Fourier and Wavelets Transforms Cintia Bertacchi Uvo http://www.mathworks.com/access/helpdesk/help/pdf_doc/wavele t/wavelet_ug.pdf Amara 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: 1 1 ; cos ; 2 1 sin

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 locations of 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, and C(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 Transform Filtering 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 Approximations or 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 transform 2- Decompose using wavelets Compare results

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