Outline. Introduction to Biosignal Processing. Overview of Signals. Measurement Systems. -Filtering -Acquisition Systems (Quantisation and Sampling)

Transcription

1 Outline Overview of Signals Measurement Systems -Filtering -Acquisition Systems (Quantisation and Sampling) Digital Filtering Design Frequency Domain Characterisations - Fourier Analysis - Power Spectral Density Analysis Time-Frequency Analysis - Spectrogram

2 What is a signal? Physical quantity, varying as a function of an independent variable, which carries information Examples of Biomedical Signals The electrocardiogram (ECG) Source:

3 Examples of Biomedical Signals The electromyogram (EMG) Examples of Biomedical Signals The electroencephalogram (EEG)

4 Applications of Biosignal Processing Types of Signals Continuous-Time and Discrete-Time Continuous Discrete V (t) V k] V ( t ) [ k Uniform (periodic) sampling. Sampling frequency f s = / T

5 Types of Signals Deterministic and Stochastic Deterministic Stochastic Typical Measurement Systems Acquisition Signal processing Interpretation Information extraction

6 Signal Acquisition System Analog-to-digital converter Signal Conditioning A/D Sensors/Transducers e.g. - Amplifier - Filter Sensors/transducers : to pick-up the physical quantity to an electrical signal Amplifier : to amplify a small-amplitude electrical signal to range of several volts Analog filter : to remove high frequencies (anti-aliasing) A/D converter : to convert continuous signal to a discretised and quantised version Filtering Ideal filters let frequency components over the passband pass through undistorted, while components at the stopband are completely cut off. Filter Types u Filter y

7 Filtering Filter properties are described in terms of frequency response and graphically in the form of a Bode plot Frequency Response Function u LTI-System y n d y d a n dt dt y dy dt d u dt d dt n m m... a n n an = b + b + m... m The Laplace Transform u + b m du dt + b m s = jω H ( jω) Filtering Bode Plot Bode Diagram H ( jω) = j ω + 2 Magnitude (db) ω u = 2 rad/sec -5 Phase (deg) Frequency (rad/sec) H ( j2) =.3536 H(j2) = 45 o bode(tf(,[ 2]))

8 Filtering Filter Bandwidth The width of the positive frequency range in which the magnitude response is greater than or equal to.77 (3 db attenuation) of the peak magnitude of a frequency plot -3 db Filtering Filter Order & Sharpness of the roll-off Source:

9 Filtering Analog Filter Implementation Analog-to-Digital (A/D) Conversion A/D Analog signal Anti-aliasing Filter Sampling Quantisation Digital signal

10 Analog-to-Digital (A/D) Conversion A/D Analog signal Anti-aliasing Filter Sampling Quantisation Digital signal x(t) Sample Analog-to-Digital (A/D) Conversion A/D Analog signal Anti-aliasing Filter Sampling Quantisation Digital signal x(t) Sample

11 Analog-to-Digital (A/D) Conversion A/D Analog signal Anti-aliasing Filter Sampling Quantisation Digital signal x(t) Sample Sampling Sampling Theorem An analog signal can be exactly recovered from its samples if the signal is sampled at a sampling rate of f s > 2f max, where f max is the highest frequency contained in the signal If Example x( t) = 3cos(5πt ) + sin(3πt ) cos(πt ) what is the minimum sampling rate which guarantees no loss information?

12 Sampling What if f s < f max Aliasing A 8 Hz-signal, sampled at Hz, appears as 2 Hz How to Prevent Aliasing Use sufficiently high sampling frequency Place an analog anti-aliasing filter in front of the sampler to reject F > F s / 2 G filter ω p ( s) = s + ω p.5.5 (a) -Hz sine + 6-Hz noise (c) prefiltered (a) with ω.5 p = 2 rad/sec time (sec) (b) 28 Hz sampled (a) time (sec) time (sec) (d) 28 Hz sampled (c) time (sec)

