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80
Signals and Spectra Chap. 2
PROOF. For periodic w(t), the Fourier series representation is valid over all time and
may be substituted into Eq. (2–12) to evaluate the normalized (i.e., R = 1)
power:
P w = ha an
c n e jnv0t 2
b i = h an
a m
c n c m
* e jnv 0t e -jmv 0t i
= an a m
c n c m
* 8e j(n-m)v 0t 9 = an a m
c n c m
*d nm = an
c n c n
*
or
P w = an ƒc n ƒ 2
(2–125)
Equation (2–124) is a special case of Parseval’s theorem, Eq. (2–40), as applied to
power signals.
EXAMPLE 2–15 AVERAGE POWER FOR A SQUARE WAVE
Using Eq. (2–125), evaluate the approximate average power for a square wave. Compare this
approximate value of the power with the exact value. See Example2_15.m for the solution.
Power Spectral Density for Periodic Waveforms
THEOREM.
For a periodic waveform, the power spectral density (PSD) is given by
ƒ
T 0 = 1/f 0 where is the period of the waveform and the {c n } are the corresponding
(f) = a ƒc 2 n d( f - nf 0 )
(2–126)
n=-q
Fourier coefficients for the waveform.
q
PROOF. Let w(t) = a -q c ne jnv 0t
. Then the autocorrelation function of w(t) is
R(t) = 8w * (t)w(t + t)9
or
q
= h a c n
*e -jnv 0t
n=-q
n=q
q
a
m=-q
c m e jmv 0(t+t) i
R(t) =
q q
a
a
n=-q m=-q
c n
*c m e jmv 0t 8e jv 0(m-n)t 9
But 8e jv0(n-m)t 9 = d nm , so this reduces to
R(t) =
q
a ƒc n ƒ 2 e jnv 0t
n=-q
(2–127)