fourier and wavelet transform

7/24/2019 fourier and wavelet transform

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Fourier and Wavelets TransformsCintia Bertacchi Uvo


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Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo

http://www.mathworks.com/access/helpdesk/help/pdf_doc/wavele

t/wavelet_ug.pdfAmara Graps (1995)

Fourier Analysis

Frequency analysis

Linear operator

Idea: Transforms time-based signals to frequency-based signals.


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Any periodic function can be decomposed to a sum of sineand cosine waves, i.e.: any periodic function f(x) can berepresented by

cos sin

where:

1

2 ;

1

cos ;

1 sin


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Basis functions: sines and cosines

Draw back: transforming to the frequency domain, timeinformation is lost. We dont know when an event

happened.


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Discrete Fourier Transform: Estimate the Fourier Transformof function from a finite number of its sample points.

Windowed Fourier Transform: Represents non periodic

signals.. Truncates sines and cosines to fit a window of particularwidth.

. Cuts the signal into sections and each section is analysedseparately.


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Example:

Windowed Fourier Transform where the window is a squarewave

. A single window width is used

. Sines and cosines are truncated

to fit to the width of the window.

. Same resolution al all locations

of the time-frequency plane.


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Wavelets Transform

. Space and frequency analysis (scale and time)

. Linear operator

A windowing technique with variable-sized regions.

. Long time intervals where more precise low-frequency information is needed.

. Shorter regions where high-frequency information isof interest.


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Basis functions: infinite number of wavelets (morecomplicated basis functions)

Variation in time and frequency (time and scale) so that the

previous example becomes:


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Definition: A wavelet is a waveform of effectively limitedduration that has an average value of zero.

Scale aspect: The signal presents a very quick local variation.

Time aspect:

Rupture and edges detection.

Study of short-time phenomena as transient processes.


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There are infinite sets of Wavelets Transforms.

Different wavelet families: Different families providedifferent relationships between how compact the basis

function are localized in space and how smooth they are.


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Vanishing Moments: if the average value ofxk (x) is zero

(where (x) is the wavelet function), for k = 0, 1, , n thenthe wavelet has n + 1 vanishing moments and polynomials ofdegree n are suppressed by this wavelet.


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Use:

Detect Discontinuities and Breakdown Points

Small discontinuity in thefunction


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. Remove noise

from time series

. Detect Long-Term Evolution

. Identify PureFrequencies

. Suppress signals


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The Continuous Wavelet Transform (CWT)

Definition: the sum over all time of the signal multiplied byscaled, shifted versions of the wavelet function :

, ,,

where:f(t) is the signal,

,, is the wavelet, andC(scale, position) are the wavelet coefficients


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Scale


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Position


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Steps to a Continuous Wavelet Transform

1. Take a wavelet and compare it to a section at thestart of the original signal.

2. Calculate C, i.e., how closely correlated the wavelet

is with this section of the signal. , ,,


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3. Shift the wavelet to the right and repeat steps 1

and 2 until youve covered the whole signal.

4. Scale (stretch) the wavelet and repeat steps 1

through 3.

5. Repeat steps 1 through 4 for all scales.


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Plot the time-scale view of the signal

x-axis is the position along the signal (time), y-axis is thescale, and the colour at eachx-y point represents themagnitude of C.

Example: From above


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From the side (3D)


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Low scale => Compressed wavelet => Rapidly changing

details => High frequency.

High scale => Stretched wavelet => Slowly changing, coarse

features => Low frequency.


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Reconstruction Inverse Discrete Wavelet Transform

Filtering and upsampling

Reconstruct the signal from the wavelet coefficients.

On Matlab:

ss = idwt(ca1,cd1,'db2');


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Approximations

or Details can bereconstructedseparately from

their coefficientvectors.


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Report:

Choose a data series

1- Apply Fourier transform

2- Decompose using wavelets

Compare results

fourier and wavelet transform

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