Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Some Characteristics of Early Color Vision
The basic material for this discussion is chapter 4 of Brian Wandell's Foundations of Vision,
which will be handed out in class. This page is just a brief outline of the aspects of the topic that
we will focus on
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
INTRODUCING THE ZTRANSFORM
After reading this section, you may want to look at Chapter 1, "Signal Processing Basics," in the
User's Guide for the Matlab Signal Processing Toolbox. There are a few things that we have not
covered, and will not cover  for
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
DFT
FFT
WaveletTransformation
(ext. sources: nbtwiki.net)
including:
5 MATLAB examples
BIOSIGNALVERARBEITUNG
VO 6
%
%
%
%
The frequency in Hz.
The length of the wave in seconds.
Sampling frequency of the wave.
the phase of the wave.
plot(t,wave);
% Gener
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Definition: Biosignal
Many measurement systems for biosignals are low cost and easy to
apply.
Biosignals:
 direct information about its source/sources
 can be acquired often in a noninvasive way
 are typically influenced by physiological and pathophys
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
linear:
y2[n]
x2[n]
superpositon
(y1[n] + y2[n])
y1[n]
x1[n]+x2[n]
Digital filter
causal LTI system
Digitale filter LTI systems
Digitale Filter
x1[n]
y[n]
shifted output
x1[n+n0]
shifted input
time invariant:
causal: the output does not depend on future i
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
BIOSIGNALVERARBEITUNG
can be represented by a ascending series of closed
Space
is built up by a set of orthogonal wavelets
Both index conventions are equivalent!
There exist other representations with reversed indexing:
The space
subspaces
Wavelet Transfo
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Examples
Fourier Transform:
from continuous to discrete
Orthogonal Transformations
Sampling, Quantization Noise
Representation of Analog Signals
Topics:
BIOSIGNAL PROCESSING
Positive phase angle shifts the function
towards the left, earlier time points.
P
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
format short
format compact
echo on
% A.1 Having saved audio1.mat in an accessible folder, we load it:
% MATLAB version:
% load('audio1');
% octave version:
load force 'audio1.mat'
% A.2 Now we plot the vector S:
plot(S,'b')
pause
% A.3 Given that S i
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
C IS 558 / Linguistics 525
C omputer Analysis and Modeling of Biological Signals and Systems
Homework 2
The MATLAB fft function implements the Discrete Fourier Transform given by the equation
where N = length(x).
In MATLAB terms, this means that for a rea
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
C IS 558 / Linguistics 525
C omputer Analysis and Modeling of Biological Signals and Systems
Homework 3
Due: 2/14/2003
Linear ShiftInvariant Systems
1. (Oppenheim and Shafer 1989 p. 68). The system T in the figure below is known to be timeinvariant. When
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
The DFT and the FFT
Calculating the DFT
The equations for the DFT (Discrete Fourier Transform) and inverse DFT, using Matlabstyle
indices, are given below:
Discrete Fourier Transform
(Matlabstyle indices)
Inverse Discrete Fourier
Transform
(Matlabstyle
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Causal Moving Average (FIR) Filters
We've discussed systems in which each sample of the output is a weighted sum of (certain of
the) the samples of the input.
Let's take a causal "weighted sum" system, where causal means that a given output sample
depends
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Towards the Discrete Fourier Transform
We spent some time earlier on the first slogan of signal processing:
The response of a linear shiftinvariant system S to an arbitrary input x is the
convolution of x with the impulse response of S.
This turned out t
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
2. If the color matching matrix for primaries Palt is called Malt, then that the equivalent spectrum
to any 'light' for Palt would be
Palt*Malt*light
This result is another (31element) sampled spectrum, just like 'light' was.
Since 'M' is the color match
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
format compact
format short
echo on
clc
% Generate a complex exponential spiral
% Start with a point on the unit circle
a = pi/4;
z = cos(a) + i*sin(a);
pause
% Move it in a bit
z = .99*z;
pause
% Make the exponential sequence
n = 1:100;
x = z.^n;
% Plot
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Impulse Response and Convolution
Digital signal processing is applied linear algebra? This is easy to grasp for color matching,
where we have fixed dimensions of 1 (number of test lights), 3 (number of primary lights, number
of photopigments), and 31 (num
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Linear Algebra Review
Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier.
Gilbert Strang
Most of digital signal processing can be seen as applied linear algebra. Therefore w e begin with
this brief review of li
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
C IS 558 / Linguistics 525
C omputer Analysis and Modeling of Biological Signals and Systems
Homework 1
D ue: 2/1/2005
Things to learn:
Arithmetic on scalars, vectors and matrices; for loops; simple use of plot( ) and imagesc(
).
Running a file of command
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
format short
format compact
echo on
% A.1 Having saved audio1.mat in an accessible folder, we load it:
% MATLAB version:
load('audio1');
% octave version:
% load force 'audio1.mat'
% A.2 Now we plot the vector S:
plot(S,'b')
pause
% A.3 Given that S i
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
C IS 558 / Linguistics 525
C omputer Analysis and Modeling of Biological Signals and Systems
Homework 2
Due: 2/15/2005
Color Matching
The background for this homework is the lecture on early color vision.
Load colmatch.mat in your Matlab environment. This
Computer Analysis and Modeling of Biological Signals and Systems
LING 525

Spring 2005
Short time Fourier Transform (STFT)
Gabor transformation
Analysis of nonstationary signals
Windowing and noise
Different window types / effects on signal
In time / frequency domain
Windowing
Topics:
BIOSIGNAL PROCEASSING
U (V)
U (V)
tstart
BIOSIGNAL PROC