The Story of Wavelets Theory and Engineering Applications

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The Story of WaveletsTheory and Engineering
Applications

• Time frequency representation
• Instantaneous frequency and group delay
• Short time Fourier transform Analysis
• Short time Fourier transform Synthesis
• Discrete time STFT

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Time Frequency Representation

• Why do we need it?
• Time info difficult to interpret in frequency
  domain
• Frequency info difficult to interpret in time
  domain
• Perfect time info in time domain , perfect freq.
  info in freq. domain Why?
• How to handle non-stationary signals
• Instantaneous frequency
• Group Delay

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Instantaneous Frequency Group Delay

• Instantaneous frequency defined as the rate of
  change in phase
• A dual quantity group delay defined as the rate
  of change in phase spectrum

Frequency as a function of time
Time as a function of frequency
What is wrong with these quantities???
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Time Frequency Representation in Two-dimensional
Space
TFR
Linear STFT, WT, etc.
Non-Linear
Quadratic Spectrogram, WD
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STFT
Amplitude
..
..
time
t0
t1
tk
tk1
tn
..
..
Frequency
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The Short Time Fourier Transform

• Take FT of segmented consecutive pieces of a
  signal.
• Each FT then provides the spectral content of
  that time segment only
• Spectral content for different time intervals
• ?Time-frequency representation

Time parameter
Signal to be analyzed
FT Kernel (basis function)
Frequency parameter
STFT of signal x(t) Computed for each window
centered at t? (localized spectrum)
Windowing function (Analysis window)
Windowing function centered at t?
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Properties of STFT

• Linear
• Complex valued
• Time invariant
• Time shift
• Frequency shift
• Many other properties of the FT also apply.

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Alternate Representation of STFT
STFT The inverse FT of the windowed spectrum,
with a phase factor
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Filter Interpretation of STFT
X(t) is passed through a bandpass filter with a
center frequency of Note that ?(f) itself is a
lowpass filter.
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Filter Interpretation of STFT
X
x(t)
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Resolution Issues
All signal attributes located within the local
window interval around t will appear at t in
the STFT
Amplitude
time
?k
?n
Frequency
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Time-Frequency Resolution

• Closely related to the choice of analysis window
• Narrow window ? good time resolution
• Wide window (narrow band) ? good frequency
  resolution
• Two extreme cases
• ?(T)?(t)? excellent time resolution, no
  frequency resolution
• ?(T)1? excellent freq. resolution (FT), no time
  info!!!
• How to choose the window length?
• Window length defines the time and frequency
  resolutions
• Heisenbergs inequality
• Cannot have arbitrarily good time and frequency
  resolutions. One must trade one for the other.
  Their product is bounded from below.

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Time-Frequency Resolution
Frequency
Time
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Time Frequency Signal Expansion and STFT
Synthesis
Basis functions
Coefficients (weights)
Synthesis window
Synthesized signal

• Each (2D) point on the STFT plane shows how
  strongly a time
• frequency point (t,f) contributes to the signal.
• Typically, analysis and synthesis windows are
  chosen to be identical.

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STFT Example
300 Hz 200 Hz 100Hz 50Hz
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STFT Example
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STFT Example
a0.01
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STFT Example
a0.001
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STFT Example
a0.0001
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STFT Example
a0.00001
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Discrete Time Stft