will be determined, which, as before, consists of specifying the worst case partial-band fraction and the corresponding maximum average error prob- ability. To simplify the ensuing analysis, we shall ignore the presence of the additive white Gaussian noise (AWGN) background in favor of the degra- dation produced by the jamming interference. Modification of the results to include the quiescent noise contribution would follow along the same lines as the approach taken in the previous chapter for coherent detection of these same modulations. 2.1 PERFORMANCE OF FH/MDPSK IN THE PRESENCE OF PARTIAL-BAND MULTITONE JAMMING In a differentially encoded frequency-hopped M-ary differential phase- shift-keyed (FH/MDPSK) system, the information to be transmitted in the i-th signalling interval (i 1)T,sisits is conveyed by appropriately select- ing one of M phases (2.1) and adding it to the total accumulated phase in the (i - 1)-st signalling inter- val of a constant amplitude (A), fixed frequency (assumed known at the receiver) sinusoid. Typically M = 2K with K integer, and these are the only cases we shall consider in detail. Furthermore, since the derivation of the per- formance of FH/MDPSK in the presence of a partial-band multitone jam- mer will rely largely on certain geometric relations, it is expedient to deal with both the signal and the jammer as phasors. Thus, the transmitted signal s(1)(t) in the i-th signalling interval is conveniently represented in complex form by (2.2) where OG-1) is the total accumulated phase in the (i - 1)-st signalling interval and (ii) ranges over the set {0m} of (2.1). In the presence of multitone jamming inteference as characterized in the previous chapter, a jamming tone J(t), constant in both phase and magni- tude (amplitude), is added to the transmitted signal. Since, when the jam- Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming mer "hits," he is assumed1 to be of the same frequency as the signal, then we may also represent the jammer in complex form, namely, 