563489578934

Problems 487
6–42 A bandpass WSS random process v(t) is represented by Eq. (6–129a), where the conditions of
Eq. (6–129) are satisfied. The PSD of v(t) is shown in Fig. P6–42, where f c = 1 MHz. Using
MATLAB
(a) Plot x ( f ).
(b) Plot xy ( f ).
p v (f)
1.0
2 1.5 1 0.5 0.25 0.5 0.75 1.0 1.25 1.5 1.75
f
MHz
Figure P6–42
★ 6–43 The PSD of a bandpass WSS process v(t) is shown in Fig. P6–43. v(t) is the input to a
product detector, and the oscillator signal (i.e., the second input to the multiplier) is
5 cos (v c t + u 0 ), where f c = 1 MHz and u 0 is an independent random variable with a uniform
distribution over (0, 2p). Using MATLAB, plot the PSD for the output of the product
detector.
p v (f)
3.0
2.0
1.0
–1.5 –1.0 –0.5
0.25 0.5 0.75 1.0 1.25 1.5
Figure P6–43
f
MHz
6–44 A WSS bandpass process v(t) is applied to a product detector as shown in Fig. 6–11, where
u c = 0.
(a) Derive an expression for the autocorrelation of w 1 (t) in terms of R v (t). Is w 1 (t) WSS?
(b) Use R w1 (t) obtained in part (a) to find an expression for w1 (f). (Hint: Use the
Wiener–Khintchine theorem.)