Digital Signal Processing Reference
In-Depth Information
0.6
0.33
n
= 100
0.4
0
n
= 20
0.2
n
= 5
−0.33
a
/
n
0
−0.66
−0.2
t
−0.99
n
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−0.4
0
5
10
15
20
25
30
Fig. 4.8. Rectangular pulse reconstructed with a finite number
n
of
DTFS coefficients
a
n
. Three different values
n
Fig. 4.7. DTFS coefficients
a
n
for the rectangular pulse in Example 4.7.
= 5, 20, and 100 are
considered.
lies in the shape of the rectangular pulse that includes two constant values
(1
/
3
, −
2
/
3) separated by a discontinuity within one period. The discontinuity
or the sharp transition in
w
(
t
) is accounted for by a sinusoidal function with an
infinite fundamental frequency. Generally, if a function has at least one discon-
tinuity, the CTFS representation will contain an infinite number of sinusoidal
functions.
Figure 4.7 shows the exponentially decaying value of the CTFS coefficients
a
n
. To obtain the precise waveform
w
(
t
), an infinite number of the CTFS coef-
ficients
a
n
are needed. Because of the decaying magnitude of the CTFS coeffi-
cients, however, a fairly reasonable approximation for
w
(
t
) can be obtained by
considering only a finite number of the CTFS coefficients
a
n
. Figure 4.8 shows
the reconstruction of
w
(
t
) obtained for three different values of
n
. We set
n
=
5,
20, and 100. It is observed that
w
(
t
) provides a close approximation of
w
(
t
) for
n
=
20. For
n
=
100, the approximated waveform is almost indistinguishable
from the waveform of
w
(
t
).
4.4.2 Jump discontinuity
Figure 4.8 shows that a CT function with a discontinuity can be approximated
more accurately by including a larger number of CTFS coefficients. When
approximating CT periodic functions with a finite number of CTFS coefficients,
two errors arise because of the discontinuity. First, several ripples are observed
in the approximated function. A careful observation of Fig. 4.8 reveals that, as
more terms are added to the CTFS, the separation between the ripples becomes
narrower and the approximated function is closer to the original function. The
peak magnitude of the ripples, however, does not decrease with more CTFS
terms. The presence of ripples near the discontinuity (i.e. around
t
=
1in
Fig. 4.8) is a limitation of the CTFS representation of discontinuous signals,
and is known as the
Gibbs phenomenon
.
Secondly, an approximation error is observed at the location of the disconti-
nuity (i.e. at
t
=
1 in Fig. 4.8). With a finite number of terms, it is impossible
to reconstruct precisely the edge of a discontinuity. However, it is possible to
calculate the value of the approximated function at the discontinuity. Suppose
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