Determine optimum coefficients of equalizer as a function

Question

Solve the following problem:

Consider the problem of equalizing the discrete-time equivalent channel shown in Figure. The information sequence {I_n} is binary (±1) and uncorrelated. The additive noise {ν_n} is white and real-valued, with variance N₀. The received sequence {y_n} is processed by a linear three-tap equalizer that is optimized on the basis of the MSE criterion.

a. Determine the optimum coefficients of the equalizer as a function of N₀.

b. Determine the three eigenvalues λ₁, λ₂, and λ₃ of the covariance matrix and the corresponding (normalized to unit length) eigenvectors v₁, v₂, v₃.

c. Determine the minimum MSE for the three-tap equalizer as a function of N₀.

d. Determine the output SNR for the three-tap equalizer as a function of N₀. How does this compare with the output SNR for the infinite-tap equalizer? For example, evaluate the output SNR for these two equalizers when N₀ = 0.1.