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Sec. 6–5 Bandwidth Measures 445
The RMS measure of bandwidth is often used in theoretical comparisons of communication
systems because the mathematics involved in the RMS calculation is often easier to carry out
than that for other bandwidth measures. However, the RMS bandwidth is not easily measured
with laboratory instruments.
THEOREM.
Proof.
For a wide-sense stationary process x(t), the mean-squared frequency is
We know that
f 2 1
= c -
(2p) 2 R(0) d d2 R x (t)
dt 2 `
q
R x (t) = x (f)e j2pft dt
L
Taking the second derivative of this equation with respect to t, we obtain
d 2 q
R x (t)
dt 2 = x (f)e j2pft (j2pf) 2 df
L
Evaluating both sides of this equation at t = 0 yields
d 2 R x (t)
dt 2 `
t = 0
Substituting this for the integral in Eq. (6–98), that equation becomes
-q
-q
= (j2p) 2 L
q
-q
t = 0
f 2 x (f) df
(6–99)
q
f 2 = L f 2 x (f) df
-q
q
x (l) dl
L-q
which is identical to Eq. (6–99).
=
-
1
(2p) 2 c d2 R x (t)
dt 2 d
t = 0
R x (0)
The RMS bandwidth for a bandpass process can also be defined. Here we are interested
in the square root of the second moment about the mean frequency of the positive frequency
portion of the spectrum.
DEFINITION.
If x(t) is a bandpass wide-sense stationary process, the RMS bandwidth
is
where
B rms = 23(f - f 0 ) 2
q
(f - f 0 ) 2 = (f - f 0 ) 2 x (f)
q df
L 0
P
x (l) dl Q
L0
(6–100)
(6–101)