Fourier and Wavelets TransformsCintia Bertacchi Uvo
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
http://www.mathworks.com/access/helpdesk/help/pdf_doc/wavelet/wavelet_ug.pdfAmara Graps (1995)
Fourier Analysis
Frequency analysis Linear operator
Idea: Transforms time-based signals to frequency-based signals.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Any periodic function can be decomposed to a sum of sine and cosine waves, i.e.: any periodic function f(x) can be represented by
cos sin
where:
12 ;
1cos ;
1sin
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Basis functions: sines and cosines
Draw back: transforming to the frequency domain, time information is lost. We don’t know when an event happened.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Discrete Fourier Transform: Estimate the Fourier Transform of 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 particular width. . Cuts the signal into sections and each section is analysed separately.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Example:Windowed Fourier Transform where the window is a square wave
. A single window width is used
. Sines and cosines are truncated to fit to the width of the window.
. Same resolution al all locationsof the time-frequency plane.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
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 is
of interest.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Basis functions: infinite number of wavelets (more complicated basis functions)
Variation in time and frequency (time and scale) so that the previous example becomes:
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Definition: A wavelet is a waveform of effectively limited duration 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.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
There are infinite sets of Wavelets Transforms.
Different wavelet families: Different families provide different relationships between how compact the basis function are localized in space and how smooth they are.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Vanishing Moments: if the average value of xkψ (x) is zero (where ψ (x) is the wavelet function), for k = 0, 1, …, n then the wavelet has n + 1 vanishing moments and polynomials of degree n are suppressed by this wavelet.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Use:Detect Discontinuities and Breakdown Points
Small discontinuity in the function
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
. Remove noise from time series. Detect Long-Term Evolution. Identify Pure Frequencies. Suppress signals
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
The Continuous Wavelet Transform (CWT)
Definition: the sum over all time of the signal multiplied by scaled, shifted versions of the wavelet function :
, Ψ , ,
where: f(t) is the signal,Ψ , , is the wavelet, andC(scale, position) are the wavelet coefficients
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Scale
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Position
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Steps to a Continuous Wavelet Transform
1. Take a wavelet and compare it to a section at the start of the original signal.
2. Calculate C, i.e., how closely correlated the wavelet is with this section of the signal.
, Ψ , ,
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
3. Shift the wavelet to the right and repeat steps 1 and 2 until you’ve covered the whole signal.
4. Scale (stretch) the wavelet and repeat steps 1 through 3.
5. Repeat steps 1 through 4 for all scales.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Plot the time-scale view of the signal x-axis is the position along the signal (time), y-axis is the scale, and the colour at each x-y point represents the magnitude of C.
Example: “From above”
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
“From the side (3D)”
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Low scale => Compressed wavelet => Rapidly changing details => High frequency.
High scale => Stretched wavelet => Slowly changing, coarse features => Low frequency.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Decomposition and Reconstruction
Approximations (A): low-frequency components (high-scale)Details (D): high-frequency components (low scale)
Decomposition – filtering and downsampling
On Matlab:[cA,cD] = dwt(s,’db2’);
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Reconstruction – Inverse Discrete Wavelet TransformFiltering and upsampling
Reconstruct the signal from the wavelet coefficients.
On Matlab:ss = idwt(ca1,cd1,'db2');
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Approximationsor Details can be reconstructed separately from their coefficient vectors.
Lund University / LTH / Dept. Water Res. Eng./ Cintia Bertacchi Uvo
Report:
Choose a data series
1- Apply Fourier transform2- Decompose using waveletsCompare results