25/03/08. 2 Outline 1.Introduction 2.The Fourier Transform 3.The Heisenberg Uncertainty Principle 4.The Windowed Fourier Transform 5.Wavelets History

25/03/0825/03/08

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OutlineOutline1. Introduction2. The Fourier Transform3. The Heisenberg Uncertainty Principle4. The Windowed Fourier Transform5. Wavelets History6. Wavelets Basic Theory7. The Continuous Wavelet Transform (CWT)8. The Discrete Wavelet Transform (DWT) & MRA9. Applications (1D & 2D)10. Conclusion11. Bibliography

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IntroductionIntroduction

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• Music: a “time-scale game”

Frequency of the notes

Time


Score Example

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Fourier AnalysisFourier Analysis Wavelet AnalysisWavelet Analysis

Information : Frequency of the notes

(Temporal information hidden)

Information : Frequency of the notes

Time instants of the notes


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Problem: how can we see the difference between two notes played simultaneously and two notes played one after the

other ?

A possible solution: use of the Wavelet Transform…


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( ) sin(2 . . ) sin1 ( )22 . .s t tf tf 82 5f Hz41 0f Hz

882 5f Hz41 0f Hz

Magnitude of the Fourier Transform of s(t)

4096Fs points


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82 5f Hz

41 0f Hz

Magnitude of the Wavelet Transform of s(t)

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IntroductionIntroduction( ) sin(2 . 1. )fs t t

82 5f Hz41 0f Hz 0;0.5t

0.5;1t ( ) sin(2 . 2. )fs t t

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4096Fs points82 5f Hz41 0f Hz

Magnitude of the Fourier Transform of s(t)

Gibbs phenomenon(“ringing” artifacts)

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82 5f Hz

41 0f Hz

Magnitude of the Wavelet Transform of s(t)

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The Fourier TransformThe Fourier Transform

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The Fourier TransformThe Fourier Transform• Fourier analysis (Fourier, J-B. J. - 1822) is one of the

most known signal processing tool to study stationary signals.

• It gives precise frequency information about a time-domain signal thanks to its decomposition basis along waves having precise frequency (sines and cosines).

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The Fourier TransformThe Fourier Transform• Its mathematical structure is very suitable to perform

linear filtering operations (transfer function).

• It has led to many algorithms (e.g. FFT) and many softwares, showing its actual notoriety of use.

• But, Some limitations occur when it is about analyzing signals with local discontinuities such as peaks.

• It is difficult to analyze high and low frequencies simultaneously…

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The Fourier TransformThe Fourier Transform• A “mathematical prism”: the Fourier Transform is a

mathematical operation that decomposes a function according to its frequencies, just like the prism decomposes the light.

2( ) ( ) j ftS f s t e dt

2( ) ( ) j fts t S f e df

(direct)

(inverse)

The new function S(f) shows how many sines and cosinesare present in the original function s(t).

s(t)

S(f)

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The Fourier TransformThe Fourier Transform• A global transform: It cannot analyze the local

frequency content or local regularity of a signal.

• Phase: Temporal information is hidden in the phases (offsets between sinusoids).

In the Fourier Transform, it is difficult to calculate those phase-coefficients with sufficient accuracy to recover temporal information.

• Non-causality: We must know the entire time-domain signal to be able to compute its Fourier Transform (FT).

• “On-the-fly” computation and real-time analysis are impossible !

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The Heisenberg The Heisenberg Uncertainty PrincipleUncertainty Principle

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The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle• In quantum physics, the Heisenberg uncertainty

principle states that:

– This can be stated exactly as:where is the uncertainty in position,

is the uncertainty in momentum,

h is the Planck's constant (1.0546 x 10-34 J.s).

“ The more precisely the position is determined, the less precisely the momentum is known in this instant,

and vice versa. ”Heisenberg, W., uncertainty paper, 1927.

Werner Heisenberg

4x

hx p

xxp

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The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle• In Schrödinger's wave mechanics, the quantum

mechanical wave function contains information about both the position and the momentum of the particle. The position of the particle is where the wave is concentrated, while the momentum is the typical wavelength.

• In signal processing, This is an exact counterpart to a well known result: the shorter a pulse in time, the less well defined the frequency. The width of a pulse in frequency space is inversely proportional to the width in time.

