(t + T) denotes the input PN code delayed by T with respect to an arbitrary time reference and c(t + ) denotes the cor- responding local reference code, then, when the correlation time MT is much less than the code time period pTc, the correlation (1.186) is approximately a Gaussian random variable with mean and variance (con- ditioned on the hypothesis Hi, = 0,1) given by [3], [13], [14] (1.187a) (1.187b) In (1.187), (1.188) denotes the fractional normalized (with respect to a chip) timing offset between the two codes where N is the smallest integer such that E lies in the interval (-1,1 1). Figure 1.29 illustrates the conditional mean and vari- ance of C for (T - 7)/Tc 3. The conditions under which the Gaussian Pseudonoise Code Acquisition in Direct-Sequence Receivers var (C/H)/MTC E 1C/H;/MT -3.00 -2.00 -1.00 Figure 1.29. The normalized mean and variance of the partial correlation C as a function of the relative timing offset (normalized to the chip interval) between the incoming and local codes. assumption applies to the random variable C have been discussed in Volume II. In short, for M << p, C behaves like a binomial distribution to behave like a Gaussian one. The second difference concerns the validity of the Gaussian approx- imation to the decision statistic which is the 