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Random Processes and Spectral Analysis Chap. 6
fundamental concepts in electrical engineering) can be related to the moments of an ergodic
random process. In the following summary of these relationships, x(t) is an ergodic random
process that may correspond to either a voltage or a current waveform:
1. DC value:
X dc !8x(t)9 K xq = m x
(6–74)
2. Normalized DC power:
P dc ![8x(t)9] 2 K (xq) 2
(6–75)
3. RMS value:
q
X rms ! 28x 2 (t)9 K 4(x 2 ) = 2R x (0) = x (f)df
CL
4. RMS value of the AC part:
(X rms ) ac ! 38(x(t) - X dc ) 2 9 K 3(x - xq) 2
= 3x 2 - (xq) 2 = 3R x (0) - (xq) 2
q
= C x (f) df - (xq) 2 = s x = standard deviation
L
-q
5. Normalized total average power:
q
P ! 8x 2 (t)9 K x 2 = R x (0) = x (f) df
L
-q
-q
(6–76)
(6–77)
(6–78)
6. Normalized average power of the AC part:
P ac ! 8(x(t) - X dc ) 2 9 K (x - xq) 2
= x 2 - (xq) 2 R x (0) - (xq) 2
q
= x (f) df - (xq) 2 = s 2 x = variance
L
-q
(6–79)
Furthermore, commonly available laboratory equipment may be used to evaluate the
mean, second moment, and variance of an ergodic process. For example, if x(t) is a voltage waveform,
xq can be measured by using a DC voltmeter, and s x can be measured by using a
“true RMS” (AC coupled) voltmeter. † Employing the measurements, we easily obtain the second
moment from x 2 = s 2 At higher frequencies (e.g., radio, microwave, and optical), x 2
x + (xq) 2 .
2
and s can be measured by using a calibrated power meter. That is, s 2 x = x 2 x = RP, where R is
the load resistance of the power meter (usually 50 Ω) and P is the power meter reading.
† Most “true RMS” meters do not have a frequency response down to DC. Thus, they do not measure the true
RMS value 38x 2 (t)9 = 3x 2 ; instead, they measure s.