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ˆ .sin ( )f cf F a af 1narrow peak : (t) = 1 , with =2 2 6

a af a

The signal is well localized in time.

The FT is delocalizedin frequency.

1a 1

a

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

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larger peak : (t) = 1 , with = 62 2a af a

ˆ .sin ( )f cf F a af

The signal is delocalized in time.

The FT is centered around zero, thus well localized in frequency.


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• It is a fundamental result in Fourier analysis:

“The narrower the peak of a function, the broader the Fourier transform”.

• Localizations of f and are related to the Heisenberg Uncertainty Principle which precise the link between the variances of f and .

• It constraints the product of the dispersions and , in time and in frequency, respectively.

f

ff

f

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• This can be stated exactly as: , where:

,

– They quantify the dispersions of and about their means µ and ξ, respectively, given by:

,

ˆ

1

4f f

222

1( )f t f t dt

f

2

2ˆ 2

1 ˆ( )f

f df

2f

2

f

2

2

1( )t f t dt

f

2

2

1 ˆ( )f df

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Time

Frequency

Time-frequency Heisenberg Boxes in the Fourier Basis

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~ Problem ~~ Problem ~

We wish to make Fourier local…We wish to make Fourier local…but how ?but how ?

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The Windowed Fourier The Windowed Fourier TransformTransform

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The Windowed Fourier TransformThe Windowed Fourier Transform• The Windowed Fourier Transform (Gabor, D. - 1947) is

also known as Short-Time (or -Term) Fourier Transform (STFT):

00 00

2( , ) ( )( ) j f tg t tftW x t e dt

Dennis Gabor1971 Nobel prize for the invention of holography,first suggested to make Fourier analysis local,first introduced “time-frequency” wavelets (Gabor wavelets).

Analysis window

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The Windowed Fourier TransformThe Windowed Fourier Transform• The STFT makes it possible to analyze a signal x(t) in

time and in frequency simultaneously, we talk about Time-Frequency Transform (TFT).

• The idea is to perform a Fourier Transform inside a window that will be translated along the signal.– What is a window ?: a window (or envelope) is a function

g(t), smooth, slowly variable and well localized in time. Its graphical representation is a portion of curve which delimitate an area containing oscillations.

In general, we choose the window g(t) even and real.

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The Windowed Fourier TransformThe Windowed Fourier Transform– When the window g(t) is a Gaussian function, we talk

about the Gabor Transform.

Two “gaborettes” with different oscillations

time

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The Windowed Fourier TransformThe Windowed Fourier Transform• The size of the window (temporal support) is related to

the size of the interval that will be analyzed. It doesn’t change during the process but it is filled with oscillations at different frequencies.

• Like the Fourier Transform, it is possible to reconstruct the original signal x(t) with the help of the coefficients obtained during the analysis. The inversion formula (synthesis) is immediate and is given by:

where c>0 is a numerical constant(for us, the value is not important here)

020 0 0 0 0( ) ( ) ( , )j f tx t c g t t e W t f dt df

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The Windowed Fourier TransformThe Windowed Fourier Transform

Time

Frequency

Time

Frequency

Time-frequency Heisenberg Boxes in the STFT Basis.Left: the window is narrow. Right: the window is broader.

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~ Problems ~~ Problems ~Since the size of the analysis window does not change Since the size of the analysis window does not change

during the during the STFTSTFT process, we have to make a process, we have to make a compromisecompromise when analyzing different frequencies.when analyzing different frequencies.

A A small windowsmall window allows the analysis of allows the analysis of transient transient componentscomponents of a signal, i.e. of a signal, i.e. high frequencieshigh frequencies..

A A broader windowbroader window allows the analysis of allows the analysis of low frequencieslow frequencies..

We cannot analyze high and low frequencies We cannot analyze high and low frequencies simultaneouslysimultaneously ! !

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~ Solution ~~ Solution ~

Find an analysis where the size of the Find an analysis where the size of the window varies with the frequency…window varies with the frequency…

The WaveletsThe Wavelets

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Wavelets HistoryWavelets History

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Wavelets HistoryWavelets History• We could go back in time and find origins of

wavelets in 1930. At this time, in mathematics, wavelets were used under the name “atomic decompositions” to study different functional spaces. Some researchers have developed wavelets - under the name “autosimilar Gabor functions” - to model the human visual system.

