563489578934

Problems 485
1
2
x (f) = e
N 0, ƒ f ƒ … 2B
0, f elsewhere
(a) Find the autocorrelation function for the output y(t).
(b) Find the PDF for y(t).
(c) When are the two random variables y 1 = y(t 1 ) and y 2 = y(t 2 ) independent?
6–27 A linear filter evaluates the T-second moving average of an input waveform, where the filter
output is
y(t) = 1 T L
and x(t) is the input. Show that the impulse response is h(t) = (1T ) P (tT ).
6–28 For Problem 6–27 show that
R y (t) = 1 T L
T
-T
t+(T/2)
t-(T/2)
x(u) du
a1 - |u|
T bR x(t - u) du
If R x (t) = e -|t| and T = 1 sec, plot R y (t), and compare it with R x (t).
★ 6–29 As shown in Example 6–8, the output signal-to-noise ratio of an RC LPF is given by Eq. (6–95)
when the input is a sinusoidal signal plus white noise. Derive the value of the RC product such
that the output signal-to-noise ratio will be a maximum.
6–30 Assume that a sine wave of peak amplitude A 0 and frequency f 0 , plus white noise with n (f) = N 0 2,
is applied to a linear filter. The transfer function of the filter is
1
H (f) = B (B - |f| ), |f| 6 B
L
0, f elsewhere
where B is the absolute bandwidth of the filter. Find the signal-to-noise power ratio for the filter
output.
6–31 For the random process x(t) with the PSD shown in Fig. P6–31, determine
(a) The equivalent bandwidth.
(b) The RMS bandwidth.
p x (f)
2.0
5 kHz
Figure P6–31
+5 kHz
f