 a true alarm, i.e., message detection. However, to relate PD to the per-channel, per-hop detection probability, PDI, namely, the probability that an individual I & D output exceeds the thresh- old k, is, in general, a difficult task due to the continuing overlap of the inphase and quadrature I & Ds throughout the duration of the message. Nevertheless, through continued application of union bound techniques, upper and lower bounds on PD are derived assuming a worst case out-of- synchronization condition of 1/4 of a hop inteval. In particular, after many simplifying but practical assumptions, it is shown that (1.52) where www.nown (1.53) To determine the required S/N° using the detector of Figure 1.13 assume for a basis of comparison the same false alarm rate and detection probabil- ity as would be required for the ideal filter bank combiner with perfect time synchronization. Since again there is no explicit closed form expression for PD, it will be equated to either of the two bounds in (1.52) and thereby a range of values of the required S/N° will be obtained. Proceeding as before, equating the right-hand side of (1.51) to PFA/2NhNT = PFA12 and eliminat- ing the threshold k between the resulting expression and either the upper or lower bound of (1.52) gives (1.54) where Cp=1 or 2 and (1.55) Since, typically, Ah < WhTh, then simplifying the square root in (1.54) to unity, and solving for SIN by substituting the values of either 1 or 2 for CD gives Low Probability of Intercept Communications the desired result:8 (1.56) Recall that the equivalent result for the ideal bank combiner with perfect time synchronization is given by (1.9). Calculating the uper and lower bounds on the SIN requirement for the FBC with overlapping I & Ds (recall that NhNT = WSSTM or Nh = WSSTMINT, and WhT = 1) and comparing these results with the SIN requirement of a 125-channel partial-band FBC with perfect time synchronization (see Table 1.2), it is concluded that for the parameters of the continuing example, time synchronization is essent