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Wavelets HistoryWavelets History• However, we usually take the year 1975 as the real

starting point for the discovery of wavelets, and the works of the French geophysicist engineer Jean Morlet (from École Polytechnique), who worked for The French oil company Elf-Aquitaine and who invented time-scale wavelets to analyze the sound echoes used in oil prospecting.

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Wavelets HistoryWavelets History

Oil layers of different thickness

1

2

3

Pulse : incoming signal

Echoes : reflection signals

Interferences

θ’

θ’’

θ’’’

θ θ

Multiple Reflections

Groundlevel 0

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Wavelets HistoryWavelets History• The frequencies of those echoes are related to the

thicknesses of oil layers.

• High frequencies ≡ Thin layers

• Problem: Difficulty to decorrelate all those reflection signals because they interfere a lot between them.

• To extract information, J. Morlet first uses the STFT with windows of different sizes… Unsuccessfully ;-(

• Thus, J. Morlet has a great idea: he fixes the number of oscillations inside the analysis window that he compresses or stretches like a accordion.

Wavelets were born !!!

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Wavelets Basic TheoryWavelets Basic Theory

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• What is a wavelet ? (Graphical approach)– “wavelet” → small wave !– “The simplest transient signal we can imagine” (Y. Meyer)

– A wavelet is a function that can be seen as:

Wavelets Basic TheoryWavelets Basic Theory

a fast-decaying

→ e.g. : Morlet Wavelet

oscillating waveform of finite-length

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Wavelets Basic TheoryWavelets Basic Theory• What is a wavelet ? (Mathematical context)

– Let ψ be a carefully chosen function, regular and localized. This function * will be called wavelet if it verifies the following admissibility condition in the frequency space:

where is the Fourier Transform of .

→ The integral of the wavelet is null.

1 2L L

2 20

0

ˆ ˆC d d

* L1 and L2 are the spaces of integrable functions and finite energy functions, respectively.

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Wavelets Basic TheoryWavelets Basic Theory• What is a wavelet ? (Mathematical context)

– We often want the wavelet to have (m+1) vanishing moments (oscillations):

– A sufficient admissibility condition, easier to verify, can be written as follows:

1 2 1, , ( ) 0andL L t L t dt

R

f o r( ) 0 0kt t d t k m

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Wavelets Basic TheoryWavelets Basic Theory• There exist many different functions ψ, called mother-

wavelets (prototypes).

• Some of them have explicit mathematical formulas:Morlet Wavelet Mexican-hat Wavelet

2 / 2( ) c o s (5 )tt e t 21/ 4 2 / 22( ) 1

3tt t e

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Wavelets Basic TheoryWavelets Basic Theory• Others are built upon more complex mathematical

properties:

Meyer Wavelet Daubechies Wavelet (‘db2’)

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Wavelets Basic TheoryWavelets Basic Theory• Each wavelet has its own properties:

– Symmetry: useful to avoid out-of-phase phenomenon,

– Vanishing moments: useful for compression,

– Regularity: useful to obtain smooth and regular reconstructed signals or images,

– etc.

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The Continuous Wavelet The Continuous Wavelet Transform (CWT)Transform (CWT)

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CWT (1-D)CWT (1-D)• The (continuous) wavelet transform replaces the Fourier

Transform's sinusoidal waves by a family (base atoms) generated by translations and dilatations of a mother-wavelet ψ.

where b is the translation parameter (time),

and a is the compression / dilatation parameter (scale).

,

1( )ba t

b

a

t

a

0,a b

:0 1 Compre oa ssi n :1 Dila na tatio

0 : Left Shifb ting0 : Right Shifb ting

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CWT (1-D)CWT (1-D) Wavelet Time-Shifting (b parameter)

Time

Amplitude

-10 -5 0 +5 +10

10b 0b 10b

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CWT (1-D)CWT (1-D) Wavelet Compression / Dilatation (a parameter)

1/ 2a

1a 2a

Compression

Dilatation

Base scale

In order to have the same energy at each scale In order to have the same energy at each scale aa, the , the wavelet is modified in amplitude ( ).wavelet is modified in amplitude ( ).

