Wavelets and Affine Distributions A Time-Frequency Perspective - … · spectrogram and scalogram...

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INSTITUT FÜRNACHRICHTENTECHNIK UNDHOCHFREQUENZTECHNIK

Wavelets and Affine DistributionsA Time-Frequency Perspective

Franz Hlawatsch

Institute of Communications and Radio-Frequency EngineeringVienna University of Technology

– 2 –WAMA-04 Cargèse, France

OUTLINE

• The notion of time-frequency analysis

• Linear and quadratic time-frequency analysis

• Short-time Fourier transform and wavelet transform;spectrogram and scalogram

• Constant-bandwidth analysis vs. constant-Q analysis

• The affine class

• Affine time-frequency smoothing

• Hyperbolic time-frequency localization

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OUTLINE









The notion of time-frequency (TF) analysis

3


Auditory perception as TF analysis


The TF plane

• Visualize time-frequency location/concentrationof signal x(t):

4


OUTLINE









Linear TF analysis

• TF analysis: Measure contribution of TF point to signal

• General approach: Inner product of with “test signal”or “sounding signal” located about :

LTFR = Linear TF Representation

5


Linear TF synthesis

• TF synthesis (inversion of LTFR): Recover (“synthesize”) signal from

• General approach:

• Problem: How to construct test (analysis) functions and synthesis functions ?

is represented as superposition of TF localized signal components, weighted by “TF coefficient function”


Quadratic TF analysis

• TF analysis: Measure “energy contribution” of TF point to signal

• Simple approach:

• Want QTFR to distribute signal energy over TF plane:

• Problem: How to construct test (analysis) functions ?

QTFR = Quadratic TF Representation

“TF energy distribution”

6


Construction of analysis/synthesis functions

• Problem: Construct family of analysis functionssuch that is localized about TF point

• Systematic approach: derived from “prototype function” via unitary “TF displacement operator” :

• Same for synthesis functions :

• Two classical definitions of :– TF shift

– TF scaling (compression/dilatation) + time shift


Two classical definitions of operator U

• TF shift:

• TF scaling + time shift:

7


OUTLINE









Short-Time Fourier Transform (STFT)

• Recall TF shift:

• ⇒ LTFR = STFT:

STFT = FT of local (windowed) segment of x (t ):

8


STFT signal synthesis

• Recall STFT analysis:

• STFT signal synthesis:

is weighted superposition of TF shifted versions of


Wavelet Transform (WT)

• Recall TF scaling + time shift:

• ⇒ LTFR = WT:

9


WT signal synthesis

• Recall WT analysis:

• WT signal synthesis:

is weighted superposition of TF scaled and time shiftedversions of


Spectrogram and scalogram

• Recall LTFR → QTFR:

• STFT → spectrogram:

• WT → scalogram:

10


OUTLINE









STFT and constant-BW filterbank: analysis

• STFT analysis as convolution:

• ⇒ Filterbank interpretation/implementation:

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STFT and constant-BW filterbank: synthesis

• STFT synthesis as convolution:



Spectrogram analysis as constant-BW filterbank

• Spectrogram analysis as convolution:


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STFT / spectrogram: example


WT and constant-Q filterbank: analysis

• WT analysis as convolution:


13


WT and constant-Q filterbank: synthesis

• WT synthesis as convolution:



Scalogram analysis as constant-Q filterbank

• Scalogram analysis as convolution:


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WT / scalogram: example


STFT / spectrogram vs. WT / scalogram

STFT / spectrogram WT / scalogram

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Good-bye and hello

• Good-bye to:

– STFT

– spectrogram

– constant-BW analysis

• Hello to:

– affine class of QTFRs

– Wigner distribution and Bertrand distribution

– hyperbolic TF localization


OUTLINE








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Axiomatic (covariance-based) definition of WT

• Generic LTFR expression:

• Covariance of LTFR to TF scalings + time shifts:

• Can show that covariant LTFRs are given by WT


Axiomatic (covariance-based) definition of the affine classof QTFRs

• Generic QTFR expression:

• Covariance of QTFR to TF scalings + time shifts:

