Timefrequency Map
Shorttime Fourier Transform (STFT)
: Spectrogram, Sonogram
A short data window centered at time
t
with time duration
T
has the frequency bandwidth of
approximately
B
T
≅
1
. Thus STFT features that all spectral estimates have the same(constant)
resolution bandwidth.
WignerVille Method
The spectral density function of a nonstationary process
x t
(
) is defined as:
S
f t
R
t e
d
E x t
x t
e
d
xx
xx
j
f
j
f
(
, )
( , )
(
) (
)
=
=
−
+
−
−∞
∞
−
−∞
∞
∫
∫
τ
τ
τ
τ
τ
π τ
π τ
2
2
2
2
But, in practice, it is never possible to compute an ensemble average. Instead, we get, for a
single sample function
x t
(
) , the Wigner distribution given by
S
R
xx
xx
j
f
j
f
f
t
t e
d
x t
x t
e
d
(
, )
( , )
(
)
(
)
=
=
−
+
−
− ∞
∞
−
− ∞
∞
∫
∫
τ
τ
τ
τ
τ
π τ
π τ
2
2
2
2
A disadvantage is that, although the above integral is centered at time
t
, it covers an infinite
range of
and so depends on the character of
τ
x t
( ) far away from the local time
t
. Therefore
it does not describe the truly local behavior of
x t
(
) at time t. Because of the continuing nature
of harmonic waves, it is impossible to have a local spectral density. In other words, Wigner
Ville method tries to break down a signal into its harmonics, which are global functions that
go on for ever. This is a fundamental uncertainty principle for timedependent spectra. In
addition, high resolution cannot be obtained simultaneously in time and frequency.
Wavelet Analysis
A transient signal is broken down into a series of local basis functions called
wavelets
. Each
wavelet is located at a different position on the time axis and is local in the sense that it
decays to zero when sufficiently far from its center. Any particular local features of a signal
can be identified from the
scale
and
position
of the wavelets in which it is decomposed.
Wavelets are a powerful tool for presenting local features of a signal. When the size and
shape of a wavelet are exactly the same as a section of the signal, the wavelet transform gives
a maximum absolute value, a
property which can be used to detect transients in a signal.
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 Spring '09
 .
 ........., Wavelet, Discrete wavelet transform, C. W. Lee, Haar Family of Wavelets

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