From Fourier Series to Analysis of Non-stationary Signals - X

Transcription

1 From Fourier Series to Analysis of Non-stationary Signals - X prof. Miroslav Vlcek December 14, 216

2 Contents Examples and MATLAB project 1 Examples and MATLAB project 2

3 Contents Examples and MATLAB project 1 Examples and MATLAB project 2

4 Transforms Fourier transform gives the spectrum of the whole time-series Which is OK if the time-series is stationary. But what if it is not? We need a technique that can march along a timeseries and analyze spectral content in different time instants. Short Time Fourier Transform, Wavelet Transform

5 STFT and non-stationarity signal 1 Remove the mean of a signal 2 Make moving average filtering 3 Do segmentation of a signal using window functions 4 Perform Fourier transform 5 Do not forget the interpolation

6 Moving average filtering Examples and MATLAB project Assume we have a EEG signal corrupted with noise Set the mean to zero Apply moving average filter to the noisy signal (use filter order=3 and 5) The higher filter order will remove more noise, but it will also distort the signal more (i.e. remove the signal parts also) So, a compromise has to be found (normally by trial and error)

7 Example 1: Moving average filtering % moving average filtering clear load( zdroj.mat ); x=eeg(3).data(:,1); N=length(x); % length of average window is 3 for i=1:n-2, y3(i)=(x(i)+x(i+1)+x(i+2))/3; end y3(n-1)=(x(n-1)+x(n))/3; y3(n)=x(n)/3; %length of average window is 5 for i=1:n-4, y5(i)=(x(i)+x(i+1)+x(i+2)+x(i+3)+x(i+4))/5; end

8 Example 1: Moving average filtering y5(n-3)=(x(n-3)+x(n-2)+x(n-1)+x(n))/5; y5(n-2)=(x(n-2)+x(n-1)+x(n))/5; y5(n-1)=(x(n-1)+x(n))/5; y5(n)=x(n)/5; subplot(3,1,1), plot(x, r ); title( original EEG ) subplot(3,1,2), plot(y3, g ); title( EEG signal with averaging of length 3 ) subplot(3,1,3), plot(y5, b ); title( EEG signal with averaging of length 5 ) print -depsc figureeeg

9 Example 1: Moving average filtering 2 original EEG EEG signal with averaging of length EEG signal with averaging of length

10 Example 2: Meyer wavelets % Set effective support and grid parameters. lb = -8; ub = 8; n = 124; % Meyer wavelet and scaling functions. [phi,psi,x] = meyer(lb,ub,n); subplot(211) plot(x,psi), grid title( Meyer wavelet ) subplot(212) plot(x,phi), grid title( Meyer scaling function )

11 Example 2: Meyer wavelets 1.5 Meyer wavelet Meyer scaling function Figure: Meyer wavelets

12 Example 3: Discrete Wavelet Transform 1 Load in EEG signal with load( zdroj.mat ); name= dmey ; name2= db1 ; signal=eeg(3).data(:,1); s=signal; ls = length(s); 2 The Discrete Wavelet Transform is provided by MATALB under a variety of extension modes. We can perform a one-stage wavelet decomposition of the signal for example with [ca1,cd1] = dwt(s,name); 3 Here ca1 is a vector of the approximation coefficient and cd1 the detailed coefficient. 4 You can plot these coefficients.

13 Example 3: Discrete Wavelet Transform 5 Perform one step reconstruction of ca1 and cd1. a1 = upcoef( a,ca1,name,1,ls); d1 = upcoef( d,cd1,name,1,ls); 6 Then you can plot the EEG signal approximation and invert directly decomposition of EEG signal using coefficients a = idwt(ca1,cd1, db1,ls);

14 Example 3: Discrete Wavelet Transform approximation a detail d

15 Example 3: Discrete Wavelet Transform 5 approximation a detail d detail d detail d

16 Some preliminary info about EEG Figure: Brain waves within -3 Hz

17 MATLAB project 1 Develope file for computing STFT and wavelet transform for an EEG signal

18 Step 1: Load signal, input parameters N = 128; okno = hanning( N+1); okno = okno( 1:N); fs = 1; prekryv = N-1; lupa = 64; zacatek = 1; delka = 1; load( zdroj.mat ); signal=eeg(3).data(:,1);

19 Step 2: FFT and moving average f =128*(1:124)/248; ax = abs(fft(signal,248)); signal = signal( zacatek: zacatek+delka-1); % here include moving average disp( Computing spectrum... ) X = rozsekej( signal, N); % Segmentation X = diag( okno) * X; % Windowing [r,c] = size( X); X = [X; zeros( lupa, c)];

20 Step 3: DFT, Spectrogram Examples and MATLAB project spektrum = fft( X); % spectrum spektrum = spektrum( 1:64, :); % the first 64 components RS = real( spektrum); IS = imag( spektrum); [numbercomponents, numbersamples] = size( RS); figure(1); colormap( jet); subplot(311) plot(signal); grid; title( Original signal ); subplot(312) plot(f,ax(1:124), LineWidth,2, color,[ 1]); title( Frequency content ) subplot(313) imagesc( abs( spektrum)); title( STFT ); print -depsc figure1eeg

21 Signal with 5 Hz noise Examples and MATLAB project 2 Original signal x 15 Frequency content STFT

22 Signal without 5 Hz noise Examples and MATLAB project 2 signal with 5 Hz noise reduced x 15 Fourier transform STFT

23 Step 4: Wavelet decomposition clear % Compute WT of an EEG signal load( zdroj.mat ); signal=eeg(3).data(:,1); ls=length(signal); name= dmey ; [ca1,cd1] = dwt(signal,name); a1 = upcoef( a,ca1,name,1,ls);

24 Step 4: A typical scalogram of EEG signal=a1; wlen=64; Bsup=1; Logsc=; wavtype = gaus4 ; wavtype2 = mexh ; figure(1); h = msgbox( Wavelet transform computation..., Please wait ); W = cwt(signal, 1:wlen/2, wavtype, abslvl ); close( h); WW = abs( W);

25 Step 4: A typical scalogram of EEG if Bsup == 1, WW = WW - min( min ( WW)); end; if Logsc == 1, WW = log1( WW+eps); end; subplot(2,1,1) plot(signal, LineWidth,1., color,[1 ]) subplot(2,1,2) imagesc(ww); colormap(jet);

26 A lorry passing in front of a microphone.6 Orange lorry.wav time(sec)

27 MATLAB project - frequency-time stamp of a vehicle In order to consult your noise measurement and compiling the final project, we will meet on Tuesday, January 4, 217. Do not hesitate to contact me any time later. I expect your project on a noise analysis for a real objects - vehicles, engines,... to be delivered by February 17th. Merry Christmas!