Digital Signal Processing Reference
In-Depth Information
Time signals
Magnitude Spectra
1
1
0.5
δ
(
f
+
f
o
)
0.5
δ
(
f
−
f
o
)
0.5
0.5
0
t
, sec
0
f
, Hz
−2
f
o
=2
−0.5
−0.5
−1
−1
−1
−0.5
0
0.5
1
−5
0
5
Fig. 1.15
Sine and cosine (with the same f
o
) have identical magnitude spectra
using the result from the previous example. Similarly,
:
Ff
sin
ð
x
o
t
Þg¼
1
2j
d
ð
f
f
o
Þ
1
2
d
ð
f
þ
f
o
Þ
Hence, the magnitude spectra of sin(x
o
t) and cos(x
o
t) are identical, as shown in
Fig.
1.15
. The phase spectra, however, would be different.
Fourier Transform of Periodic Signals
A periodic signal x(t) can be represented by a Fourier Series Expansion according
to:
x
ð
t
Þ¼
X
1
X
k
e
þ
j2npf
o
t
;
k
¼1
where {X
k
} are the FS coefficients. Taking the Fourier transform of both sides
yields:
(
)
Ff
x
ð
t
Þg¼F
X
1
X
k
e
þ
j2npf
o
t
k
¼1
¼
X
¼
X
1
1
X
k
F
e
þ
j2npf
o
t
X
k
d
ð
f
kf
o
Þ:
k
¼1
k
¼1
Hence, the FT of a periodic signal x(t) with period T
o
is a sum of frequency
impulses at integer multiples of the fundamental frequency f
o
(i.e., at f = kf
o
),
weighted by the FS coefficients.
Example The complex Fourier series expansion and Fourier transform of the
square wave shown in Fig.
1.16
are given respectively by:
x
ð
t
Þ¼
X
X
ð
f
Þ¼
X
X
k
d
1
1
2
sinc
k
2
f
k
4
e
j
2
pkt
and
k
¼1
|{z}
X
k
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