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444
Random Processes and Spectral Analysis Chap. 6
or
a S N b =
out
2A 0
2
RC
N 0 [1 + (2pf 0 RC) 2 ]
The reader is encouraged to use calculus to find the value of RC that maximizes the SNR.
It is RC = 1(2pf 0 ). Thus, for maximum SNR, the filter is designed to have a 3-dB bandwidth
equal to f 0 .
6–5 BANDWIDTH MEASURES
Several bandwidth measures were defined in Sec. 2–9, namely, absolute bandwidth, 3-dB
bandwidth, equivalent bandwidth, null-to-null (zero-crossing) bandwidth, bounded bandwidth,
power bandwidth, and the FCC bandwidth parameter. These definitions can be applied
to evaluate the bandwidth of a wide-sense stationary process x(t), where x ( f ) replaces
|H( f )| 2 in the definitions. In this section, we review the equivalent bandwidth and define a
new bandwidth measure: RMS bandwidth.
Equivalent Bandwidth
For a wide-sense stationary process x(t), the equivalent bandwidth, as defined by Eq. (2–192),
becomes
q
1
B cq =
x (f) df =
R x(0)
(6–96)
x (f 0 ) L0
2 x (f 0 )
where f 0 is the frequency at which x (f) is a maximum. This equation is valid for both bandpass
and baseband processes. (f 0 = 0 is used for baseband processes.)
RMS Bandwidth
The RMS bandwidth is the square root of the second moment of the frequency with respect to
a properly normalized PSD. In this case, f, although not random, may be treated as a random
q
variable that has the “density function” x (f)> - q x (l) dl. This is a nonnegative function
in which the denominator provides the correct normalization so that the integrated value of
the ratio is unity. Thus, the function satisfies the properties of a PDF.
DEFINITION.
If x(t) is a low-pass wide-sense stationary process, the RMS bandwidth is
where
B rms = 3f 2
q
q
f 2 = f 2 x (f)
q df = L f 2 x (f) df
-q
q
L -q
J x (l) dl K
x (l) dl
L-q
L-q
(6–97)
(6–98)