Biomedical Engineering Reference
In-Depth Information
where the star denotes complex conjugation. Parseval's theorem is a special case of
the Plancherel theorem and states:
N
1
n
=
0
|
x
[
n
]
|
−
N
1
k
=
0
|
X
[
k
]
|
−
1
N
2
2
=
.
(1.33)
2
as the portion of
signal power carried by the complex exponential component of frequency indexed
by
k
. If we process real signals, then the complex exponential in Fourier series come
in conjugate pairs indexed by
k
and
N
1
N
|
[
]
|
Due to the Parseval's theorem (1.33) we can think of
X
k
2. Each of the components
of the pair carries half of the power related to the oscillation with frequency
f
k
=
−
k
for
k
∈
1
,...,
N
/
k
N
F
s
(
F
s
is the sampling frequency). To recover the total power of oscillations at frequency
f
k
we need to sum the two parts. That is:
N
2
1
2
P
(
f
k
)=
|
X
[
k
]
|
+
|
X
[
N
−
k
]
|
(1.34)
For real signals the power spectrum is often displayed by plotting only the power
(1.34) for the positive frequencies.
1.4.6 Z-transform
A more general case of the discrete signal transformation is the
Z
transform. It is
especially useful when considering parametric models or filters.
In a more general context, the
Z
transform is a discrete version of the Laplace
transform. For a discrete signal
x
[
n
]
the
Z
transform is given by:
∞
n
=
0
x
[
n
]
z
−
n
X
(
z
)=
Z
{
x
[
n
]
}
=
(1.35)
Ae
i
φ
is a complex number. The discrete Fourier transform is a special case
of the
Z
transform.
Properties of the
Z
transform:
where
z
=
Linearity:
Z
{
a
1
x
1
[
n
]+
a
2
x
2
[
n
]
}
=
a
1
X
1
(
z
)+
a
2
X
2
(
z
)
Time translation:
z
−
k
X
Z
{
x
[
n
−
k
]
}
=
(
z
)
Transform of an impulse:
Z
{
δ
[
n
]
}
=
1
The transform of an impulse together with the time translation yields:
z
−
n
0
{
[
−
]
}
=
Z
δ
n
n
0
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