The signal y[n] is the output of an LTI system with input x[n], which is zero-mean white...

The signal y[n] is the output of an LTI system with input x[n], which is zero-mean white noise. The system is described by the difference equation

(a) What is the z-transform Φ_yy(z) of the autocorrelation function Φ_yy[n]?

Sometimes it is of interest to process y[n] with a linear filter such that the power spectrum of the linear filter’s output will be flat when the input to the linear filter is y[n].This procedure is known as “whitening” y[n], and the linear filter that accomplishes the task is said to be the “whitening filter” for the signal y[n]. Suppose that we know the autocorrelation function Φ_yy[n] and its z-transform Φ_yy(z), but not the values of the coefficients a_k and b_k.

(b) Describe a procedure for finding a system function Hw(z) of the whitening filter.

(c) Is the whitening filter unique?