Assuming that xn is a wide-sense stationary random process

Question

Suppose that a signal x(n) is recorded and that, due to measurement errors, there are outliers in the data, i.e., for some values of n there is a large error in the measured value of x (n). Instead of eliminating these data values, suppose that we perform a minimum mean-square interpolation as follows. Given a "bad" data value at time n = n₀, consider an estimate for x (no) of the form

(a) Assuming that x(n) is a wide-sense stationary random process with autocorrelation sequence r_x(k), find the values for a and b that minimize the mean-square error

(b) If r_x(k) = (0.5)^|k|, evaluate the mean-square error for the interpolator found in part (a).

(c) Discuss when it may be better to use an estimator of the form or explain why such an estimator should not be used.

(d) Given an autocorrelation sequence r_x(k), derive the Wiener-Hopf equations that define the optimum filter for interpolating x(n) to produce the best estimate of x(n₀) in terms of the 2 p data values

x(n₀ - 1), x(n₀ - 2), ... , x(n₀ - p) and x(n₀ + 1), x(n₀ + 2), ... , x(n₀ + p)

(e) Find an expression for the minimum mean-square error for your estimate in part (d).