lities by Z, and locates the value of Z, say Zmax which yields the max- imum, say Pmax, of this product. Then, from (2.47) and (2.48), the worst case jammer strategy corresponds to a partial-band fraction (2.49) and maximum bit error probability (2.50) Although, for arbitrary M, the single integral of (2.46) cannot be evalu- ated in closed form, the special case of 2-ary MDPSK (often just abbrevi- ated as DPSK) allows one to obtain the simple result (2.51) or for FH/DPSK in partial-band noise jamming (2.52) Directly differentiating (2.52) with respect to p and equating the result to zero yields the worst case performance (2.53) (2.54) Differentially Coherent Modulation Techniques Figure 2.7 illustrates ZP (M) versus Z in dB for M = 2, 4, 8, and 16. Table 2.2 provides the corresponding values of Zmax and Pmax from which the asymptotic performance of FH/MDPSK for worst case partial-band noise jamming is computed using (2.49) and (2.50). Comparing Tables 2.1 and 2.2, we observe that for all values of M, FH/MDPSK is more sensitive to worst case partial-band multitone jamming than it is to worst case partial-band noise jamming. Pmax = 2.171 Pmax . 7471 ZPs(M) P max =.3176 Pmax ...1839 max Zmax = 4.88 dB Zmax = 9.64 dB Zmax = O dB Figure 2.7. A plot of ZP (M) versus Z in dB. Performance of DQASK in the Presence of Additive White Gaussian Noise Table 2.2 Asymptotic performance of FH/MDPSK for worst case partial-band noise jamming. Worst case partial-band fraction Maximum average bit error probability Kp(Zmax) .1839 1.000 .1839 .3176 1.245 .2118 .7471 3.076 .4269 2.171 9.204 1.1579 2.3 PERFORMANCE OF DQASK IN THE PRESENCE OF ADDITIVE WHITE GAUSSIAN NOISE Having now discussed the jamming performance of the most classical dif- ferentially coherent modulation technique, namely FH/MDPSK, we turn our attention to the SS application of differentially coherent detection to multiple-amplitude-and phase-shift-keying (MAPSK). In particular, we introduce the concept of differentially coherent detection of differentially phase-encod