Digital Signal Processing Reference
In-Depth Information
impulse response
h
(
t
) is equated, the frequency-domain equivalent
H
(
ω
) can be
used to test for the conditions of causality:
F
{
h(t)
}=
H(ω)
The properties of the Fourier transform can be used to make some initial con-
clusions about the causality of a system.
1. If
h
(
t
) is real,
H(
−
ω)
=
H(ω)
∗
, where
∗
indicates the complex conjugate
[LePage, 1980]. Since time-domain waveforms are always real, this is a
necessary requirement.
2. If
h
(
t
) is real and odd,
H
(
ω
) is imaginary and odd [O'Neil, 1991].
3. If
h
(
t
) is real and even,
H
(
ω
) is real and even [O'Neil, 1991].
As a reminder, odd and even functions obey the following rules: Let
f
(
t
)be
a real-valued function of a real variable. Then
f
is
even
if
f(t)
=
f(
−
t)
and
odd
if
−
f(t)
=
f(
−
t)
This means that if
h
(
t
) is even or odd, there will exist a nonzero value for
t<
0,
which violates the definition of causality as expressed in equation (8-14a).
Another useful property is that the vector space of all real-valued functions
is the direct sum of the subspaces of even and odd functions. In other words,
every function can be written uniquely as the sum of an even function and an
odd function:
f(x)
+
f(
−
x)
2
f(x)
−
f(
−
x)
2
f(x)
=
f
e
(x)
+
f
o
(x)
=
+
Therefore, a causal function [where
h
(
t
)
0 for
t<
0] must be the sum of an
even function
h
e
(
t
) and an odd function
h
o
(
t
).
=
2
h(t)
+
h(
−
t)
+
2
h(t)
−
h(
−
t)
1
1
h(t)
=
h
e
(t)
+
h
o
(t)
=
(8-15)
Consequently, for
h
(
t
) to be causal,
h
(
t
) must be composed of
both
odd and even
functions. Therefore, based on Fourier transform properties 2 and 3 above,
H
(
ω
)
must have both a real and an imaginary part.
For an impulse response h
(
t
)
to be
real and causal
,
H
(
ω
)
must be complex and satisfy the complex-conjugate rule
shown in condition 1 above
.
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