13 Anti-Aliasing Filter Avoid designing an anti-aliasing filter too tight against half the sampling frequency In practice, the cut-off frequency (f c ) of the filter may be selected a little above f max. Then, select sampling rate f s > 3f c Digital Filtering General form y( k) = a + b y( k ) a y( k 2)... a + b u( k ) b 2 m n u( k m) z-transform y( k n) +... Y ( z) U ( z) b + b z + a z b a m m = = n n z z B( z) A( z) H(z) - Transfer function

14 Digital Filtering IIR and FIR filters Infinite Impulse Response (IIR) filter: Response is a function of both current and past inputs and past outputs y( k) = n i= Finite Impulse Response (FIR) filter: Response is a function of current and past inputs only i y( k) = a y( k i) + m i= b u( k i) i m i= b u( k i) i Digital Filtering IIR vs FIR filters Pros Cons IIR Requiring a much lower filter order than FIR filters to achieve a specific frequency criterion Nonlinear phase characteristics FIR Always being stable Having linear phase shifts less efficient in terms of computer time and memory than IIR because of their nonrecursive nature

15 Digital Filtering Design ω p Passband edge frequency ω s Stopband edge frequency Passband and stopband attenuation Transition width(s) Some Digital Filters Source:

16 Some MATLAB Tools for filter design Frequency Response/Filter Analysis: freqz, fvtool FIR: fir, fir2, firls, remez IIR: butter, cheby, cheby2, ellip, yulewalk Filter Design & Analysis GUI: fdatool FDATool

17 Frequency Domain Characterisation Why frequency domain? In physiological systems information is often more readily expressed in the frequency domain Noise and information are often more easily separated in frequency domain than in time domain Some algorithms are more efficient when implemented in the frequency domain, e.g. computation of convolution in frequency domain Frequency Domain Analysis The Fourier Transform (FT) method Any signal may be presented as a series of sinusoids of different frequencies, each with an amplitude and phase. Continuous Time Fourier Transform X ( f ) j πft = x( t) e 2 dt j 2πft x( t) = X ( f ) e df 2π

18 Frequency Domain Analysis Example π 3π 5π x( t) =.452cos t.45cos t.62cos t Harmonics, 3 and 5.2 x(t). -. Magnitude Time (sec) π / 2 3π / 2 5π / 2 frequency (rad/sec) Frequency Domain Analysis Example EEG signal Source: D. Simpson, Tutorial on Biomedical Signal Processing

19 Frequency Domain Analysis Discrete Fourier Transform (DFT) The frequency spectrum of a signal is numerically computed using its time samples, providing a discrete spectrum X ( k) = N n= x[ n] e 2πnk j N x[ n] = N N n= X ( k) e 2πnk j N fs frequencies fk = k, k =,,..., N N In practice, DFTs are computed using a high-speed algorithm, known as the Fast Fourier Transform (FFT) Frequency Domain Analysis Example: FFT with N=24, f s = Hz Time(sec) Amplitude Frequency (Hz) Amplitude f = f s / N Frequency (Hz) Ts=.; %sampling time Ts t=[:499]*ts; %time vector x=cos(2*pi*t/)+.5*sin(2*pi*t/5); %signal x N=24; % no. of computational points X=abs(fft(x,N)); %magnitude of FFT of x X=X(:N/2+); F=[:N/2]/N/Ts; figure; subplot(2);plot(t,x); xlabel('time(sec)') subplot(22);plot(f,x); xlabel('frequency (Hz)'); ylabel('amplitude')

20 Frequency Domain Analysis Power Spectral Analysis describes how the power of a signal is distributed with frequency Power/Hz E = Parseval s theorem x( t) 2 dt = X ( f ) 2 df where E the average energy of x(t) Frequency (Hz) Power Spectrum P x ( f ) = FT of the autocorrelation function of x[n] Power Spectral Estimation Estimate PS using FFT Pˆ ( f x k ) = X ( f N k f ) s 2 The periodogram method where X ( f k ) is the DFT of x[n]at f k fs = k, k =,,..., N N

