Digital Signal Processing Reference
In-Depth Information
However, just because
H
(
ω
) is complex obviously does not guarantee causal-
ity. Equation (8-15) shows how the negative time components of the odd and
even functions must cancel each other out to ensure a causal response. Since
the odd and even values of
h
(
t
) are generated from imaginary and real parts of
H
(
ω
), respectively, causality requires a specific relationship between
Re
[
H
(
ω
)]
and
Im
[
H
(
ω
)].
To demonstrate the relationship between
Re
[
H
(
ω
)] and
Im
[
H
(
ω
)], consider
a simple causal system with the impulse response
h(t)
=
u
s
(t)e
−
pt
, where
u
s
(t)
is a unit step function with a value of 0 for
t
≤
0 and a value of 1 for
t>
0
and
p>
0. Following equation (8-15), the odd and even functions of
h
(
t
) can be
written:
1
2
u
s
(t)e
−
pt
1
2
u
s
(
−
t)e
pt
h
e
(t)
=
+
1
1
2
u
s
(t)e
−
pt
2
u
s
(
−
t)e
pt
h
o
(t)
=
−
The functions
h
e
(t)
,
h
o
(t)
, and
h
(
t
) and are plotted in Figure 8-15. Note that
when
h
o
(t)
=
h
e
(t)
for
t>
0 and
h
o
(t)
=−
h
e
(t)
for
t<
0, the output
h
(
t
)is
zero for
t<
0 and therefore causal. This allows the odd function to be written
in terms of the even component:
h
o
(t)
=
sgn
(t)h
e
(t)
(8-16)
1.0
1.0
0.8
0.8
0.6
0.6
h
e
(
t
)
h
o
(
t
)
0.4
0.4
0.2
0.2
t
1.0
0.5
0.5
1.0
−
−
−
0.2
t
−
1.0
−
0.5
0.0
0.5
1.0
−
0.4
1.0
0.8
0.6
h
(
t
) =
h
e
(
t
)
+
h
o
(
t
)
0.4
0.2
t
−
1.0
−
0.5
0.0
0.5
1.0
Figure 8-15
The odd and even functions are summed to produce the final causal impulse
response
h
(
t
).
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