Post-processing using Matlab (Advanced)

OvGU! Vorlesung « Messtechnik »!

Post-processing using Matlab (Advanced)! Dominique Thévenin! Lehrstuhl für Strömungsmechanik und Strömungstechnik (LSS)! [email protected]!

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Noise filtering (1/2)!  We have discussed the last time how to filter (random) noise!  Execute again filternoise_demo.m!  Considering a signal composed of the superposition of a sinus wave at frequency f1=1 Hz with amplitude A1, plus random noise with amplitude A2!  Random noise is generated under Matlab using rand (value between 0 and 1)!

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Noise filtering (2/2)!  For random noise, something like h=h+A2*(rand(size (h))-0.5);  Test different cut-off frequencies for your filter, from 50 down to 2 Hz for example!  Test different amplitudes, for example!  A1=2, A2=1: signal to noise amplitude = 4!  A1=1, A2=2: signal to noise amplitude = 1

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 Observations?!

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Understanding better filtering! Download and execute filter_demo.m Check the script to superpose signals with frequency of 1 Hz (f1) and 100 Hz (f2):  Both frequencies are well separated (factor 100)!  The signal looks like that:

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Filtering (1/3)!  One of the most useful application of FFT is filtering!  You can decide freely which frequency domain should be filtered out!!  Typically, you will use a low-pass filter, in order to remove « experimental noise ». Noise is normally always located at high frequencies...!  In our present example, we consider that « noise » is the signal at 100 Hz, while the signal at 1 Hz is the « information » that should be acquired... !

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Filtering (2/3)!  In order to filter the signal at a given frequency, it is necessary to set the corresponding part of the DFT H(f) to 0!  But the DFT is imaginary...!  Therefore, we must of course set to 0 both the real and the imaginary component!!  Either separately (working on HR and HI)!  or directly (working on H)!

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Filtering (3/3)!

low frequency! keep!! high frequency! suppress by filtering !

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Script for filtering!  Identify in your script the correct frequency domain (we decide: filtering everything from 50 Hz up to the Nyquist frequency)!  Code the cut-off frequency in a flexible manner using a variable...!  Set the corresponding part of the DFT to 0 (complex value)!  Compute the inverse FFT using ifft, and look at the result, by comparison with the original signal!!

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Energy Spectral Density (ESD)!  The energy spectral density describes how the energy of a signal is distributed with frequency. It is nothing else than the square of the magnitude of the Fourier transform!

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Power Spectral Density (PSD)!  If the global mean value of a signal is not 0 and has not been removed, the ESD tends toward infinity for long sequences!

 Indeed, for such a case, it will become at some point even impossible to compute the DFT!  As an alternative, it is still possible to compute the Power Spectral Density (PSD)!  Physically, the computation of the PSD uses an overlapping segmentation of the original signal, followed by corresponding DFT computations and final re-assembling by averaging!

1!

N!

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Practical ESD/PSD computation!  Relies on the complex script structure spectrum  See help spectrum for more information!  Several different methods have been coded to compute numerically a PSD!  It will first be necessary to define an appropriate handle to receive the information, for example through !  HWelch=spectrum.welch;  Afterwards, the PSD of the time signal h(t) can be computed as!  HPSD=psd(HWelch,h,’Fs’,f)  PSD are only necessary for very long signals with a high level of noise: you should not encounter this too often at first...! 11

Practical PSD computation!

h(t)

sampling

h(kΔt)

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Hi(kΔf)

segmentation

averaging

hi(kΔt)

DFT

PSD!

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Script for PSD!  Download and execute psd_demo.m!  Considering a signal with a frequency of 10 Hz, plus noise!  Code the cut-off frequency in a flexible manner using a variable. Choose then first for example 50 Hz.!  Compare the efficiency of both methods (basic filtering, PSD determination)!  Solution: psd_demo.m!

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Where do we stand?! We know how to carry out frequency analysis and filtering   Using DFT! Wavelet will be soon an alternative...   But DFT can do even more!

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Correlation!  It is often essential to determine the correlation between two signals!  This means: « how similar are the two signals, determined in a quantitative manner ?»!  The DFT can be used in an extremely efficient manner to determine this correlation based on a frequency analysis!  This is in particular due to the mathematical property:!  DFT(correlation between h(t) and g(t))=N H*(f) G(f)!  Means: the DFT of the correlation is the “product” of the DFTs of the two signals, multiplied by the signal length (N)!  As a consequence, it is sufficient to compute the individual DFTs to quantify the correlation!! 15

Normalizing the correlation!  Obviously, the best possible correlation should be obtained for two identical signals!  The general equation delivers in this case:!  DFT(correlation between h(t) and h(t))=N H*(f) H(f)! !