0

1

-1

-2

f

f

21

a

22

Amplitude

Time

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CWT (1-D)CWT (1-D)

• The CWT of a function is defined by:

, ,( , ) ( ) ( )f a b a bC a b f f t t dt

,

1( )a b

t bt

aa

0,a b

2 ( )f L

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CWTCWT• The previous function is centered around b. If the frequency center of ψ is

η, then the frequency center of the dilated function is η /a.

• Its time spread is proportional to a. Its frequency spread is proportional to the inverse of a.

Example of Heisenberg boxes of wavelet atoms.

0

,a b

t

ω

0 0,a b

a

0a

a

0a

ta0 ta

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CWT (1-D)CWT (1-D)• At the finer scales, more Heisenberg boxes can be placed

side to side because there is a better time resolution.

Time

Frequency

Scale

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CWT (1-D)CWT (1-D)• Properties

– The wavelet transform has thus a time-frequency resolution which depends on the scale a. Under the (admissibility) condition:

– It is a complete, stable and redundant representation of the signal; in particular, the wavelet transform is left invertible.

2

0

ˆC d

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CWT (1-D)CWT (1-D)• Scalogram

– If η denotes the frequency center of the base wavelet, then the frequency center of a dilated wavelet is ξ = η /a.

– The scalogram of a signal is defined by:

– The normalized scalogram is defined by:

2

2, , ,f f fS b C a b C b

,Nf fS S b

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CWT (1-D)CWT (1-D)• What it looks like to perform the CWT in 1-D ?

– Mother wavelet ψ: Morlet wavelet

– Signal to be analyzed (e.g. 40 points):

– The signal to be analyzed has voluntary been generated with a portion of curve very close to the shape of the mother-wavelet in order to underline the computation and the value of the wavelet coefficients.

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• For the Morlet wavelet, the analysis window is Gaussian but to simplify the understanding of the animation, we will model the window by a simple rectangle.

• The CWT is a continuous transform, thus we should perform the algorithm on each point of the signal. Of course, we will only show some steps !

Rectangular analysis window

CWT (1-D)CWT (1-D)

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Time

The wavelet is centered at the beginning of the signalπ : signal x wavelet (inside the window)

∫ π → one wavelet coefficient: The wavelet is shifted to the right

Signal to be analyzed

0b 10b 25b 40b

)0(1,fC )0(1,1fC )5(1,2fC )0(1,4fC

End of the algorithm for a=1 → 1st row of coefficients.

,( , ) ( ) ( )f a bC a b f t t dt

: 1scale a

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ScalogramScalogram

Time (b)

Scale (a) Amplitude1a

10 25 40

1

0

+

(1,0)C (1,10)C (1,25)C (1,40)C

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Same operations → Scale Change → Wavelet Dilatation.

2a

Time

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ScalogramScalogram

Time (b)

Scale (a) Amplitude2a

10 25 40

1

0

+

(1,0)C (1,10)C (1,25)C (1,40)C

2

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1/ 2a

ETC…ETC…

Time

Same operations → Scale Change → Wavelet Compression

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The Discrete Wavelet The Discrete Wavelet Transform (DWT)Transform (DWT)

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DWT (1-D)DWT (1-D)• The DWT has been designed to perform fast algorithms

by discretizing the continuous form.

• For convenience, in the discretization, we restrict a and b to the following dyadic values:

– where are the sampling steps,

– where (resolution: 2j, frequency: 2-j).

• Thus, the set of functions constitutes an orthonormal basis of :

2 , 2j ja b k 0, 0

2( , )j k 2Z

/ 2 2, ( ) 2 (2 ), ( , )j j

j k x x k j k Z

2, ; ( , )j k j k Z

2L R

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DWT (1-D)DWT (1-D)• The DWT is a spatial-frequency decomposition that provides

a flexible multiresolution analysis of the image. In one dimension, the aim of the wavelet transform is to represent the signal as a superposition of wavelets.

• Let f(x) be a discrete signal, its wavelet decomposition is then:

– where j and k are integers. This ensures that the signal is decomposed into normalized wavelets at octave scales (when an octave is reached, the frequency doubles).

, ,,

( ) ( )j k j kj k

f x c x

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DWT (1-D)DWT (1-D)• For an iterated wavelet transform, additional coefficients aj,k

are required at each scale.

• At each scale aj,k and aj-1,k describe the approximations of the function f at resolution 2j and at the coarser resolution 2j-1, respectively.

