Biomedical Engineering Reference
In-Depth Information
2. Create a vector
h
:
⎧
⎨
1for
i
=
1
,
(
n
/
2
)+
1
h
(
i
)=
2for
i
=
2
,
3
,...,
(
n
/
2
)
(2.65)
⎩
0for
i
=(
n
/
2
)+
2
,...,
n
3. Calculate the element-wise product of
X
and
h
:
Y
=
X
·
h
4. Calculate the inverse FFT of
Y
.
This algorithm is implemented in MATLAB Signal Processing Toolbox as
hilbert
. The analytic signal can be presented in the form:
x
a
(
t
)=
A
(
t
)
cos
(
φ
(
t
))
(2.66)
where
A
(
t
)=
|
x
a
(
t
)
|
is called the instantaneous amplitude, φ
(
t
)=
arg
(
x
a
(
t
))
and
))
dt
is called the instantaneous frequency
1
.
The concepts of instantaneous amplitude and frequency are useful in case of sim-
ple signals without overlapping frequency components, e.g., in
Figure 2.12
a-c; the
instantaneous amplitude recovers the modulation of the chirp, and instantaneous fre-
quency recovers its linearly changing frequency. However, the instantaneous mea-
sures turn out to be meaningless, if the signal is more complicated, e.g., if there are
two structures with different frequencies present at the same moment. Figure 2.12 d-f
illustrates such a case. The signal in panel d) is composed of two signals analogous
to that in panel a), but shifted in frequency by a constant. Its instantaneous amplitude
(panel e) and frequency (panel f) display fast oscillatory behavior.
d
arg
(
x
a
(
t
1
2π
f
(
t
)=
2.4.2
Analytic tools in the time-frequency domain
2.4.2.1
Time-frequency energy distributions
Signal energy can be computed in time or in frequency domain as:
Z
∞
Z
∞
2
dt
2
df
E
x
=
∞
|
x
(
t
)
|
=
∞
|
X
(
f
)
|
(2.67)
−
−
2
are interpreted as energy density. The idea of sig-
nal energy density can be extended to the time-frequency space. The time-frequency
energy density ρ
x
(
2
or
and the quantities
|
x
(
t
)
|
|
X
(
f
)
|
should represent the amount of signal energy assigned to a
given time-frequency point
t
,
f
)
(
t
,
f
)
,andfulfill the conditions:
1. The total energy is conserved:
Z
∞
Z
∞
E
x
=
ρ
x
(
t
,
f
)
dtd f
(2.68)
−
∞
−
∞
1
Note, that in case of a stationary signal, e.g.,
x
=
B
cos
(
2π
ft
)
these definitions retrieve amplitude
A
(
t
)=
B
and frequency φ
(
t
)=
f
of the signal.
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