!

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!= N ||H(f)||2!

 As a consequence, the definition of the normalized correlation (value between -1 and 1, with 0=no correlation at all) between a known, reference function h(t) and a test function g(t) becomes:!  Cor(h(t),g(t))=iDFT(H*(f) G(f) / (||H(f)|| ||G(f)||))!

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Script to compute correlation!  Download and execute correl_demo.m!  We will now introduce a second time signal g, identical in length and discretization to h, but containing different information!

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Script to compute correlation!  Test in particular:!   g = h   g = -h   g = 2*h   g = 2*h + 13   g = (5-2*h).^2   g = 1   g = h+A2*(rand(size(h))-0.5) with A2 from 0.005 to 0.5!  Observations?!  Solution: correl_demo.m!

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Compute correlation in Matlab!  As a direct alternative, Matlab allows also the computation of correlations using the function!  corrcoef  The information we need is contained outside of the diagonal! C(h1,h1)

!C(h1,h2)!

C(h2,h1)

!C(h2,h2)!

 Implemented for comparison in your script (see help corrcoef for more information)!

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Introducing Wavelets (1/2)!  Wavelets have been defined in the 50s!  as a complement to classical Fourier analysis!  Why extend the Fourier analysis at all?!

 Drawback of FFT: by transformation, the time information is completely lost when obtaining the frequency spectrum!  It is thus impossible to know at which time an event took place! 20  A choice is needed: either time or frequency, not both!!

Introducing Wavelets (2/2)!  If the signal does not change (in a statistical sense) with time, this is not a problem!!  This was the case in all our examples up to now.!  This covers also most simple applications, and 50% of more complex applications.!  But when considering signals with drift, monotonous increase/decrease, abrupt changes, and near the boundaries of the time signal, changes in time become significant, and cannot be analyzed with FFT.!  Time variations must then be taken into account!!

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Short-time Fourier analysis!  To solve this problem, D. Gabor introduced the short-time Fourier analysis!  It is simply a DFT relying on windowing the signal!  The window has a constant, user-defined size!

 We now have an analysis both in time and frequency!

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 But using a constant-size window limits the possible resolution in both dimensions

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Wavelet analysis!  As a final evolution, the wavelet analysis introduces windowing on variable-sized regions!  Long window sizes are employed where accurate low-frequency information is required, shorter windows are used where highfrequency information is important.!

 We now have an analysis both in time and frequency (or scale)!

  With an optimally-adapted window size! 23

Advantage of Wavelet analysis!  Wavelets can reveal aspects of data that DFT completely miss (trends, breakdown points, discontinuities, fractal nature...)!

wavelet time point 100!

fft

time point 100!

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Fourier analysis vs. Wavelet (1/2)!  The Fourier analysis breaks up a signal into sine waves of various frequencies!  The function « sin » is the basis function for decomposition!  This function extends from -∞ to +∞, and is thus not well suited for a local analysis!

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Fourier analysis vs. Wavelet (2/2)!  The Wavelet transform uses other basis functions, called wavelet functions!  These wavelet functions are local in space!  Different families of wavelet functions exist!  The decomposition is then similar to FFT!

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Scaling / shifting wavelet functions!  Scaling!

 Shifting!

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Understanding wavelet output! Frequency=1/(time scale)! low f !

high f !

Time-information is not lost!!

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First hands-on: noisy signal!  For this basic example, we use a pre-defined noisy signal of Matlab, comprising 1000 points with a sampling period of 1 (« s »).!  From Matlab, type!  load noissin  plot noissin

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Wavelet analysis (1/2)!  The wavelet analysis is realized with the instruction cwt (for Continuous Wavelet Transform)!  From Matlab, type!  c = cwt(noissin,1:48,’db4’) Wavelet basis function! (here Daubechies type 4)! scales to consider! signal to process!

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Wavelet analysis (2/2)!  The variable c contains the output of the wavelet analysis.!  For a direct graphical representation, add a last flag!  c = cwt(noissin,1:48,’db4’,’plot’);

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Choosing the scales!  The choice of the scales is essential for the analysis!  All scales must be real, positive, increasing and are bounded by the length of the signal. Try:!  c = cwt(noissin,2:2:128,’db4’,’plot’)  Periodicity is now visible! (about 5 periods)!

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Further 1D examples!  Use the Wavelet GUI wavemenu  Choose wavelet 1-D  Load File/Example Analysis/Basic Signals  Frequency Breakdown (2 different frequencies in time)!

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Image processing!  Use the Wavelet GUI wavemenu  Choose SWT Denoising 2-D  Load File/Example Analysis/Indexed Images  Choose noisywoman  Try to modify the parameters for a better result!

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