Generalize the fading problem of binary non-coherent FSK signaling to the M-ary case. Let the ith hypothesis be of the form

i = 1, 2,......., M; 0 ‰¤ t ‰¤ T_s

where G_i is Rayleigh, θ_i is uniform in (0, 2Ï€), E_i is the energy of the unperturbed ith signal of duration T_s, and |ω_i - ω_j| >> T^-1_s, for i ‰  j, so that the signals are orthogonal. Note that G_i cos θ_i and -G_i sin θ_i  are Gaussian with mean zero; assume their variances to be σ².

(a) Find the likelihood ratio test and show that the optimum correlation receiver is identical to the one shown in Figure 11.8(a) with 2M correlations, 2M squarers, and M summers where the summer with the largest output is chosen as the best guess (minimum P_E) for the transmitted signal if all E_i's are equal. How is the receiver structure modified if the E_i 's are not equal?

(b) Write down an expression for the probability of symbol error.

Transcribed Image Text:

2E cos(@,t + 0;) + n(t) Н, : у() %3D G;