Biomedical Engineering Reference
In-Depth Information
where
h
is the window. If the window has a finite energy then the STFT can be
reversed:
Z
∞
Z
∞
1
E
h
e
i
2π
tf
dud f
x
(
t
)=
F
x
(
u
,
f
;
h
)
h
(
t
−
u
)
(2.82)
−
∞
−
∞
R
∞
−
2
dt
. Thus we can view the STFT as a tool that decomposes the
signal into waveforms,
time-frequency atoms
, of the form:
where
E
h
=
|
h
(
t
)
|
∞
e
i
2π
fu
h
t
,
f
(
u
)=
h
(
u
−
t
)
(2.83)
This concept is illustrated in
Figure 2.15.
Each atom is obtained by translation of a
single window
h
and its modulation with frequency
f
.
STFT can be used to obtain the distribution of energy in the time-frequency space.
If the window function has a unit energy (i.e.,
E
h
1) then a squared modulus of
STFT, a
spectrogram
, is an estimator of energy density (
Figure 2.16)
:
=
e
−
i
2π
fu
du
Z
∞
2
h
∗
(
S
x
(
t
,
f
)=
x
(
u
)
u
−
t
)
(2.84)
−
∞
Spectrogram conserves the translations in time:
y
(
t
)=
x
(
t
−
t
0
)
⇒
S
y
(
t
,
f
;
h
)=
S
x
(
t
−
t
0
,
f
;
h
)
(2.85)
and in frequency:
e
i
2π
f
0
t
y
(
t
)=
x
(
t
)
⇒
S
y
(
t
,
f
;
h
)=
S
x
(
t
,
f
−
f
0
;
h
)
(2.86)
Spectrogram is a quadratic form; thus in the representation of multi-component sig-
nals the cross-terms are also present:
y
(
t
)=
x
1
(
t
)+
x
2
(
t
)
⇒
S
y
(
t
,
f
)=
S
x
1
(
t
,
f
)+
S
x
2
(
t
,
f
)+
2
Re
{
S
x
1
,
x
2
(
t
,
f
)
}
(2.87)
where
F
x
2
(
(2.88)
The formula (2.88) shows that the cross-terms are present only if the signal compo-
nents are close enough in the time-frequency space, i.e., when their individual STFTs
overlap. The time and frequency resolution are determined by the properties of the
window
h
. The time resolution can be observed as the width of the spectrogram of
the Dirac's delta:
S
x
1
,
x
2
(
t
,
f
)=
F
x
1
(
t
,
f
)
t
,
f
)
e
−
i
2π
t
0
f
h
(
)=
(
−
)
⇒
(
,
)=
(
−
)
.
x
t
δ
t
t
0
S
x
t
f
;
h
t
t
0
(2.89)
The frequency resolution can be observed as the width of the spectrogram of the
complex sinusoid.
e
i
2π
f
0
t
e
−
i
2π
tf
0
H
x
(
t
)=
⇒
S
x
(
t
,
f
;
h
)=
(
f
−
f
0
)
(2.90)
It is clear from (2.89) that the shorter duration of time window
h
, the better is the
time resolution. But due to the uncertainty principle (1.41) the shorter the window
in time, the broader is its frequency band; thus from (2.90) the poorer the frequency
resolution.
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