ECE 461 Fall 2006Differential Phase Modulation and Detection• Consider MPSK signaling on an ideal AWGN channel with phase offset φ(t) that may changewith time. Over N symbol intervalsr(t) =NXn=1√Eejθng(t − (n − 1)Ts) ejφ(t)+ w(t) .Suppose φ(t) changes slowly with time so that we can assume that it is constant over two consec-utive symbol intervals.• For standard MPSKθn= θmn=2π(mn− 1)M, mn∈ {1, 2, . . . , M } .We know that this scheme performs poorly if we cannot estimate φ at the receiver. But ifφ(t) changes slowly, a differential modulation approach can be taken where the sequence {θn} isgenerated from {mn} asθn− θn−1= δmn=2π(mn−1)M(with θ0= 0) .• Sufficient statistics for demodulation over the N-symbol interval are given byRn= hr(t), g(t − (n − 1)Ts)i, n = 1, 2, . . . , N.Note that under the slowly changing phase assumptionRn=√E ejθnejφn+ Wnwhere φn≈ φn−1for all n.• Since the information about mnis contained in the phase difference between Rnand Rn−1, it isconvenient to form the statistics:Yn=RnR?n−1√E≈√Eejδmn+ Xn, n = 2, 3, . . . , N (with Y1= R1)whereXn= Wne−jφne−jθn−1+ W?n−1e−jφne−jθn+WnW?n−1√ESince {Rn} can be recovered from {Yn}, there is no loss of optimality if we use {Yn} in place of{Rn} for detection.• The statistics {Yn} are related to the symbols {mn} in the same way as in standard PSK withcoherent detection, except that {Xn} is not a sequence of i.i.d. CN(0, N0) random variables. Thefact that {Xn} are correlated implies that symbol-by-symbol detection is not optimum, even if g(t)satisfies the zero-ISI condition and the symbols are independent. The MAP (or MPE) detector forthis problem is quite complicated and impractical, and hence the following suboptimum detectoris used.cV.V. Veeravalli, 2006 1• Differential Detector. This detector makes a decision on mnbased on Ynalone. In particular, ˆmnis chosen via a minimum distance criterion asˆmn= arg minm|yn−√Eejδm|2“pretending” that Xnis a zero-mean PCG random variable.• Performance of Differential Detector. At high SNR, we may ignore the noise cross-term in theequation for Xn, and conclude that Xnis approximately CN(0, 2N0). Then it is clear that theperformance of DPSK is worse than PSK with coherent demodulation by approximately 3 dB.An exact analysis of the performance of the differential detector can be done in the special casesof DBPSK and DQPSK (see textbook). For DBPSK, we get the surprisingly simple expression:Pb=12exp(−γb) .cV.V. Veeravalli, 2006