Transcript Lesson 3

M.Tech. (CS), Semester III, Course B50
Functional Brain Signal
Processing: EEG & fMRI
Lesson 3
Kaushik Majumdar
Indian Statistical Institute
Bangalore Center
[email protected]
Impulse Response Filtering
(1)
Original signal
Filtered signal
Impulse response
Convolution
This is in time domain, but filters are frequency specific
and therefore should be specified in the frequency
domain.
Fourier Transform

F ( x(t )) 

x(t ) exp( j 2 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

x(t ) 
a
n 
n
cos(2 nt )  bn sin(2 nt ) where

an 
 x(t ) cos(2 nt )dt,


bn 
 x(t )sin(2 nt )dt

Signal Decomposition into Simpler
Orthonormal Components
exp(j6πt)
exp(j2πt)
Component drawings are
not authentic
Real EEG signal
exp(j4πt)
Signal will have to be stationary
and square integrable.
Generalization to Laplace
Transform

L( x(t )) 
 x(t ) exp(st )dt

Where s is a complex
number
Discrete Laplace transform = Z transform
Ld ( x(m)) 
where


m 

x(m) exp( sm)   x(m) z  m
exp( s)  z
m 
1
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)
Parks and McClelland, 1972
H() in a Low Pass Filter
Put z = F in H(z), where F
is normalized frequency.
Majumdar, 2013
Frequency and Magnitude
Response
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.
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)
Filter Coefficients (cont.)
(4)
If condition (2) holds then (4) becomes
If condition (3) holds then (4) becomes
Rao and Gejji, 2010
An Implementation
Design a 17 tap linear phase low pass filter
with a cutoff frequency
.
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