Page 241 - Probability, Random Variables and Random Processes

CHAP. 61 ANALYSIS AND PROCESSING OF RANDOM PROCESSES 233

Verify Eq. (6.72).
From Eq. (6.71), we have

Taking the Fourier transform of Rdk), we obtain

Letting k + i - 1 = n, we get

The discrete-time system shown in Fig. 6-6 consists of one unit delay element and one scalar
multiplier (a < 1). The input X(n) is discrete-time white noise with average power a2. Find the
spectral density and average power of the output Y(n).

Fig. 6-6

From Fig. 6-6, Y(n) and X(n) are related by
Y(n) = aY(n - 1) + X(n)
The impulse response h(n) of the system is defined by
h(n) = ah(n - 1) + 6(n)

Solving Eq. (6.149), we obtain
h(n) = anu(n)
where u(n) is the unit step sequence defined by

Taking the Fourier transform of Eq. (6.150), we obtain