• Can show that covariant QTFRs are given by

AC = Affine Class

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The affine class of QTFRs

• Affine class of QTFRs:

• 2-D “kernel” specifies QTFR of the AC

• Scalogram is a member of the AC; its kernel is separable:

• Expression of AC QTFRs in terms of signal's FT:


Affine class and affine group

• TF scaling + time shift:

• Affine time transformation (“clock change”)

• Composition of clock changes is another clock change:

• ⇒ is unitary representation of the affine group:

– Set:

– Group operation:

– Neutral element:

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OUTLINE









The Wigner-Ville Distribution (WVD)

• Prominent member of the AC: the WVD

• Properties of the WVD:– Covariant to TF scaling and time shift (of course) – Covariant to frequency shift ⇒ not constant-Q

– Real for any (real or complex) signal x(t)

– Marginal properties: e.g.,

– Localization properties: e.g.,for

– Many more…

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Interference terms in the WVD

Interference/cross term

t

ff

t

Source: P. Flandrin, Temps-fréquence. Hermes, Paris, 1993


Constant-BW smoothing of the WVD

t

f

Smaller/less interference terms

Poorer TF resolution


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AC expression in terms of WVD

• Any QTFR of the AC can be expressed in terms of the WVD:

where is related to and by FTs

• If is a smooth function, then is a smoothedversion of

• Smoothing causes…– smaller/less interference terms– poorer TF resolution

Affine (constant-Q) smoothing, different from constant-BW smoothing shown on previous slide!


Affine (constant-Q) smoothing of the WVD

• Recall:

• Smoothing function at various TF positions:

Smoothing function

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Affine smoothing: example



t

f



Scalogram as smoothed WVD

• Recall scalogram:

• Expression of scalogram as smoothed WVD:

Smoothing function is WVD of wavelet:

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Affine WVD smoothing and constant-Q analysis

• Scalogram as smoothed WVD:


Constant-BW vs. affine (constant-Q) smoothing

t

f



Source: P. Flandrin, Temps-fréquence. Hermes, Paris, 1993 t

f

t

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OUTLINE









Doppler-tolerant signals

• TF scaling / Doppler effect:

• “Doppler-tolerant” signal = eigenfunction of :

• Solution: “hyperbolic impulse”

• Group delay:

Hyperbola in the TF plane

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Example: Bat sonar signals

f

t

HUNTING APPROACH PURSUIT CAPTURE

Source: P. Flandrin


Hyperbolic TF localization

• Want AC QTFR to satisfy hyperbolic TF localization property:

• Not satisfied by WVD !

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The Bertrand P0 distribution

• The hyperbolic TF localization property is satisfied by the (unitary) Bertrand P0 distribution

• The Bertrand P0 distribution is a central member of the AC. It satisfies several important properties (besides the hyper-bolic TF localization property).

with


Bertrand P0 distribution as generator of the AC

• Any QTFR of the AC can be expressed in terms of the Bertrand P0 distribution:

• Special case: scalogram

where is related to

Smoothing function is BER of wavelet:

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Mellin transform and hyperbolic marginals

• Recall hyperbolic impulse

• Mellin transform:

• Hyperbolic marginal property:

• Not satisfied by WVD… but satisfied by Bertrand P0 distri-bution !

Integrate ACx(t,f) over TF hyperbola t=c/f


Application: TF analysis of gravitational wave

Idealized Matched BertrandReassignedspectrogram

WVD Spectrogram Scalogram

Source: E. Chassande-Mottin and P. Flandrin, On the time-frequency detection of chirps. Appl. Comp. Harm. Anal., 6(9): 252-281, 1999.

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Conclusion

• Linear and quadratic TF analysis

• Short-time Fourier transform and spectrogram

• Wavelet transform and scalogram

• Filterbank interpretation: constant-BW analysis versus constant-Q analysis

• Scaling/shift covariance and affine class of QTFRs

• Wigner-Ville distribution and affine smoothing

• Doppler tolerance and hyperbolic impulses

• Hyperbolic TF localization and Bertrand P0 distribution

• Mellin transform and hyperbolic marginal property


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