Functional Brain Signal Processing: EEG & fMRI Lesson 3

1. M.Tech. (CS), Semester III, Course B50 Functional Brain Signal Processing: EEG & fMRILesson 3 Kaushik Majumdar Indian Statistical Institute Bangalore Center kmajumdar@isibang.ac.in

2. Impulse Response Filtering (1) Original signal Impulse response Convolution Filtered signal This is in time domain, but filters are frequency specific and therefore should be specified in the frequency domain.

3. Fourier Transform n takes integer values. Let x(t) be a periodic signal and square integral of x(t) over the whole real line converges. Then x(t) can be expressed as where

4. Signal Decomposition into Simpler Orthonormal Components exp(j4πt) exp(j6πt) exp(j2πt) Component drawings are not authentic Real EEG signal Signal will have to be stationary and square integrable.

5. Generalization to Laplace Transform Where s is a complex number Discrete Laplace transform = Z transform where

6. Convolution under Z Transform • under z transform will become (just like Fourier transform): Y, S, Z are z transform for y, s, z respectively. Designing a filter is all about finding a suitable h(i) and therefore finding a suitable H(z). Latter is mathematically more convenient.

7. Inverse Z Transform h(i) can be recovered from H(z) by inverse z transform C is a closed curve lying within the convergence of H(z)

8. Parks and McClelland, 1972 H() in a Low Pass Filter Put z = F in H(z), where F is normalized frequency.

9. Majumdar, 2013 Frequency and Magnitude Response

10. Rao and Gejji, 2010 Finite Impulse Response (FIR) Filter h(k) is filter coefficient or tap, N is filter order. Amplitude response |H(w)| of a 17 tap FIR filter (thick line) has been plotted against the circular frequency w.

11. Filter with Real Coefficients For N odd H(0) will have to be real and (2) For N even H(0) will have to be real and (3)

12. Filter Coefficients (cont.) (4) If condition (2) holds then (4) becomes If condition (3) holds then (4) becomes

13. Rao and Gejji, 2010 An Implementation Design a 17 tap linear phase low pass filter with a cutoff frequency .

14. Implementation (cont.) Pass band Stop band

15. Implementation (cont.) Phase response of the 17 tap FIR filter with respect to circular frequency.

16. Implementation (cont.)

17. Implementation (cont.) Getting back the h(n)s by applying iDFT on H(k)s

18. Implementation (cont.)

19. Infinite Impulse Response (IIR) Filters for EEG Processing

20. Butterworth Filter

21. Butterworth Filter: Amplitude Response

22. Butterworth Filter (cont.)

23. Butterworth Filter (cont.)

24. References • Proakis and Manolakis, Digital signal processing: principles, algorithms and applications, 4e, Dorling Kindersley India Pvt. Ltd., 2007. Section 5.4.2 and Chapter 10. • Majumdar, A brief survey of quantitative EEG analysis (under preparation), Chapter 2, 2013. • Rao and Gejji, Digital signal processing: theory and lab practice, 2e, Pearson, New Delhi 2010.

25. Exercise • Design low-pass, high-pass and band-pass filters by using Filter Design toolbox in MATLAB. • Learn how to correct phase distortion by the filtfilt command in MATLAB.

26. THANK YOUThis lecture is available at http://www.isibang.ac.in/~kaushik