Digital Signal Processing Reference
In-Depth Information
where sgn
(t)
=
1. Now the causal impulse response can be
written in terms of only the even component:
1 and sgn
(
−
t)
=−
h(t)
=
h
e
(t)
+
sgn
(t)h
e
(t)
(8-17)
In the frequency domain,
H
(
ω
) can also be written in terms of the even function
using the property that multiplication in the time domain is the same as con-
volution in the frequency domain and
F{
sgn
(t)
}=−
j(
1
/πω)
using the signal
processing convention (
a
=
0,
b
=−
2
π
) of equation (8-1a):
πω
∗
H
e
(ω)
(8-18)
Equation (8-18) can be simplified by using the definition of a
Hilbert transform
,
which is the convolution of a function
g
(
ω
) and 1
/πω
:
{
h
e
(t)
}
)
=
H
e
(ω)
−
j
1
H(ω)
=
F
{
h
e
(t)
}+
(
F
{
sgn(t)
}∗
F
∞
g(ω
)
ω
−
ω
1
πω
=
1
π
dω
g(ω)
=
g(ω)
∗
(8-19)
−∞
Therefore, equation (8-18) can be written in terms of the Hilbert transform of
the even function:
1
πω
∗
H
e
(ω)
H
e
(ω)
=
(8-20)
H
e
(ω)
H(ω)
=
H
e
(ω)
−
j
where
H
e
(ω)
denotes the Hilbert transform of
H
e
(ω)
.
Equation (8-20) demonstrates two very important properties of a system that
will produce a real, linear, and causal response in the time domain.
1. The imaginary part of the frequency response is determined by the Hilbert
transform of the real part.
Knowledge of the real part is sufficient to define
the entire function
.
2. Causality can be tested by performing the Hilbert transform of the real part
and ensuring that it is identical to the imaginary part.
Note that the real part of a causal signal can be derived from its imaginary
part also. It is interesting to note that the Kramers-Kronig relations mentioned
in Chapter 6 are another form of the Hilbert transforms that relate the real and
imaginary parts of the complex dielectric permittivity to each other.
∞
ω
ε
r
(ω
)
(ω
)
2
2
π
ε
(ω)
=
−
ω
2
dω
1
+
(6-34a)
0
∞
−
ε
r
(ω
)
(ω
)
2
2
ω
π
1
ε
(ω)
=
−
ω
2
dω
(6-34b)
0
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