The output ynnbspof the smoothing filter in example 115 has

Question

The output Y_n of the smoothing filter in Example 11.5 has µ_Y = 1 and autocorrelation function

Y_n is the input to another smoothing filter with impulse response

The output of this second smoothing filter is W_n.

(a) What is µ_W, the expected value of the output of the second filter?

(b) What is the autocorrelation function of W_n?

(c) What is the variance of W_n?

(d) Wn is a linear function of the original input process X_n in Example 11.5:

What is the impulse response g_i of the combined filter

Example 11.5

A wide sense stationary random sequence X n with µ_X = 1 and autocorrelation function R_X [n] is the input to the order M -1 discrete-time moving-average filter h_n where

For the case M = 2, find the following properties of the output random sequence Y_n: the expected value µ_Y, the autocorrelation R_Y [n], and the variance Var[Yn].