Suppose we observe a random process Z (t) (without any noise) over a time interval (–∞, t. Based on this observation, we wish to predict the value of the same random process at time t + t_o. That is, we desire to design a filter with impulse response, h (t), whose output will be an estimate of Z (t + t_o):

(a) Find the Wiener– Hopf equation for the optimum (in the minimum mean square error (MMSE) sense) filter.

(b) Find the form of the Wiener filter if R_ZZ (t) = exp (– |τ|).

(c) Find an expression for the mean square error E [Ɛ² (t)] = E (Z (t + t_o) – Y (t)) ²].

Y(t) = Z(t+t) = . h(u)Z(t-u)du. %3D