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Functional Brain Signal Processing: EEG & fMRI
Lesson 3
Kaushik Majumdar
Indian Statistical Institute Bangalore Center
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