.70) Finally, recognizing that (2.69) resembles the decision variables for a coher- ent QASK receiver whose carrier demodulation reference signals are in error by n(i-1) radians, we can immediately write down an expression for V2 sin (word (i-I)Ts K- Level Phase Quantizer Modulator - V2 sin (wot-ta) Phase Estimator s(1;8) (i-I)Ts (i-1) Delay Ts vc(i) (i-I)Ts V2 COS (wor-ta) Phase Modulator uciii (i-1)Ts K Level Quantizer Figure 2.9. An alternate and equivalent implementation of the receiver shown in Figure 2.8. Performance of DQASK in the Presence of Additive White Gaussian Noise the symbol error probability of differentially coherent detected QASK (conditioned on the (i - 1)-st symbol SNR), namely, (2.71) where the probability density function (pdf) of the nor- malized phase n(i-1) in the (i - 1)-st signalling interval, is given by [2] xQ[-V21c 0; elsewhere (2.72) with the (i - 1)-st transmission interval symbol SNR 2(i-1) defined by (2.73) and Q(x) denoting, as in previous chapters, the Gaussian probability inte- gral. Also from (51) of [8], with replaced by n(i-1), X - - (2.74) In (2.74), the sum over j is for values 1, =3, (K 1) while the sums over k and l are for values 0, =2, 4, (K - 2). Also, (2.75) where (2.76) is the average symbol SNR of the QASK-K signal set with average power S defined in (2.56). Finally, the average symbol error probability, Ps, is obtained by averag- ing (2.71) over the pdf of y(i-1) To obtain this pdf, we note that, for a given K, the (i - 1)-st symbol signal power (A(i-1))2 ranges over K(K + 2)/8 dif- ferent values. K/2 of these values correspond to signal points on the diago- nal of any quadrant and occur with probability 4/K2. The remaining Differentially Coherent Modulation Techniques K(K - 2)/8 values correspond to off-diagonal signal points either above or below the diagonal of any quadrant and occur with probability 8/K2. Thus, 2(i-1) takes on the discrete set, R, of values 2(m2 + n2)/2; m, n = 1, (K - 1);msnandt the corresponding pdf is then m² ²²; =1,3,...,(K- m,n =1,3,..(