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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
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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)
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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
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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
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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
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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.
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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)
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H() in a Low Pass Filter
Put z = F in H(z), where F is normalized frequency.
Parks and McClelland, 1972
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Frequency and Magnitude Response
Majumdar, 2013
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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
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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)
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Filter Coefficients (cont.)
If condition (2) holds then (4) becomes(4)
If condition (3) holds then (4) becomes
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An Implementation
Design a 17 tap linear phase low pass filter with a cutoff frequency .
Rao and Gejji, 2010
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Implementation (cont.)
Pass band
Stop band
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Implementation (cont.)
Phase response of the 17 tap FIR filter with respect to circular frequency.
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Implementation (cont.)
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Implementation (cont.)
Getting back the h(n)s by applying iDFT on H(k)s
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Implementation (cont.)
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Infinite Impulse Response (IIR) Filters for EEG Processing
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Butterworth Filter
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Butterworth Filter: Amplitude Response
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Butterworth Filter (cont.)
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Butterworth Filter (cont.)
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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.
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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.
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THANK YOU
This lecture is available at http://www.isibang.ac.in/~kaushik