• At each scale, the coefficients cj,k describe the difference between one approximation and the other.

• In order to obtain the coefficients cj,k and aj,k at each scale, and position, a scaling function is needed that is similarly defined to the previous equation.

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DWT (1-D)DWT (1-D)

A single stage wavelet analysis and synthesis in one dimension.

h: low-pass analysis filter, : low-pass synthesis filter

g: high-pass analysis filter, : high-pass synthesis filter

h

g

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DWT (2-D)DWT (2-D)• To extend the wavelet transform to two dimensions, it is

just necessary to separate filter and downsample in the horizontal and vertical directions.

• This produces four subbands at each scale. Denoting the horizontal frequency first and then the vertical frequency second, this produces low-low (LL), low-high (LH), high-low (HL) and high-high (HH) image subbands.

• By recursively applying the same scheme to the LL subband, a multiresolution decomposition can be achieved.

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DWT (2-D): Multiresolution Analysis (MRA)DWT (2-D): Multiresolution Analysis (MRA)

One stage of the 2-D DWT decomposition.

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DWT (2-D): Multiresolution Analysis (MRA)DWT (2-D): Multiresolution Analysis (MRA)

Horizontal details

Vertical details Diagonal details

Approximationat level 1

2nd resolution level

1st resolution level

Normal layout of the 2D-DWT.At each scale, the subbands are sensitive to frequencies at that scale

and the LH, HL and HH subbands are sensitive to horizontal, vertical and diagonal frequencies respectively. The sizes of frequency bands will decrease as the decomposition goes on.

1HH

1LH

1HL

2LL

2HL

2LH

2HH

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ApplicationsApplications

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ApplicationsApplicationsWavelets are very efficient to solve 3 classical

problems in signal processing

Analysis

Denoising

Compression

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ApplicationsApplications• 1-D Analysis (linear chirp)

Linear evolution of the frequency with time,by a simple observation of the scalogram

ESD

sig = fmlin(256,0.1,0.4)(Shannon normalized)

“Linear chirp”: signal which frequency varies linearly with time.

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ApplicationsApplications• 1-D Denoising

– Definition: Recover a signal from observations corrupted by an additive noise (N samples, standard deviation: ).

– Principle: We transform the signal into the wavelet domain then we select, by a thresholding method (‘hard’ or ‘soft’), coefficients from which the final signal is reconstructed in the time-domain, using an inverse wavelet transform.

– Universal Threshold (Donoho, D., Stanford University)

– Example: Electric consumption signal denoising

2 log( )T N

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ApplicationsApplications• 1-D Denoising (Electric consumption signal)

Linear evolution of the frequency with the time,by a simple observation of the scalogram

Noised signalNoised signal

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ApplicationsApplications• 1-D Denoising (Electric consumption signal)

Linear evolution of the frequency with the time,by a simple observation of the scalogram

Wavelet denoised signalWavelet denoised signal

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ApplicationsApplications• 2-D Compression

– Definition: Reduce the size of data while maintaining their integrity and quality as high as possible.

– Principle: Close to the denoising principle. Wavelets are, in general, able to concentrate in few non-null coefficients, the most important part of the energy of a signal, hence the compression phenomena.

– Example: Image Compression (JPEG vs. JPEG 2000)

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ApplicationsApplications• 2-D Compression (JPEG vs. JPEG 2000)

– JPEG: based on the Discrete Cosine Transform (DCT)• 8x8 blocks analysis of the image,

• “Pixellization” effects ( more and more visible as the compression ratio increases !).

– JPEG 2000: based on the Wavelet Transform• Global analysis of the image,

• Very good quality even at a high compression ratio,

• Possibility to calculate the size of the compressed image,

• Progressive display of the image during the reconstruction.