21 Power Spectral Estimation Windowing Extracting a segment of a signal in time is the same as multiplying the signal with a rectangular window Source: Power Spectral Estimation Spectral Leakage Original Signal Window - Windowed Signal Sine frequency Hz + noise Power/frequency (db/hz) Frequency (khz)

22 Power Spectral Estimation How to reduce leakage Using a window type which smoothes the edges of the signal Commonly used windows: Hamming windows Hanning windows Barlett windows Blackman windows Kaiser windows Sine frequency Hz + noise Modified periodogram method Power/frequency (db/hz) Frequency (khz) Power Spectral Estimation Estimate PS by Welch s method Divides the data several segments, possibly overlapping, performs an FFT N each segment, computes the power spectrum, then averages these spectra Averaging Overlap Averaging It reduces noise in the estimated power spectra in exchange for reducing the frequency resolution Source:

23 Power Spectral Estimation Example Periodogram Power Spectral Density Estimate 5 Power/frequency (db/hz) Power/frequency (db/hz) Frequency (Hz) Welch Power Spectral Density Estimate % Generate data fs = ; % Sampling frequency t = (:.3*fs)./fs; % 3 samples A = [2 8]; % Sinusoid amplitudes f = [5;4]; % Sinusoid frequencies xn = A*sin(2*pi*f*t) + 5*randn(size(t)); % Periodogram Hs = spectrum.periodogram('rectangular'); psd(hs,xn,'fs',fs,'nfft',24); % Welch s spectral estimate for 3 segments with % 5% overlap Hs = spectrum.welch('rectangular',5,5); figure; psd(hs,xn,'fs',fs,'nfft',24); Frequency (Hz) Time-Frequency Analysis Fourier analysis assumes stationarity of the signal, e.g. its statistical properties do not change Many signals - particularly those of biological origin - are not stationary. Extract both time and frequency information from the signal

24 Short-Term Fourier Transform (STFT) The Spectrogram Divide signal into stationary segments. Perform Fourier transform on a window of data that slides along the time axis time-frequency function that describes the frequency distribution near time τ STFT( f, τ ) = FT( x( t) w( t τ )) The spectrogram is given by the magnitude of the STFT of the function { x t) } Spectrogra m ( = STFT ( f, τ ) 2 Short-Term Fourier Transform (STFT) The Spectrogram A linear chirp, DC and crossing 5 Hz at t= sec

25 Short-Term Fourier Transform (STFT) The Spectrogram The time-frequency tradeoff: we cannot simultaneously obtain arbitrarily high resolution along both the time and frequency axes. Shortening the data length (N) to improve time resolution will reduce frequency resolution ( f s /N) N=5 N=8 N = 8 N = Frequency 2 8 Frequency Time Time MATLAB Code Ts=.25; fs=/ts; M=5/Ts; %Sampling time %sampling frequency %M = Data length over 5 sec % Generate Data t=:ts:(m-)*ts; y=[cos(2*pi**t) cos(2*pi*25*t) cos(2*pi*5*t) cos(2*pi**t)]; t=[:length(y)-]*ts; %Time vector nfft=[5 8]; %Window widths %Plot the spectrogram for window widths of 5 and 8 for n=:2 N=nfft(n); %Tested window width (N) figure; %Call a new figure specgram(y,n,fs,n,[]); %Plot the spectrogram using current N title(['n = ' num2str(n)]) %Add title end

26 More methods Noise reduction - Optimal filters - Adaptive noise cancellation Time-Frequency -Wigner-Ville - Wavelets Signals and Systems - Correlation and coherence - Modelling (black box vs. physiological) Pattern recognition and classification -Rule-based - Statistical methods -Learning Suggested Reading Oppenheim, A.V. and Schafer, R.W. Discrete-Time signal Processing. Prentice Hall: Engle-wood Cliffs, NJ, 989. Semmlow, J.L. Biosignal and Biomedical Image Processing: MATLAB-Based Applications. CRC, 24. Rangayyan, R.M. Biomedical Signal Analysis: A Case-Study Approach. Wiley-IEEE Press, 2. Bruce, E.N. Biomedical Signal Processing and Signal Modeling. Wiley-Interscience, 2.