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72
Signals and Spectra Chap. 2
2. If w(t) is real and even [i.e., w(t) = w(-t) ],
Im [c n ] = 0
(2–91)
3. If w(t) is real and odd [i.e., w(t) = -w(-t)],
Re [c n ] = 0
4. Parseval’s theorem is
1
T 0 La
(See Eq. (2–125) for the proof.)
a+T 0
|w(t)| 2 dt =
n=q
a
n=-q
|c n | 2
(2–92)
5. The complex Fourier series coefficients of a real waveform are related to the quadrature
Fourier series coefficients by
c n = d
[See Eqs. (2–96), (2–97), and (2–98).]
1
2 a n - j 1 2 b n, n 7 0
a 0 , n = 0
1
2 a -n + j 1 2 b -n, n 6 0
(2–93)
6. The complex Fourier series coefficients of a real waveform are related to the polar
Fourier series coefficients by
(2–94)
[See Eqs. (2–106) and (2–107).]
Note that these properties for the complex Fourier series coefficients are similar to those of
the Fourier transform as given in Sec. 2–2.
Quadrature Fourier Series
1
2 D lw n , n 7 0
c n = d D 0 , n = 0
1
2 D lw -n -n, n 6 0
The quadrature form of the Fourier series representing any physical waveform w(t) over the
interval a 6 t 6 a + T 0 is
w(t) =
n=q
a a n cos nv 0 t +
n=0
n=q
a b n sin nv 0 t
n=1
(2–95)