Consider a stable LTI system with a real input x[n], a real impulse response h[n], and o...

Consider a stable LTI system with a real input x[n], a real impulse response h[n], and output y[n]. Assume that the input x[n] is white noise with zero mean and variance σ² x. The system function is

where we assume the a_ks and b_ks are real for this problem. The input and output satisfy the following difference equation with constant coefficients:

If all the a_ks are zero, y[n] is called a moving-average (MA) linear random process. If all the b_ks are zero, except for b₀, then y[n] is called an autoregressive (AR) linear random process. If both N and M are nonzero, then y[n] is an autoregressive moving-average (ARMA) linear random process.

(a) Express the autocorrelation of y[n] in terms of the impulse response h[n] of the linear system.

(b) Use the result of part (a) to express the power density spectrum of y[n] in terms of the frequency response of the system.

(c) Show that the autocorrelation sequence Φ_yy[m] of an MA process is nonzero only in the interval |m| ≤ M.

(d) Find a general expression for the autocorrelation sequence for an AR process.

(e) Show that if b₀ = 1, the autocorrelation function of an AR process satisfies the difference equation

(f) Use the result of part (e) and the symmetry of Φ_yy[m] to show that

It can be shown that, given Φ_yy[m] for m = 0, 1, . . . , N, we can always solve uniquely for the values of the a_ks and σ²x for the random-process model. These values may be used in the result in part (b) to obtain an expression for the power density spectrum of y[n]. This approach is the basis for a number of parametric spectrum estimation techniques. (For further discussion of these methods, see Gardner, 1988; Kay, 1988; and Marple, 1987.)