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, 21

2 Contents Stationary and non-stationary 1 Stationary and non-stationary 2 3

3 Contents Stationary and non-stationary 1 Stationary and non-stationary 2 3

4 Contents Stationary and non-stationary 1 Stationary and non-stationary 2 3

5 Stationary and non-stationary Continuous system Discrete system u(t)... input (control) vector x(t)... state vector y(t)... output vector Linear state variable system ẋ(t) = A(t) x(t) + B(t) u(t) y(t) = C(t) x(t) + D(t)u(t) u(n)... input (control) vector x(n)... state vector y(n)... output vector Linear state variable system x(n + 1) = M(n) x(n)+ N(n) u(n) y(n) = C(n) x(n) + D(n)u(n) A(t) system matrix (n n) B(t) matrix of inputs (n r) C(t) matrix of outputs(m n) D(t) matrix of outputs(m r) M(n) system matrix N(n) matrix of inputs C(n) matrix of outputs D(n) matrix of outputs

6 Stationary and non-stationary Continuous system Discrete system u(t)... input (control) vector x(t)... state vector y(t)... output vector Linear state variable system ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + Du(t) u(n)... input (control) vector x(n)... state vector y(n)... output vector Linear state variable system x(n + 1) = M x(n)+ N u(n) y(n) = C x(n) + Du(n) A system matrix (n n) B matrix of inputs (n r) C matrix of outputs (m n) D matrix of outputs (m r) M system matrix N matrix of inputs C matrix of outputs D matrix of outputs

7 Stationary and non-stationary Non-stationary signals differential/difference equations with time-varying coefficients Airy s functions Ai(t) = 1 3 Bi(t) = 1 3 ÿ(t) t y(t) = t [ I 1/3 ( 2 3 t3/2 ) I 1/3 ( 2 3 t3/2 )] t [ I 1/3 ( 2 3 t3/2 ) + I 1/3 ( 2 3 t3/2 )]

8 Stationary and non-stationary Stationary and non-stationary 1.8 ℜ(Ai(t)) ℜ(Bi(t)) prof. Miroslav Vlcek 4 t 2 Lecture 1 2

9 Stationary and non-stationary Stationary signals differential/difference equations with constant coefficients ÿ(t)+ω 2 y(t) = Harmonic wave (periodic functions) cos(ω t) sin(ω t)

10 Stationary and non-stationary Stationary and non-stationary 1.5 sin(π t) cos(π t) prof. Miroslav Vlcek t 1 Lecture 1 2 3

11 About stationarity A deterministic signal is said to be stationary if it can be written as a discrete sum of cosine waves or exponentials : x(t) = A + x(t) = N k= N N A k cos(2πkf t +Φ k ) (1) k=1 C k exp(2πjkf t +Φ k ) (2) i.e. as a sum of elements which have constant instantaneous amplitude and instantaneous frequency.

12 About stationarity In the random case, a signal {x(n)} is said to be wide-sense stationary (or stationary up to the second order) if its variance is independent of time σ 2 = E[(x µ) 2 ] = 1 N 1 (x µ) 2 (n) N n=

13 About stationarity The autocorrelation function for a discrete process of length N {x(n)} with known mean µ and variance σ, xx (n, n, m) xx (n, n+ m) = 1 Nσ 2 N (x(n) µ)(x(n+m) µ) n=1 depends only on the time distance n+m.

14 ...and non-stationarity A signal is said to be non-stationary if one of these fundamental assumptions is no longer valid, e.g. either variance σ, or autocorrelation xx (n, n, m), or both are time-varying. 1 For example, a finite duration signal, and in particular a transient signal (for which the length is short compared to the observation duration), is non-stationary. 2 Speech and EEG are non-stationary signals.

15 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

16 Moving average filtering 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)

17 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

18 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

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

20 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 )

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

22 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.

23 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);

24 Example 3: Discrete Wavelet Transform approximation a detail d

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

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

27 MATLAB project 1 Simplify the moving average using declaration of the vector y(1:1:n+4)=; and create MATLAB function for average. 2 Develope file for computing STFT and wavelet transform for an EEG signal

28 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);

29 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)];

30 Step 3: DFT, Spectrogram 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

31 Signal with 5 Hz noise 2 Original signal x 15 Frequency content STFT

32 Signal without 5 Hz noise 2 signal with 5 Hz noise reduced x 15 Fourier transform STFT

33 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);

34 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);

35 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);

36 A typical scalogram of EEG

37 MATLAB project Finish the task from the last lesson - create your own graphic user interface which provides Haar wavelet approximation of a signal. Prepare your graphic user interface for analysis of EEG signal using DFT and Wavelets. Write the report and prepare presentation of your EEG analysis. We will meet on January 4, 211. You are expected to present your results. Merry Christmas!