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Presents. The Story of Wavelets. Robi Polikar Dept. of Electrical & Computer Engineering Rowan University. The Story of Wavelets. Technical Overview But…We cannot do that with Fourier Transform…. Time - frequency representation and the STFT Continuous wavelet transform

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The Story of Wavelets

Robi PolikarDept. of Electrical & Computer EngineeringRowan University

The Story of Wavelets

  • Technical Overview

    • But…We cannot do that with Fourier Transform….

    • Time - frequency representation and the STFT

    • Continuous wavelet transform

    • Multiresolution analysis and discrete wavelet transform (DWT)

  • Application Overview

    • Conventional Applications: Data compression, denoising, solution of PDEs, biomedical signal analysis.

    • Unconventional applications

    • Yes…We can do that with wavelets too…

  • Historical Overview

    • 1807 ~ 1940s: The reign of the Fourier Transform

    • 1940s ~ 1970s: STFT and Subband Coding

    • 1980s & 1990s: The Wavelet Transform and MRA

What is a Transformand Why Do we Need One ?

  • Transform: A mathematical operation that takes a function or sequence and maps it into another one

  • Transforms are good things because…

    • The transform of a function may give additional /hidden information about the original function, which may not be available /obvious otherwise

    • The transform of an equation may be easier to solve than the original equation (recall your fond memories of Laplace transforms in DFQs)

    • The transform of a function/sequence may require less storage, hence provide data compression / reduction

    • An operation may be easier to apply on the transformed function, rather than the original function (recall other fond memories on convolution).

Jean B. Joseph Fourier


December, 21, 1807

“An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids”

J.B.J. Fourier

  • Complex function representation through simple building blocks

    • Basis functions

  • Using only a few blocks  Compressed representation

  • Using sinusoids as building blocks  Fourier transform

    • Frequency domain representation of the function

How Does FT Work Anyway?

  • Recall that FT uses complex exponentials (sinusoids) as building blocks.

  • For each frequency of complex exponential, the sinusoid at that frequency is compared to the signal.

  • If the signal consists of that frequency, the correlation is high  large FT coefficients.

  • If the signal does not have any spectral component at a frequency, the correlation at that frequency is low / zero,  small / zero FT coefficient.

FT At Work

FT At Work




FT At Work


FT At Work

Complex exponentials (sinusoids) as basis functions:


An ultrasonic A-scan using 1.5 MHz transducer, sampled at 10 MHz

Stationary and Non-stationary Signals

  • FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. Why?

  • Stationary signals consist of spectral components that do not change in time

    • all spectral components exist at all times

    • no need to know any time information

    • FT works well for stationary signals

  • However, non-stationary signals consists of time varying spectral components

    • How do we find out which spectral component appears when?

    • FT only provides what spectral components exist , not where in time they are located.

    • Need some other ways to determine time localization of spectral components

Stationary and Non-stationary Signals

  • Stationary signals’ spectral characteristics do not change with time

  • Non-stationary signals have time varying spectra


Stationary vs. Non-Stationary


Perfect knowledge of what

frequencies exist, but no

information about where

these frequencies are

located in time


Shortcomings of the FT

  • Sinusoids and exponentials

    • Stretch into infinity in time, no time localization

    • Instantaneous in frequency, perfect spectral localization

    • Global analysis does not allow analysis of non-stationary signals

  • Need a local analysis scheme for a time-frequency representation (TFR) of nonstationary signals

    • Windowed F.T. or Short Time F.T. (STFT) : Segmenting the signal into narrow time intervals, narrow enough to be considered stationary, and then take the Fourier transform of each segment, Gabor 1946.

    • Followed by other TFRs, which differed from each other by the selection of the windowing function

Short Time Fourier Transform(STFT)

  • Choose a window function of finite length

  • Place the window on top of the signal at t=0

  • Truncate the signal using this window

  • Compute the FT of the truncated signal, save.

  • Incrementally slide the window to the right

  • Go to step 3, until window reaches the end of the signal

  • For each time location where the window is centered, we obtain a different FT

    • Hence, each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information






Signal to

be analyzed

FT Kernel

(basis function)

STFT of signal x(t):

Computed for each

window centered at t=t’



Windowing function

centered at t=t’




STFT at Work

STFT At Work

STFT At Work


  • STFT provides the time information by computing a different FTs for consecutive time intervals, and then putting them together

    • Time-Frequency Representation (TFR)

    • Maps 1-D time domain signals to 2-D time-frequency signals

  • Consecutive time intervals of the signal are obtained by truncating the signal using a sliding windowing function

  • How to choose the windowing function?

