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Random Processes and Spectral Analysis Chap. 6
Example 6–7 OUTPUT AUTOCORRELATION AND PSD
FOR A RC LOW-PASS FILTER
A RC low-pass filter (LPF) is shown in Fig. 6–9. Assume that the input is an ergodic random
process with a uniform PSD:
Then the PSD of the output is
x (f) = 1 2 N 0
y (f) = ƒ H(f) ƒ 2 x (f)
which becomes
y (f) =
1
2 N 0
1 + (f/B 3dB ) 2
(6–88)
where B 3db = 1/(2pRC). Note that y (B 3dB )> y (0) is 2 , so B 3dB = 1(2pRC) is indeed the 3-dB
bandwidth. Taking the inverse Fourier transform of y(f), as described by Eq. (6–88), we obtain
the output autocorrelation function for the RC filter.
R y (t) = N 0 |>(RC)
e-|t (6–89)
4RC
The normalized output power, which is the second moment of the output, is
1
P y = y 2 = R y (0) = N 0
4RC
(6–90)
Furthermore, the DC value of the output, which is the mean value, is zero, since †
e
Y DC = m y = lim (6–91)
C e: 0 y (f) df = 0
e 7 0 L-e
The variance of the output is also given by Eq. (6–90) because s 2 y = y 2 - m 2 y , where m y = 0.
R
x(t)
C
y(t)
1
H(f)=
f
1+ j(
B 3dB
Figure 6–9
(
1
where B 3dB =
2pRC
RC LPF
† When the integral is nonzero, the value obtained is equal to or larger than the square of the mean value.