Biomedical Engineering Reference
In-Depth Information
Translation:
For any real number
t
0
:
e
−
2π
it
0
f
X
z
(
t
)=
x
(
t
−
t
0
)
⇒
Z
(
f
)=
(
f
)
Modulation:
For any real number
f
0
:
e
2π
f
0
x
z
(
t
)=
(
t
)
⇒
Z
(
f
)=
X
(
f
−
f
0
)
Scaling:
For all non-zero real numbers
a
:
X
f
a
1
z
(
t
)=
x
(
at
)
⇒
Z
(
f
)=
|
a
|
The case
a
=
−
1 leads to the time-reversal property, which states:
z
(
t
)=
x
(
−
t
)
⇒
Z
(
f
)=
X
(
−
f
)
Conjugation:
)
∗
The
∗
symbol throughout the topic denotes the complex conjugation.
)
∗
z
(
t
)=
x
(
t
⇒
Z
(
f
)=
X
(
−
f
Convolution theorem:
y
(
t
)=(
x
z
)(
t
)
⇔
Y
(
f
)=
X
(
f
)
·
Z
(
f
)
(1.30)
˙
This theorem works also in the opposite direction:
Y
(
f
)=(
X
H
)(
f
)
⇔
y
(
t
)=
x
(
t
)
·
z
(
t
)
(1.31)
˙
This theorem has many applications. It allows to change the convolution op-
eration in one of the dual (time or frequency) spaces into the multiplication in
the other space. Combined with the FFT algorithm the convolution theorem
allows for fast computations of convolution. It also provides insight into the
consequences of windowing the signals, or applications of filters.
1.4.5 Power spectrum: the Plancherel theorem and Parseval's theorem
Let's consider the
x
[
n
]
as samples of a voltage across a resistor with the unit re-
2
sistance
R
R
is the power dissipated by that resistor. By
analogy in the signal processing language a square absolute value of a sample is
called
instantaneous signal power
.
If
X
=
1. Then the
P
=
x
[
n
]
/
[
k
]
and
Y
[
k
]
are the DFTs of
x
[
n
]
and
y
[
n
]
, respectively, then the Plancherel
theorem states:
N
1
n
=
0
x
[
n
]
y
∗
[
n
]=
−
N
1
k
=
0
X
[
k
]
Y
∗
[
k
]
−
1
N
(1.32)
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