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ApplicationsApplications• 2-D Compression (JPEG vs. JPEG 2000)

JPEG 2000(Ratio 43:1)

JPEG(Ratio 43:1)

Original image256*256

24-Bit RGB

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ApplicationsApplications• 2-D Compression - Performance

Original image512*512

24-Bit RGB

786 Ko

1st compression(Ratio 75:1)

10.6 Ko

3rd compression(Ratio 300:1)

2.6 Ko

2nd compression(Ratio 150:1)

5.3 Ko

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ApplicationsApplications• Others Fields

– Geophysics (seism detection),

– Medicine (ECG, EEG,…),

– Satellite imagery,

– Video encoding (divX 4/5),

– Internet traffic modeling,

– Cryptography,

– Optoelectronics,

– Biometrics,

– …

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ConclusionConclusion

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Conclusion: What we have to rememberConclusion: What we have to remember New Technique in signal processing (Birth : 1975, Morlet, J.),

Local analysis of a signal at different scales:

« Wavelets = Mathematical Microscope »

Wavelet Transform → Time-scale analysis (time:b, scale:a)

Transformation of a signal into numerical coefficients:

« Multiresolution analysis (MRA) »

Adaptability to the different components of a signal:

« Wavelet Coefficients: Cf (a,b) »

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Conclusion: What we have to rememberConclusion: What we have to remember There exist many wavelets and one can construct his own wavelet,

Many possible applications in various fields,

Nowadays, several variants of the wavelets (ridgelets, curvelets, contourlets, bandelets, etc.)

JPEG 2000 - 500 octets(wavelets)

Let It Wave - 500 octets(bandelets)

Original Image(~ 100 Ko)

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Conclusion: Wavelets’ Hall of FameConclusion: Wavelets’ Hall of Fame• Jean Morlet

• Alex Grossmann

Ecole Polytechnique (X-1954)

French Geophysicist Engineer (Elf-Aquitaine)

Inventor of the Time-scale Wavelets

Mathematician

1984: proved that the inverse wavelet transform is exact (ε=0)

Wavelets ≡ Fourier local

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Conclusion: Wavelets’ Hall of FameConclusion: Wavelets’ Hall of Fame• Yves Meyer

• Ingrid Daubechies

Fellow Professor at École Normale Supérieure de Cachan (ENS-CMLA)

French Academy of Sciences Member since 1993

Discovered orthogonal wavelets (Meyer wavelet)

Professor at Princeton University, Department of Mathematics

Discovered a compactly-supported wavelets family

(Daubechies wavelets, ‘dbX’)

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Conclusion: Wavelets’ Hall of FameConclusion: Wavelets’ Hall of Fame• Stéphane Mallat

Professor at École Polytechnique (CMLA)

Invented a fast transform algorithm (with Y. Meyer)

Start-up: “Let It Wave” → Wavelets Applications in imaging

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Bibliography: Books & PapersBibliography: Books & Papers

[1] : Barbara Burke Hubbard, « Ondes et ondelettes – La saga d’un outil mathématique », Belin Pour la Science – 2000.

[2] : Michel Misiti, Yves Misiti, Georges Oppenheim, Jean-Michel Poggi,« Les ondelettes et leurs applications », Hermes Science Publications – 2003.

[3] : Ingrid Daubechies, « Ten Lectures on Wavelets », SIAM – 1999.

[4] : Stéphane Mallat, « A Wavelet Tour of Signal Processing, 2nd Edition »,Academic Press – 1999.

[4] : Stéphane Jaffard, Yves Meyer, Robert D. Ryan, « Wavelets – Tools for Science & Technology », SIAM – 2001.

[5] : Nicolas Morizet, « Initiation aux ondelettes », Revue de l’Electricité et de l’Electronique (REE) – 2006. REE 2006 Best Paper Award.

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Bibliography: WebsitesBibliography: Websites

[1] : La Recherche: http://www.larecherche.fr – édition du 01/02/2005.

[6] : Wavelet Digest: http://www.wavelet.org

[3] : Première Start-Up sur les « bandelettes » : http://www.letitwave.fr

[4] : Forum sur les ondelettes: http://www.ondelette.com

[5] : Un enseignement sur les ondelettes: http://www.tsi.enst.fr/tsi/enseignement/ressources/mti/ondelettes/Sommaire.htm

[7] : An excellent MATLAB Toolbox to practice wavelet analysis: http://perso.wanadoo.fr/francois.auger/tftb.html

[8] : The Wavelet Tutorial – An introduction to wavelet analysis: http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html

[9] : JPEG – JPEG 2000: http://www.dspworx.com/primer_jpeg2000.htm

French Websites

English Websites

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Documents

25/03/08. 2 Outline 1.Introduction 2.The Fourier Transform 3.The Heisenberg Uncertainty Principle 4.The Windowed Fourier Transform 5.Wavelets History