Determine the optimum coefficients

Question

Solve the following problem:

Consider the channel model shown in Figure . {ν_n} is a real-valued white noise sequence with zero-mean and variance N₀. Suppose the channel is to be equalized by a DFE having a two-tap feed forward filter (c₀, c-₁) and a one-tap feedback filter (c₁). The {c_i} are optimized using the MSE criterion.

a. Determine the optimum coefficients and their approximate values for N₀

b. Determine the exact value of the minimum MSE and a first-order approximation appropriate to the case N₀

c. Determine the exact value of the output SNR for the three-tap equalizer as a function of N0 and a first-order approximation appropriate to the case N₀

d. Compare the results in (b) and (c) with the performance of the infinite-tap DFE.

e. Evaluate and compare the exact values of the output SNR for the three-tap and infinitetap DFE in the special cases where N₀ = 0.1 and 0.01. Comment on how well the three-tap equalizer performs relative to the infinite-tap equalizer.