Calculate the autocorrelation function

Question

Response to the following problem:

The Gauss-Markov model for a time-varying channel is given by h(m + 1) = √ 1 - αh(m) + αw(m + 1)
where {w(m)} is a sequence of iid CN (0, 1) random variables independent of h(0) ∼ CN (0, 1). The sampling time is Ts. The coherence time of this channel is controlled by the choice of parameter α.

1. Calculate the autocorrelation function of the sequence {h(m)} denoted by R_h(m).

2. Define coherence time as that corresponding to Rh (m) = 0.5. Determine the value of α in terms of T_s and the coherence time T_c.

3. Suppose that {h(m} is transmitted from the receiver to the transmitter with a delay of T_s. The transmitter predicts the value of h(m), say hˆ (m), from the past samples h(m - n) and h(m - n - 1). Thus

hˆ(m) = b₁h(m - n) + b₂h(m - n - 1)

where the prediction coefficients b1 and b2 are determined to minimize the MSE

E[|e|²] = E[|h(m) - hˆ(m)|²]

Determine b₁ and b₂ that minimize MSE.