2.11. Normalized signal point constellation for QASK-16. Figure 2.12. A vector diagram representation of the signal and jammer in the i-th and (i - 1)-st transmission intervals (signal point in first quadrant). Differentially Coherent Modulation Techniques takes on values with Prob. 1/4 A1 = V18 with Prob. 1/2 (2.87) with Prob. 1/4. From the results of the previous section, we observe that signal point will be correctly detected if 0 < R2 cos(A2 - 01) 2 2 < R2 sin(A2 - 01) < 8. (2.88) The boundaries on the inequalities in (2.88) correspond to the x and y coor- dinates of the decision region indicated by the shaded area in Figure 2.12. Expanding the sine and cosine of the difference angle A2 - A1 and noting, from Figure 2.12, that R2 sin A2 3 + sin OJ B sin 01 B sin 0 1 n + sin (2.89) the inequalities of (2.88) can be rewritten as + 2BA1 cos OJ + (2.90) Dividing the numerator and denominator of each inequality in (2.90) by A1 Performance of FH/DQASK in the Presence of Partial-Band Multitone Jamming and simplifying results in + B sin OJ) (2.91) Let Pco(Aj;B,A1) denote the conditional probability of correctly de- tecting signal point 1 for given Aj, B, and A. . Then, from (2.91), for values of 01 such that <X1,3(Aj;B,A1) < 2 and Pco(Aj;B,A1) (2.92) <Y1,3(+);,B,A1)< 8 all other values of AJ in (0, 2), where we have defined the generalized functions (1+2(A) Bsine,+-(joose,-isin i,j==1,13. (2.93) Further defining (2.94) then, for a > 0, -00<X<a a<X<00 (2.95) otherwise. Differentially Coherent Modulation Techniques In view of (2.95), we may rewrite (2.92) as Pco(Aj;B,A1) = G(X1,3(A);B,A1) - 2) - G(X1,3(Aj;B,A1)) 1 G(Y1,3(Aj;,B,A1) - 2)]. (2.96) As our next example, consider the problem of correctly detecting signal point 10 of Figure 2.11. The vector diagram describing this situation is given in Figure 2.13. Noting again that the shaded area corresponds to the correct decision region, analogous to (2.88), we than have -2<R2c(2-01)<0 2 < R2 sin(A2 - < 00. (2.97) Once again expanding the sine and cosine of the difference 