= = e e (2.3) where OJ is a random phase uniformly distributed in the interval (0,2). Thus, in any hop interval which is hit by the jammer (the probability of this occur- ring is the partial-band fraction p), the signals on which a decision for the i- th signalling interval is to be based are given (in complex form) by (2.4) Assuming a receiver structure that is optimum in the absence of the jam- mer, i.e., it employs the optimum decision rule for MDPSK against wideband noise, then in the presence of the on-tune jammer this rule would result in the estimate (2.5) where k is such that (2.6) Then if On is indeed the true value of (), a symbol (phase) error is made, i.e., whenever (2.7) Without loss in generality, we shall, for convenience, rotate the actual trans- mitted signal vectors by TM radians SO that the possible transmitted signal phases of (2.1) become (2.8) Finally, letting Q2TnIM; = 0; +1, =2, =(M - 2)/2, denote the prob- ability of the error event in (2.7), namely, (2.9) and noting that since we have assumed the absence of an AWGN back- ground, the probability of error in hop intervals which are not hit by the jam- mer is zero, then the average symbol error probability for MDPSK in the The assumption of on-tune jamming is made solely to simplify the analysis, as has been done in previous chapters. Both the analytical technique and the sensitivity of the results that follow from its application depend heavily on this assumption. Some evidence of this statement will be discussed at the end of this section. Differentially Coherent Modulation Techniques presence of multitone jamming is given by P.(M) (2.10) where the summation on n ranges over the set n = 0, 1, +2,...,- ..., 2)/2, M/2. Since OJ is uniformly distributed, we can recognize the symmetry (2.11) Also if On = 0 is transmitted, then, from (2.4), Y(i-1) and Yi are identical vec- tors. Equivalently, (2.12) and, from (2.9), (2.13) Thus, using (2.11) and (2.13), P (M) of (2.10) simplifies to (2.14) Finally, using the relation between average symbol and