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76
Signals and Spectra Chap. 2
PROOF.
w(t) =
Taking the Fourier transform of both sides, we obtain
where the integral representation for a delta function, Eq. (2–48), was used.
This theorem indicates that a periodic function always has a line (delta function) spectrum,
with the lines being at f = nf 0 and having weights given by the c n values. An illustration
of that property was given by Example 2–5, where c 1 = -jA/2 and c -1 = jA/2 and the other
c n ’s were zero. It is also obvious that there is no DC component, since there is no line at f = 0
(i.e., c 0 = 0). Conversely, if a function does not contain any periodic component, the spectrum
will be continuous (no lines), except for a line at f = 0 when the function has a DC component.
It is also possible to evaluate the Fourier coefficients by sampling the Fourier transform
of a pulse corresponding to w(t) over a period. This is shown by the following theorem:
THEOREM.
n=q
a c n e jnv 0t
n=-q
W(f) =
L
q
-q
n=q
= a
n=-q
n=q
a a c n e jnv0t b e -jvt dt
n=-q
c n
L
q
-q
e -j2p(f-nf n=q
0)t
dt = a
n=-q
c n d(f - nf 0 )
If w(t) is a periodic function with period T 0 and is represented by
w(t) =
n=q
a
n=-q
h(t - nT 0 ) =
n=q
a
n=-q
c n e jnv 0t
(2–110)
where
w(t), ƒtƒ 6 T 0
2
h(t) = c
0, t elsewhere
(2–111)
then the Fourier coefficients are given by
c n = f 0 H(nf 0 )
(2–112)
where H(f) = [h(t)] and f 0 = 1/T 0 .
Proof. w(t) = h(t - nT 0 ) = h(t) * d(t - nT 0 ) (2–113)
q
a
n=-q
q
a
n=-q
where * denotes the convolution operation. Thus,
q
w(t) = h(t) * a d(t - nT 0 )
n=-q
(2–114)