Investigate the use of diversity to improve the performance of binary non-coherent FSK signaling over the flat-fading Rayleigh channel. Assume that the signal energy E_sis divided equally among N sub paths, all of which fade independently. For equal SNRs in all paths, the optimum receiver is shown in Figure 11.13.

(a) Referring to Problem 6.37 of Chapter 6, show that Y₁ and Y₂ are chi-squared random variables under either hypothesis.

(b) Show that the probability of error is of the form 

where 

(c) Plot P_E versus for N = 1, 2, 3,..., and show that an optimum value of N exists that minimizes P_E for a given SNR.

Transcribed Image Text:

R1,1 Subpath No. 1 System of Figure 9.8 (a) R2, 1? Y1 R1,22 Subpath No. 2 System of Figure 9.8 (a) Choose largest 2 Σ Y2 R1, N Subpath No. N System of Figure 9.8 (a) N-1 P, = a" E (N+/-)a-a 1 PE = aN Σ (1 – a) j=0