Page 252 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES

Delay
T
I
Fig. 6-8
6.64. Verify Eq. (6.67).
Hint: Proceed as in Prob. 6.24.

6.65. Suppose that the input to the discrete-time filter shown in Fig. 6-9 is a discrete-time white noise with
average power a2. Find the power spectral density of Y(n).
Ans. SY(Q) = oZ(l + az + 2a cos R)

delay
Fig. 6-9

6.66. Using the Karhunen-Loeve expansion of the Wiener process, obtain the Karhunen-Lobe expansion of the
white normal noise.
Hint: Take the derivative of Eq. (6.1 75) of Prob. 6.39.

where W, are independent normal r.v.'s with the same variance a'.

6.67. Let Y(t) = X(t) + W(t), where X(t) and W(t) are orthogonal and W(t) is a white noise specified by Eq. (6.43)
or (6.45). Let $,(t) be the eigenfunctions of the integral equation (6.86) and 1, the corresponding eigenvalues.
(a) Show that 4,(t) are also the eigenfunctions of the integral equation for the Karhunen-Loeve expansion
of Y(t) with Ry(t, s).
(b) Find the corresponding eigenvalues.
Hint: Use the result of Prob. 6.58.
Ans. (b) An + a2

6.68. Suppose that

where Xn are r.v.'s and o, is a constant. Find the Fourier transform of X(t).
Ans. X(w) = C 2nXn6(w - no,)
n