Wide-sense stationary random process

Reference no: EM131834787

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).

(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).

Reference no: EM131834787

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