Time

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The Story of WaveletsTheory and Engineering
Applications

• Time frequency resolution problem
• Concepts of scale and translation
• The mother of all oscillatory little basis
  functions
• The continuous wavelet transform
• Filter interpretation of wavelet transform
• Constant Q filters

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Time Frequency Resolution

• Time frequency resolution problem with STFT
• Analysis window dictates both time and frequency
  resolutions, once and for all
• Narrow window ? Good time resolution
• Narrow band (wide window) ? Good frequency
  resolution
• When do we need good time resolution, when do we
  need good frequency resolution?

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Scale Translation

• Translation ? time shift
• f(t)? f(a.t) agt0
• If 0ltalt1 ?dilation, expansion ? lower frequency
• If agt1 ? contraction ? higher frequency
• f(t)?f(t/a) agt0
• If 0ltalt1 ? contraction ? low scale (high
  frequency)
• If agt1 ? dilation, expansion ? large scale (lower
  frequency)
• Scaling ? Similar meaning of scale in maps
• Large scale Overall view, long term behavior
• Small scale Detail view, local behavior

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1375,500
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The Mother of All Oscillatory Little Basis
Functions

• The kernel functions used in Wavelet transform
  are all obtained from one prototype function, by
  scaling and translating the prototype function.
• This prototype is called the mother wavelet

Translation parameter
Scale parameter
Normalization factor to ensure that allwavelets
have the same energy
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Continuous Wavelet Transform
translation
Mother wavelet
Normalization factor
Scaling Changes the support of the wavelet based
on the scale (frequency)
CWT of x(t) at scale a and translation b Note
low scale ? high frequency
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Computation of CWT
Amplitude
Amplitude
bN
b0
time
b0
bN
time
Amplitude
Amplitude
bN
b0
b0
bN
time
time
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Why Wavelet?

• We require that the wavelet functions, at a
  minimum, satisfy the following

Wave
let
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The CWT as a Correlation

• Recall that in the L2 space an inner product is
  defined as
• then
• Cross correlation
• then

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The CWT as a Correlation
wavelets

• Meaning of life
• W(a,b) is the cross correlation of the signal
  x(t) with the mother wavelet at scale a, at the
  lag of b. If x(t) is similar to the mother
  wavelet at this scale and lag, then W(a,b) will
  be large.

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Filtering Interpretation of Wavelet Transform

• Recall that for a given system hn,
  ynxnhn
• Observe that
• InterpretationFor any given scale a (frequency
  1/a), the CWT W(a,b) is the output of the filter
  with the impulse response to the
  input x(b), i.e., we have a continuum of filters,
  parameterized by the scale factor a.

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What do Wavelets Look Like???

• Mexican Hat Wavelet
• Haar Wavelet
• Morlet Wavelet

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Constant Q Filtering

• A special property of the filters defined by the
  mother wavelet is that they are so called
  constant Q filters.
• Q Factor
• We observe that the filters defined by the mother
  wavelet increase their bandwidth, as the scale is
  reduced (center frequency is increased)

w (rad/s)
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Constant Q
STFT
f0 2f0 3f0 4f0
5f0 6f0
2B
4B
8B
CWT
f0 2f0 4f0
8f0
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Inverse CWT
provided that
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Properties of Continuous Wavelet
Transform

• Linearity
• Translation
• Scaling
• Wavelet shifting
• Wavelet scaling
• Linear combination of wavelets

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Example
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Example
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Example