on SNR or on the thresholds N1 and N2. In the next section, we summarize these results beginning with a brief presentation of Albert's integral equation approach. 1.7.3 Probability of False Alarm and Average Test Duration in the Absence of Signal Let {x}} represent a sequence of observables which form a stationary Markov process with transitions governed by the probability distribution function F(x1x;-1). Denote by di, i = 1,2 a pair of decisions which is to be made about 31 Actually, Albert's work considered a far more general sequential test than the sequential prob- ability-ratio test of Wald. Our interest, however, is only in Wald's test, which is a special case of Albert's results corresponding to stationary increments in the log-likelihood ratio. SAMPLE AT RATE I/B SQUARE-LAW - RECEIVED SIGNAL ENVELOPE (BANDWIDTH = B) PLUS NOISE DETECTOR 72 <Zi<n, INCREMENT i BY 1 LOCAL Zi VI N2 DOUBLE PN CODE THRESHOLD GENERATOR COMPARISON Zizn, SIGNAL PRESENT Figure 1.46. Block diagram of a small SNR sequential detection PN acquisition system. Pseudonoise Code Acquisition in Direct-Sequence Receivers F(x,x;-1) and do the decision to defer making either d1 or d2. The test is con- ducted by first choosing an arbitrary starting point X0 (later on we shall set x0 = 0) and making one of the decisions di with probabilities (X0), i = 0,1, 2. If either d or d2 is made the test continues and the element X1 is drawn using the distribution F(x1)x) Once again one of the decisions is made with the set of probabilities (XX), i = 0,1,2 and the test either terminates or X2 is drawn using the distribution F(x2/x1). This process is continued until either d1 or d2 is made. To guarantee that this occurs with unit probability in a finite number of trials, it must be assumed that there exists an integer M and some p < 1 such that for all m > M the inequality (1.264) is satisfied for all x0. Using the foregoing model Albert [28] shows that the probability Pi(x0) that the test ends with decision d or d2 satisfies the integral equation (1.265) an