Wavelets
Review Fourier Transform and Short Term Fourier Transform ( Read the tutorial posted
on the web and/or read from either Sayood or Salomon).
The Problems with Fourier Transform
The Fourier transform hides information about time. It gives unambiguous information
about the frequencies that the signal contains but it does not say at what times these
frequencies occur. As a result, two signals, one stationary and the other nostationary,
containing the same frequencies will give the same frequency spectrum. Every frequency
is considered a global characteristic of the signal. A discontinuity in the local part of the
signal is translated in the frequency spectrum over the entire time domain – a local
characteristic becomes a global characteristic. This does not mean that the information
regarding time is totally lost, it becomes embedded in the ‘phases’ of the components and
this is the reason we can reconstruct the original time signal faithfully.
The lack of time information makes Fourier transform error prone. If a signal is received
correctly for hours and gets corrupted for only a few second, it totally destroys the signal
because the frequencies injected
spread over the entire time domain and the errors
become global.
A qualitative explanation of why Fourier transform fails to capture time information is
the fact that the set of basis functions ( sines and cosines) are infinitely long and the
transform picks up the frequencies regardless of where it appears in the signal.
Uncertainty Principle
The time and frequency domains are complimentary. If one is local, the other is global.
For an impulse signal, which assumes a constant value for a very brief period of time, the
frequency spectrum is infinite whereas if a sinusoidal signal extends over infinite time, its
frequency spectrum is a single vertical line at the given frequency. We can always
localize a signal or a frequency but we cannot do that simultaneously. If the signal has a
short duration, its band of frequency is wide and vice versa.
Heisenberg’s uncertainty principle was enunciated in the context of quantum physics
which stated that the position and the momentum of a particle cannot be precisely
determined simultaneously.
This principle is also applicable to signal processing where
the precise statement is as follows.
Let
g
(
t
) be a function with the property
1
)
(
2
=
∫
∞
∞
−
dt
t
g
. Then
∫
∞
∞
−
−
)
)
(
)
(
2
2
dt
t
g
t
t
m
(
∫
∞
∞
−
Π
≥
−
2
2
2
16
1
)
)
(
)
(
(
dt
f
G
f
f
m
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where
t
denote average values of
t
and
f
,and
G
(
f
) is the Fourier transform of
g
(
t
).
m
m
f
,
ShortTerm Fourier Transform (STFT)
(Read the tutorial posted in the course web page.)
The STFT is an attempt to alleviate the problems with FT. It takes a nonstationary signal
and breaks it down into “windows” of signals for a specified short period of time and
does FT on the window by
considering
the signal to consist of repeated windows over
time. This gives a better picture of the frequency content of the signal over the segment
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 Spring '11
 Mukherjee
 Digital Signal Processing, wA, Wavelet, Discrete wavelet transform, Haar

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