Determine the error rate performance

Question

Solve the following problem:

Consider a binary communication system for transmitting a binary sequence over a fading channel. The modulation is orthogonal FSK with third-order frequency diversity (L = 3). The demodulator consists of matched filters followed by square-law detectors. Assume that the FSK carriers fade independently and identically according to a Rayleigh envelope distribution. The additive noises on the diversity signals are zero-mean Gaussian with autocorrelation functions

The noise processes are mutually statistically independent.

a. The transmitted signal may be viewed as binary FSK with square-law detection, generated by a repetition code of the form

1 → c₁ = [1 1 1], 0 → c₀ = [0 0 0]

Determine the error rate performance P_bh for a hard-decision decoder following the square-law-detected signals.

b. Evaluate P_bh for Y¯ c = 100 and 1000.

c. Evaluate the error rate Pbs for Y¯ c = 100 and 1000 if the decoder employs soft-decision decoding.

d. Consider the generalization of the result in (a). If a repetition code of block length L (L odd) is used, determine the error probability P_bh of the hard-decision decoder and compare that with Pbs, the error rate of the soft-decision decoder. Assume γ 1.