i-th signalling interval is the trans- mitted signal of (2.58) to which the channel has added a random phase shift O and an AWGN which has the usual narrowband representation (repeated here for convenience) n(t) = V2[N((t)cos(wot+0)- N(t)sin(wot 0)], (2.61) where Nc(t) and Ns(t) are statistically independent low-pass Gaussian noise processes with single-sided power spectral density N w/Hz. Thus, the received signal in the i-th signalling interval is of the form = = (2.62) The receiver first performs in-phase and quadrature carrier demodulation with a pair of quadrature reference signals of known frequency wo but unknown phase The results of these demodulations are then passed through integrate-and-dump (I & D) filters whose outputs are given by4 = ATcos(A) + 0a) + nc cos - NS , (2.63) where (2.64) The receiver next generates the equivalent envelope and phase of the I & D outputs, namely, (2.65) 4Without any loss in generality, we shall set 0 = 0 for simplicy of notation. Performance of DQASK in the Presence of Additive White Gaussian Noise Finally, the differential phase n(i) - n(i-1) is formed and used to produce the in-phase and quadrature decision variables Vii cos (n (i) - n(i-1) and - V(i) sin (n(i) - n(i-1) upon which K-level decisions (âm and bn) are made. At this point, it is convenient to redraw Figure 2.8 in its equivalent form illustrated in Figure 2.9 by recognizing that = n(i-1) Vc(i) Vs(i) - = -Vii n(i) cos n(1) sin n(i-1) Vs(i) (cii) (2.66) Figure 2.9 has the advantage of resembling a conventional coherent QASK receiver [8] with a noisy carrier demodulation reference and thus its error probability performance can be obtained almost by inspection. In particu- lar, from (2.63) and (2.66), we obtain the decision variables U(i) = - - - - (2.67) Letting (2.68) and using (2.59), we can rewrite (2.67) as UATss(a1s11 = = - cos + N2 sin ((i-1) U(i) = ATcos( + + nc cos(A(i-1) + n&-1) + NS sin(A(i-1)+ nä-1) = N 1 sin (i-1) + N2 cos where N1 = ns cos A(i-1) - nc sin ((i-1) N2 = NS sin ((i-1) + nc cos ((i-1) (2