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

2. Impulse Response Filtering 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. (1)

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

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

6. Convolution under Z Transform (1) 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. H() in a Low Pass Filter Put z = F in H(z), where F is normalized frequency. Parks and McClelland, 1972

9. Frequency and Magnitude Response Majumdar, 2013

10. 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. Rao and Gejji, 2010

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

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

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

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. 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.)

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 YOU This lecture is available at http://www.isibang.ac.in/~kaushikhttp://www.isibang.ac.in/~kaushik