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

Impulse Response Filtering

Original signal Impulse response

ConvolutionFiltered signal

This is in time domain, but filters are frequency specific and therefore should be specified in the frequency domain.

(1)

Fourier Transform

( ( )) ( ) exp( 2 )F x t x t j nt dt

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

( ) cos(2 ) sin(2 )n nn

x t a nt b nt

where

( ) cos(2 ) , ( )sin(2 )n na x t nt dt b x t nt dt

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

Generalization to Laplace Transform

( ( )) ( ) exp( )L x t x t st dt

Where s is a complex number

Discrete Laplace transform = Z transform

( ( )) ( ) exp( ) ( ) md

m m

L x m x m sm x m z

where1exp( )s z

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.

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)

H() in a Low Pass Filter

Put z = F in H(z), where F is normalized frequency.

Parks and McClelland, 1972

Frequency and Magnitude Response

Majumdar, 2013

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

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)

Filter Coefficients (cont.)

If condition (2) holds then (4) becomes(4)

If condition (3) holds then (4) becomes

An Implementation

Design a 17 tap linear phase low pass filter with a cutoff frequency .

Rao and Gejji, 2010

Implementation (cont.)

Pass band

Stop band

Implementation (cont.)

Phase response of the 17 tap FIR filter with respect to circular frequency.

Implementation (cont.)

Implementation (cont.)

Getting back the h(n)s by applying iDFT on H(k)s

Implementation (cont.)

Infinite Impulse Response (IIR) Filters for EEG Processing

Butterworth Filter

Butterworth Filter: Amplitude Response

Butterworth Filter (cont.)

Butterworth Filter (cont.)

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.

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.

THANK YOU

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