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Random Processes and Spectral Analysis Chap. 6
and
f 0 ! f
q
x (f)
= f q df
L 0
P x (l) dl Q
L0
(6–102)
As a sketch of a typical bandpass PSD will verify, the quantity given by the radical in
Eq. (6–100) is analogous to s f . Consequently, the factor of 2 is needed to give a reasonable
definition for the bandpass bandwidth.
Example 6–9 EQUIVALENT BANDWIDTH AND RMS
BANDWIDTH FOR A RC LPF
To evaluate the equivalent bandwidth and the RMS bandwidth for a filter, white noise may be
applied to the input. The bandwidth of the output PSD is the bandwidth of the filter, since the
input PSD is a constant.
For the RC LPF (Fig. 6–9), the output PSD, for white noise at the input, is given by Eq.
(6–88). The corresponding output autocorrelation function is given by Eq. (6–89). When we substitute
these equations into Eq. (6–96), the equivalent bandwidth for the RC LPF is
B eq =
R y(0)
2 y (0)
= N 0>4RC
2A 1 2 N 0B
= 1
4RC hertz
Consequently, for a RC LPF, from Eq. (2–147), B 3dB = 1(2pRC), and
(6–103)
B eq = p 2 B 3dB
(6–104)
The RMS bandwidth is obtained by substituting Eqs. (6–88) and (6–90) into Eq. (6–98). We obtain
B rms = T
L
q
-q
f 2 y (f)df
R y (0)
1
= C 2p 2 RC L
q
-q
f 2
(B 3dB ) 2 + f 2 df
(6–105)
Examining the integral, we note that the integrand becomes unity as f → ;q, so that the value of
the integral is infinity. Thus, B rms =qfor an RC LPF. For the RMS bandwidth to be finite, the
PSD needs to decay faster than 1| f | 2 as the frequency becomes large. Consequently, for the RC
LPF, the RMS definition is not very useful.
6–6 THE GAUSSIAN RANDOM PROCESS
DEFINITION.
A random process x(t) is said to be Gaussian if the random variables
x 1 = x(t 1 ), x 2 = x(t 2 ), Á , x N = x(t N )
(6–106)
have an N-dimensional Gaussian PDF for any N and any t 1 , t 2 , ..., t N .