 bit error probabili- ties, namely, (2.15) the average bit error probability for MDPSK in the presence of multitone jamming is given by (2.16) Actually, the relation in (2.15) holds as an equality only for orthogonal sig- nal sets [5]. However, for low signal-to-jammer ratios, the right-hand side of (2.15) becomes a tight upper bound for the average bit error probability per- formance of FH/MDPSK. For binary DPSK (M = 2), the equality in (2.15) is exact. We shall see shortly that, for the evaluation of Q2TnIM, it is convenient to renormalize the problem in terms of the ratio of jamming power per tone J = J/Q to signal power S. Let B2 denote this ratio, i.e., (2.17) Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming Then, recalling from the previous chapter that the number of hop slots N in the total hop frequency band Wss is (2.18) then, the partial-band fraction p can be expressed in terms of B2 and the bit energy-to-jammer noise spectral density ratio EIN by (2.19) Using (2.19), we can rewrite (2.16) in the form (2.20) Before proceeding to the evaluation of = 1, 2, , M/2 we make the final observation that the per tone jamming-to-signal power ratio B2 can also be expressed in terms of the vector definitions of the signal and tone jamming interference. Since from (2.2) and (2.3) the signal power and jam- mer power per tone are given by (2.21) then equivalently from (2.17) we have that (2.22) 2.1.1 Evaluation of Q2nnIm In view of (2.22), Q2TTIIM of (2.9) may be restated in the normalized form Q2nn/M = - 0} (2.23) where (2.24) Zi-1e-ieRei (2.25) Differentially Coherent Modulation Techniques Note that, in obtaining (2.25) from (2.4), we have substituted for (ii) its assumed true value, namely, On = 2n/M - 2n0, and, since OJ is uniformly distributed, we have arbitrarily established the symmetry OG-1 = -nn/M = -no. Figure 2.1 is a graphical representation of (2.25) where we have further introduced the notation (2.26) Thus, using (2.25) and (2.26), arg(Z¹ - Z(i-1)) - = = nov2 - (nov(1) - 2n0 = 4/