    • What shape? Rectangular, Gaussian, Elliptic…?

    • How wide?

      • Wider window require less time steps  low time resolution

      • Also, window should be narrow enough to make sure that the portion of the signal falling within the window is stationary

      • Can we choose an arbitrarily narrow window…?

Selection of STFT Window

Two extreme cases:

  • W(t) infinitely long:  STFT turns into FT, providing excellent frequency information (good frequency resolution), but no time information

  • W(t) infinitely short:

     STFT then gives the time signal back, with a phase factor. Excellent time information (good time resolution), but no frequency information

    Wide analysis window poor time resolution, good frequency resolution

    Narrow analysis windowgood time resolution, poor frequency resolution

    Once the window is chosen, the resolution is set for both time and frequency.

Heisenberg Principle

Frequency resolution: How well two spectral components can be separated from each other in the transform domain

Time resolution: How well two spikes in time can be separated from each other in the transform domain

Both time and frequency resolutions cannot be arbitrarily high!!! We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals

The Wavelet Transform

  • Overcomes the preset resolution problem of the STFT by using a variable length window

  • Analysis windows of different lengths are used for different frequencies:

    • Analysis of high frequencies Use narrower windows for better time resolution

    • Analysis of low frequencies  Use wider windows for better frequency resolution

  • This works well, if the signal to be analyzed mainly consists of slowly varying characteristics with occasional short high frequency bursts.

  • Heisenberg principle still holds!!!

  • The function used to window the signal is called the wavelet

The Wavelet Transform

A normalization


Translation parameter, measure of time

Scale parameter, measure of frequency

Signal to be analyzed

Continuous wavelet transform of the signal x(t) using the analysis wavelet (.)

The mother wavelet. All kernels are obtained by translating (shifting) and/or scaling the mother wavelet

Scale = 1/frequency

High frequency (small scale)

Low frequency (large scale)

WT at Work

WT at Work

WT at Work

WT at Work

Matlab Demos on CWT

Discrete Wavelet Transform

  • CWT computed by computers is really not CWT, it is a discretized version of the CWT.

  • The resolution of the time-frequency grid can be controlled (within Heisenberg’s inequality), can be controlled by time and scale step sizes.

  • Often this results in a very redundant representation

  • How to discretize the continuous time-frequency plane, so that the representation is non-redundant?

    • Sample the time-frequency plane on a dyadic (octave) grid

Discrete Wavelet Transform

  • Dyadic sampling of the time –frequency plane results in a very efficient algorithm for computing DWT:

    • Subband coding using multiresolution analysis

    • Dyadic sampling and multiresolution is achieved through a series of filtering and up/down sampling operations
































Discrete Wavelet TransformImplementation



Half band high pass filter

Half band low pass filter



2-level DWT decomposition. The decomposition can be continues as long as there are enough samples for down-sampling.










DWT - Demystified

Length: 512

B: 0 ~ 



Length: 256

B: 0 ~ /2 Hz

Length: 256

B: /2 ~  Hz



d1: Level 1 DWT




Length: 128

B: 0 ~  /4 Hz


Length: 128

B: /4 ~ /2 Hz





d2: Level 2 DWT





Length: 64

B: 0 ~ /8 Hz

Length: 64

B: /8 ~ /4 Hz


d3: Level 3 DWT


Level 3 approximation


Implementation of DWT on MATLAB

Choose wavelet

and number

of levels

Load signal

Hit Analyze



Approx. coef.

at level 5

Level 1 coeff.

Highest freq.


Applications of


Applications of Wavelets

  • Compression

  • De-noising

  • Feature Extraction

  • Discontinuity Detection

  • Distribution Estimation

  • Data analysis

    • Biological data

    • NDE data

    • Financial data


  • DWT is commonly used for compression, since most DWT are very small, can be zeroed-out!



