Consider a stable linear time-invariant system with a real input x[n], a real impulse response h[n], and
Question:
Consider a stable linear time-invariant 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 σ2x. The system function is
where we assume the αk’s and bk’s are real for this problem. The input and output satisfy the following difference equation with constant coefficients:
If all the αk’s are zero, y[n] is called a moving-average (MA) linear random process. If all the bk’s 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 Фyy[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 Фyy[m] to show that
It can be shown that, given Фyy[m] for m = 0, 1, …. , N, we can always solve uniquely for the values of the αk’s 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.
Step by Step Answer:
Discrete Time Signal Processing
ISBN: 978-0137549207
2nd Edition
Authors: Alan V. Oppenheim, Rolan W. Schafer