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Random Processes and Spectral Analysis Chap. 6
6–32 If x(t) is a real bandpass random process that is wide-sense stationary, show that the definition of
the RMS bandwidth, Eq. (6–100), is equivalent to
B rms = 23f 2 - (f 0 ) 2
where f 2 is given by Eq. (6–98) or Eq. (6–99) and f 0 is given by Eq. (6–102).
6–33 In the definition for the RMS bandwidth of a bandpass random process, f 0 is used. Show that
f 0 =
1
2pR x (0) a dR N x (t)
b `
dt t = 0
where RN x (t) is the Hilbert transform of R x (t).
★ 6–34 Two identical RC LPFs are coupled in cascade by an isolation amplifier that has a voltage
gain of 10.
(a) Find the overall transfer function of the network as a function of R and C.
(b) Find the 3-dB bandwidth in terms of R and C.
6–35 Let x(t) be a Gaussian process in which two random variables are x 1 = x(t 1 ) and x 2 = x(t 2 ).
2 2
The random variables have variances of s 1 and s 2 and means of m 1 and m 2 . The correlation
coefficient is
r = (x 1 - m 1 )(x 2 - m 2 )/(s 1 s 2 )
Using matrix notation for the N = 2-dimensional PDF, show that the equation for the PDF of x
reduces to the bivariate Gaussian PDF as given by Eq. (B–97).
6–36 A bandlimited white Gaussian random process has an autocorrelation function that is specified
by Eq. (6–125). Show that as B → q, the autocorrelation function becomes
R n (t) = 1 2 N 0d(t).
★ 6–37 Let two random processes x(t) and y(t) be jointly Gaussian with zero-mean values. That is,
(x 1 , x 2 ,..., x N , y 1 , y 2 ,..., y M ) is described by an (N + M)-dimensional Gaussian PDF. The crosscorrelation
is
R xy (t) = x(t 1 )y(t 2 ) = 10 sin (2pt)
(a) When are the random variables x 1 = x(t 1 ) and y 2 = y(t 2 ) independent?
(b) Show that x(t) and y(t) are or are not independent random processes.
6–38 Starting with Eq. (6–121), show that
C y = HC x H T
(Hint: Use the identity matrix property, AA -1 A -1 A I, where I is the identity matrix.)
★ 6–39 Consider the random process
x(t)A 0 cos (v c t + u)
where A 0 and ω 0 are constants and u is a random variable that is uniformly distributed over the
interval (0, p2). Determine whether x(t) is wide-sense stationary.
★ 6–40 Referring to Prob. 6–39, find the PSD for x(t).
★ 6–41 Referring to Prob. 6–39, if u is uniformly distributed over (0, 2p), is x(t) wide-sense
stationary?