2 - 1/1 (2.27) and, hence, Q2nn/M = Pr{|42-441 > 0}==1 = - Pr{|42 - 41/00} (2.28) Consider the product (asterisk denotes complex conjugate) = RR2ei(42-41). (2.29) Figure 2.1. A graphical representation of (2.25). Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming The above product can also be written in the form (2.30) Thus, using (2.29) and (2.30) in (2.28) results in the equivalent relation - Oje-jno] < 0}. (2.31) Equation (2.31) can be given a geometric interpretation as in Figure 2.2. Here the vector OQ (a line drawn from point O to either point Q) represents Figure 2.2. A geometric interpretation of (2.31). Differentially Coherent Modulation Techniques the complex number whose argument is required in (2.31), i.e.,. (2.32) Thus, in terms of the geometry in Figure 2.2, (2.31) may be written in the alternate form Q2nn/M =1 Pr{point Q is within the 20 wedge} =1 - pr{point Q lies along the line AB}. (2.33) Considering separately the cases where point P falls outside and inside the 20 wedge, which expressed mathematically corresponds to the inequal- ities (see Figure 2.3) (2.34) then after much routine trigonometry, it can be shown that Q2nn/M Q2no 1; B 1 (2.35) where u(B) is the unit step function and (2.36) [(2n 1)A] sin [ (2n-1)8]=RS Figure 2.3. The geometry needed to establish (2.34). Performance of FH/MDPSK in the Presence of Partial-Band Multitone Jamming Note that (2.37) For n = 1, the appropriate result analogous to (2.35) is O<B<B1 B = 1 (2.38) (2.39) Finally, for n = M/2, we have the result 0<B<BM12-1 B 1 1 BM/2-1 (2.40) Also, lim Q = = 1. (2.41) As an example, Figure 2.4 is a plot of Q2no;n = 1, 2, 3, 4, 8 versus B for M = 16. These probabilities are computed from (2.35), (2.38), and (2.40). Using these results in (2.20), Figure 2.5 illustrates the product (E/N) X Pb (16) versus B. This curve has a maximum value of 1.457 at B = 0.1614, which, from Differentially Coherent Modulation Techniques (2.19), corresponds to the optimal (worst case) jamming strategy. 9.597 Eb/NJ 9.597 Eb/NJ