Wavelets: A Brief TutorialLarry Cornman

Signal AnalysisOne of the goals of signal analysis is determining how well the data matches a given function or model.

Fourier AnalysisGiven certain assumptions, one can decompose a ( -periodic) signal in terms of complex exponentials (e.g., sines and cosines). 1-D Fourier series:

( ) i tf t c e ωω

ω

∞

=−∞

= ∑2

0

1 ( )2

i tc f t e dtπ

ωω π

−= ∫

2π

Fourier Analysis, cont’dFourier analysis allows for the study of the frequency characteristics of the data.A key tool for doing that is the power spectra:

2( ) cωωΦ =

Fourier Analysis, cont’dAnother tool in Fourier analysis is the Fourier transform:

( ) ( ) i tf f t e dtωω+∞

−∞

= ∫

Fourier Analysis: Examples

Pure Sinusoid - SpectraPure Sinusoid

Fourier Analysis: Examples

Sum of Four Sinusoids - SpectraSum of Four Sinusoids

Fourier Analysis: ExamplesPower spectral analysis is a major tool in the study of turbulence.

Problems with Fourier AnalysisThe signal is assumed to be periodic on the given interval – which is rarely true. Other basis functions can also be used to expand the signal – and these are often chosen to reflect expected characteristics of the data.

Problems with Fourier Analysis, cont’d

A key drawback to these approaches is that the signal is decomposed in terms of a single parameter, the frequency. So, for a non-stationary signal, any temporal information is lost (smeared).

Problems with Fourier Analysis, cont’d

Four Sinusoids – in Sequence Four Sinusoids – Spectra

Problems with Fourier Analysis, cont’dTo accommodate non-stationarity, one could use short-time windowed Fourier transforms.But this results in an equal partition in time/freq.

Frequency

Time

Finally…Wavelets!We want fairly general, local functions that can be used to analyze signals.But we also want functions that are scalable, so that they have simultaneous frequency-resolving and location-resolving capabilities.

“Scale”

Time

Wavelets: DefinitionSimilar to the Fourier series, we have a wavelet series and wavelet transform:

Wavelet Series:

, ,,

( ) ( )j k j kj k

f t c tψ+∞

=−∞

= ∑ , ( )(2 ,2 )j jj kc Wf k− −=

1/ 2 *( )( , ) ( ) ( )t bWf a b a f t dta

ψ+∞

−∞

−= ∫Wavelet Transform:

Wavelet RequirementsIn order to reconstruct the signal from it’s transform without loss of information, it must be admissible. This fact also implies that

( ) 0t dtψ+∞

−∞

=∫

This implies that the function must be oscillatory,i.e., wave-like.

ψ

Wavelet Requirements, cont’dThe other condition imposed on wavelets is regularity: the wavelet function should have some smoothness and concentration in both time and frequency domains.So, we have a localized, oscillatory function: A Wavelet!

Examples of Wavelets

Wavelet Analysis: Examples

Sum of Four Sinusoids Four, Sequential Sinusoids

Scal

e

Time

Filtering via Wavelets

Original Time Series

“Heavy” Filtering

“Light” Filtering

Application: Plume and Noise Detection

Raw BackscatterPower From REAL(Aerosol Lidar)

Wavelet Application, cont’d

Raw Data -Single Radial

Wavelet-Filtered“Plume Region”

Wavelet-Filtered“Noise Region”

Wavelet Application, cont’d

Wavelet-Filtered BackscatterPower

Noise Detection

One Wavelet “Noise Score” Total “Noise Score”

Plume Detection

Raw Data Final “Plume Score”

Wavelets: A New Tool For YourData Analysis Toolbox…

Wavelets are easy to use. Matlab has many built-in functions. Other libraries exist.A wide variety of applications in signal and image processing.Go home and try it now!