Wavelets - ee.stanford.eduosgood/Sophomore College/Wavelets.pdfWavelets: Who Gives A Damn? The Fourier Transform Has no time resolution (integral is from –∞ to ∞) Not very useful

WaveletsWaveletsBy Matt ScheferBy Matt Schefer

andandSimon BerringSimon Berring

Wavelets: Who Gives A Damn?Wavelets: Who Gives A Damn?The Fourier TransformThe Fourier Transform

Has no time resolution Has no time resolution (integral is from (integral is from ––∞ to ∞)∞ to ∞)Not very useful on nonNot very useful on non--stationary signalsstationary signals

Solution #1: Short Term FTSolution #1: Short Term FTShort Term Fourier TransformShort Term Fourier Transform

Computes FT over a small windowComputes FT over a small windowWindow is usually a GaussianWindow is usually a GaussianCannot provide localization in both time and frequencyCannot provide localization in both time and frequency

Visual Example of STFTVisual Example of STFT

The Uncertainty PrincipleThe Uncertainty Principle

As frequency As frequency resolution increases, resolution increases, time resolution time resolution decreases (and vice decreases (and vice versa)versa)

Solution #2: WaveletsSolution #2: Wavelets1.1. A set of building blocks (set of basis functions)A set of building blocks (set of basis functions)

All functions generated from the “mother wavelet” by scaling All functions generated from the “mother wavelet” by scaling and translationand translation

2.2. Satisfy the Satisfy the multiresolutionmultiresolution conditionconditionBy making expansion signals half as wide, we can represent a By making expansion signals half as wide, we can represent a larger set of functions (including the original set)larger set of functions (including the original set)

3.3. Provide timeProvide time--frequency localizationfrequency localization4.4. Coefficients can be calculated efficientlyCoefficients can be calculated efficiently

For many wavelet systems, requires For many wavelet systems, requires OO((NN) time (FFT requires ) time (FFT requires OO((NN**log(log(NN))))

Commonly Used WaveletsCommonly Used WaveletsHaarHaar WaveletWavelet

MorletMorlet WaveletWavelet

Mexican Hat WaveletMexican Hat Wavelet

Representing A Function As A SumRepresenting A Function As A Sum

Fourier series use sine and cosine wavesFourier series use sine and cosine wavesWavelets can be used insteadWavelets can be used instead

where: where:

Problems with Fourier series that may be Problems with Fourier series that may be avoided by wavelets:avoided by wavelets:

NonNon--stationary signalsstationary signalsSignals with discontinuitiesSignals with discontinuities

Approximating a function using Haar Wavelets

The Continuous Wavelet TransformThe Continuous Wavelet Transform

Allows us to find the amplitude of “frequency” Allows us to find the amplitude of “frequency” components at different timescomponents at different timesPlotted in 3 dimensions (amplitude, scale, Plotted in 3 dimensions (amplitude, scale, translation)translation)

Differences with Short Term FTDifferences with Short Term FTDifferent window sizes used to measure Different window sizes used to measure different frequenciesdifferent frequencies

Example CWTExample CWT

A non-stationary signal and its corresponding Continuous Wavelet Transform

ApplicationsApplications

Data CompressionData CompressionFBI fingerprint archive (200+ terabytes)FBI fingerprint archive (200+ terabytes)

DenoisingDenoising Noisy DataNoisy DataStoring/Synthesizing Musical TonesStoring/Synthesizing Musical Tones

Data Compression / Data Compression / DenoisingDenoising

Both tasks involve removing insignificant Both tasks involve removing insignificant informationinformationProcedureProcedure

1.1. Choose a wavelet type and decomposition level. Choose a wavelet type and decomposition level. Compute decomposition of signal.Compute decomposition of signal.

2.2. Remove coefficients less than a given thresholdRemove coefficients less than a given threshold3.3. Reconstruct signal (for Reconstruct signal (for denoisingdenoising only)only)

Before After

Wavelet Image CompressionWavelet Image Compression

Wavelets JPEG

Compression Ratio – 5:1


Wavelets JPEG



Wavelets JPEG



Wavelets


RecapRecap

WaveletsWavelets

Multiple wavelet typesMultiple wavelet typesWavelet ExpansionWavelet ExpansionContinuous Wavelet Continuous Wavelet TransformTransformGood time/frequency Good time/frequency localization

FourierFourier

Complex exponentialsComplex exponentialsFourier SeriesFourier SeriesFourier TransformFourier Transform

Poor time/frequency Poor time/frequency localizationlocalization localization

Questions?Questions?

Wavelets - ee.stanford.eduosgood/Sophomore College/Wavelets.pdfWavelets: Who Gives A Damn? The Fourier Transform Has no time resolution (integral is from –∞ to ∞) Not very useful

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