ECG- Compression

Denoising Implementation in Matlab

First, analyze the signal with appropriate wavelets



(Noisy Doppler)

Denoising Using Matlab

Choose thresholding


Choose noise type

Choose thrsholds



Denosing Using Matlab

Discontinuity Detection


Discontinuity Detectionwith CWT


Application Overview

  • Data Compression

  • Wavelet Shrinkage Denoising

  • Source and Channel Coding

  • Biomedical Engineering

    • EEG, ECG, EMG, etc analysis

    • MRI

  • Nondestructive Evaluation

    • Ultrasonic data analysis for nuclear power plant pipe inspections

    • Eddy current analysis for gas pipeline inspections

  • Numerical Solution of PDEs

  • Study of Distant Universes

    • Galaxies form hierarchical structures at different scales

Application Overview

  • Wavelet Networks

    • Real time learning of unknown functions

    • Learning from sparse data

  • Turbulence Analysis

    • Analysis of turbulent flow of low viscosity fluids flowing at high speeds

  • Topographic Data Analysis

    • Analysis of geo-topographic data for reconnaissance / object identification

  • Fractals

    • Daubechies wavelets: Perfect fit for analyzing fractals

  • Financial Analysis

    • Time series analysis for stock market predictions

History Repeats Itself…

  • 1807, J.B. Fourier:

    • All periodic functions can be expressed as a weighted sum of trigonometric function

    • Denied publication by Lagrange, Legendre and Laplace

    • 1822: Fourier’s work is finally published

    • 1965, Cooley & Tukey: Fast Fourier Transform

143 years

History Repeats Itself: Morlet’s Story

  • 1946, Gabor: STFT analysis:

    • high frequency components using a narrow window, or

    • low frequency components using a wide window, but not both

  • Late 1970s, Morlet’s (geophysical engineer) problem:

    • Time - frequency analysis of signals with high frequency components for short time spans and low frequency components with long time spans

    • STFT can do one or the other, but not both Solution: Use different windowing functions for sections of the signal with different frequency content

    • Windows to be generated from dilation / compression of prototype small, oscillatory signals  wavelets

  • Criticism for lack of mathematical rigor !!!

  • Early 1980s, Grossman (theoretical physicist): Formalize the transform and devise the inverse transformation  First wavelet transform !

  • Rediscovery of Alberto Calderon’s 1964 work on harmonic analysis


  • 1984, Yeves Meyer :

    • Similarity between Morlet’s and Colderon’s work, 1984

    • Redundancy in Morlet’s choice of basis functions

    • 1985, Orthogonal wavelet basis functions with better time and frequency localization

  • Rediscovery of J.O. Stromberg’s 1980 work the same basis functions (also a harmonic analyst)

  • Yet re-rediscovery of Alfred Haar’s work on orthogonal basis functions, 1909 (!).

    • Simplest known orthonormal wavelets

Transition to the Discrete Signal Analysis

Ingrid Daubechies:

  • Discretization of time and scale parameters of the wavelet transform

  • Wavelet frames, 1986

  • Orthonormal bases of compactly supported wavelets (Daubechies wavelets), 1988

  • Liberty in the choice of basis functions at the expense of redundancy

    Stephane Mallat:

  • Multiresolution analysis w/ Meyer, 1986

    Ph.D. dissertation, 1988

  • Discrete wavelet transform

  • Cascade algorithm for computing DWT


  • Decomposition of a discrete into dyadic frequencies (MRA) , known to EEs under the name of “Quadrature Mirror Filters”, Croisier, Esteban and Galand, 1976 (!)

Transition to the Discrete Signal Analysis

Martin Vetterli & Jelena Kovacevic

  • Wavelets and filter banks, 1986

  • Perfect reconstruction of signals using FIR filter banks, 1988

  • Subband coding

  • Multidimensional filter banks, 1992


  • Equivalence of QMF and MRA, Albert Cohen, 1990

  • Compactly supported biorthogonal wavelets, Cohen, Daubechies, J. Feauveau, 1993

  • Wavelet packets, Coifman, Meyer, and Wickerhauser, 1996

  • Zero Tree Coding, Schapiro 1993 ~ 1999

  • Search for new wavelets with better time and frequency localization properties.

  • Super-wavelets

  • Matching Pursuit, Mallat, 1993 ~ 1999

New & Noteworthy

  • Zero crossing representation

    • signal classification

    • computer vision

    • data compression

    • denoising

  • Super wavelet

    • Linear combination of known basic wavelets

  • Zero Tree Coding, Schapiro

  • Matching Pursuit , Mallat

    • Using a library of basis functions for decomposition

  • New MPEG standard

The Story of Wavelets

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