n 1. (25pts) Let s[n] be a zero-mean WSS process with the autocorrelation function Rs [k] 10(1).

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1. (25pts) Let s[n] be a zero-mean WSS process with the autocorrelation function Rs [k] 10(1). We obtain noisy measurements of s[n] as follows: z[n] = s[n] +v[n], Vn = where z[n] denotes the measurements and v[n] denotes zero-mean WSS measurement noise with the auto-correlation function R₂[k] = 26[k]. Here s[n₁] and v[n₂] are uncorrelated for all n1, n2. We would like to predict s[n+ 1] using z[n] and z[n 1] as follows s[n+ 1] = a z[n] + bz[n 1] +c, where ŝ[n+ 1] is the estimate of s[n+ 1]. Here, a, b, c are the real-valued coefficients we are going to choose. (a) Find the following: E[s[n+1] z[n]], E[s[n+1] z[n-1]], E[z[n] z[n-1]], E[z[n]²], E[z[n-1]²]. (5 x 3pts) (b) Find a, b and c so that the mean-square error E[(s[n+ 1] - ŝ[n+ 1])²] is minimized. (c) Find the mean-square error associated with s[n+1] of Part 1b. (5pts) (5pts) 1. (25pts) Let s[n] be a zero-mean WSS process with the autocorrelation function Rs [k] 10(1). We obtain noisy measurements of s[n] as follows: z[n] = s[n] +v[n], Vn = where z[n] denotes the measurements and v[n] denotes zero-mean WSS measurement noise with the auto-correlation function R₂[k] = 26[k]. Here s[n₁] and v[n₂] are uncorrelated for all n1, n2. We would like to predict s[n+ 1] using z[n] and z[n 1] as follows s[n+ 1] = a z[n] + bz[n 1] +c, where ŝ[n+ 1] is the estimate of s[n+ 1]. Here, a, b, c are the real-valued coefficients we are going to choose. (a) Find the following: E[s[n+1] z[n]], E[s[n+1] z[n-1]], E[z[n] z[n-1]], E[z[n]²], E[z[n-1]²]. (5 x 3pts) (b) Find a, b and c so that the mean-square error E[(s[n+ 1] - ŝ[n+ 1])²] is minimized. (c) Find the mean-square error associated with s[n+1] of Part 1b. (5pts) (5pts)