 Vo are jointly Gaussian random variables. For the worst case situation of Figure 1.10, their joint probability density func- Interception Detectors NOISE ONLY EARLY I&D INTERVAL SIGNAL PLUS NOISE LATE 18D INTERVAL Figure 1.10. Worst case signal misalignment. tion is given by (1.28) =2W.,Tw(1+3) (1.29) (1.30) The parameter y in (1.30) represents the signal-to-noise ratio in the input bandwidth. Using (1.28) the detection probability of (1.27) can be expressed as (1.31) which unfortunately cannot be obtained in closed form. Nevertheless, PD can be upper (union) and lower bounded as follows: [ plvndv, (1.32) Low Probability of Intercept Communications or, equivalently, (1.33) Substituting (1.29) into (1.33) gives the desired result: (1.34) To determine the SIN required by an inteceptor using the overlapped I & D detector of Figure 1.9, for a basis of comparison, the same false alarm rate and detection probability as would be required for a non-overlapped single I & D detector with perfect time synchronization is assumed. Since the overlapped I & D does not have an explicit closed-form expression for PD, it will be equated to both of the two bounds in (1.34) and thereby a bounding range of values for the required S/N will be obtained. Proceeding as above, the right-hand side of (1.26) is equated to PFA/2, and the threshold k is eliminated between the resulting expression and either the upper or lower bound of (1.34) with the result (1.35) where CD=1or2, and (1.36) Since, typically l < WSSTM, the square root of (1.35) simplifies to unity. Finally, solving for S/N° and substituting the values of 1 or 2 for CD gives the desired result: (1.37) Interception Detectors Recall that the equivalent result for the detector with a single non-overlap- ping I & D (and perfect synchronization) is given by (1.4) together with (1.5). As an example of the application of (1.37), consider the parameters of the continuing example of Section 1.2.1 (WssTM =8x109. = For the performance criteria of PD = 0.9 and PFA=10-6, = the lower and upp