Digital Signal Processing Reference
In-Depth Information
(1)
Absolutely integrable
. The area under one period of
x
(
t
)
is finite, i.e.
x
(
t
)
d
t
<
∞
.
(4.65)
T
0
(2)
Bounded variation
. The periodic signal
x
(
t
) has a finite number of maxima
or minima in one period.
(3)
Finite discontinuities
. The period
x
(
t
) has a finite number of discontinuities
in one period. In addition, each of the discontinuity has a finite value.
The above conditions are known as the Dirichlet conditions.
†
If these condi-
tions are satisfied, it is guaranteed that perfect reconstruction is obtained from
the CTFS coefficients except at a few isolated points where the function
x
(
t
)
is discontinuous. The first condition is also known as the
weak
Dirichlet con-
dition, whereas the second and third conditions are known as
strong
Dirichlet
conditions. Most practical signals satisfy these three conditions. Examples of
the CT functions that violate these conditions are included in the following
discussion.
Example 4.23
Determine whether the following functions satisfy the Dirichlet conditions:
(i)
h
(
t
)
=
tan(
π
t
);
(4.66)
(ii)
g
(
t
)
=
sin(0
.
5
π/
t
) for 0
≤
t
<
1 and
g
(
t
)
=
g
(
t
+
1);
(4.67)
−
2
m
−
1
−
2
m
12
<
t
≤
2
=
(iii)
x
(
t
)
(4.68)
−
2
m
−
2
−
2
m
−
1
02
<
t
≤
2
+
for
m
∈
Z
,
0
≤
t
<
1
,
and
x
(
t
)
=
x
(
t
+
1)
.
Solution
(i) The CT function
h
(
t
) is plotted in Fig. 4.22(a). We now proceed to determine
if
h
(
t
) satisfies the Dirichlet conditions. Condition (1) is violated because
0
.
5
x
(
t
)
d
t
=
tan(
π
t
)d
t
=∞
.
T
0
−
0
.
5
This is also apparent from the waveform of tan(
π
t
), plotted in Fig. 4.22(a),
where the waveform approaches
∞
at each discontinuity. Condition (2) is
satisfied as there are only one maximum and one minimum within a single
period of
h
(
t
). Condition (3) is violated. Although there is only one discontinuity
within a single period of
h
(
t
), the magnitude of the discontinuity is infinite.
(ii) The CT function
g
(
t
) is plotted in Fig. 4.22(b). Condition (1) is sat-
isfied as the area enclosed by
g
(
t
)
is finite. Condition (2) is violated as an
†
These conditions were derived by Johann Peter Gustav Lejeune Dirichlet (1805-1859), a
German mathematician.
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