Signal Processing
Using a Computer Algebra System
Alex Lobregt
a.lobregt@fnt.hvu.nl
Hogeschool van Utrecht
Netherlands
Abstract
The
overall philosophy is that we can, using a Computer Algebra System (CAS), shift
the emphasis from acquiring skills to gaining insight.
A major item in signal processing is the introduction of Fourier Series. The
traditional way to introduce the Fourier series is to derive the Fourier coefficients
from the series:
Introducing these series this way, will place a high demand on the calculating
skills of the students attending the course. The students involved should be
completely familiar with:
?dd/even functions.
?eatures of convergence.
?nderstanding of complex numbers.
?asics of trigonometry.
?armonic functions.
?ntegration by parts
However, even to
better than average students the combination of these subjects proved to be
a formidable obstacle. Consequently the periods would be slowed down by lengthy
calculating exercises, leaving insufficient time for acquiring a thorough understanding
of Fourier Series and their applications.
For some years a ?erive?practical exercise,
and more recently the use of the Ti89 calculator, has been attached to the Fourier
Courses, enhancing the students?enthusiasm for the subject considerably.
The illustrations they can produce definitely contribute to this effect.
In Electrical Engineering functions as shown
below, are frequently used. These periodic functions may well be approximated
by so-called Fourier series.
Example:
The CAS ?erive?allows us to proceed as follows:
The Utility INT_APPS.MTH contains the function Fourier(f(t),t,a,b,n).
?t is the independent variable.
?f(t) is a periodic function
?[a,b] is the begin and end of one period.
?n is the number of harmonic terms you wish to demand.
Example 1.
Find the first 5 harmonic terms of Sq(t) as given below. period T=2 1 0 1 2
Sq(t):=chi(0,t,1)-chi(1,t,2) Fourier(Sq(t),t,0,2,5) Simplify Derive allows you
to draw the individual harmonic functions and their summation side by side,
which gives an excellent insight into the structure of the Fourier-Series.
After this introduction the underlying theory
can be checked by calculating the Fourier-coefficients using Derive and by hand.
Until recently, any investigation of this Gibbs Phenomenon used to be impossible,
whereas in the present situation each student can perform calculations in this
field. Gibbs Phenomenon. Use the square wave of Sq(t) of example 1. Express
the Fourier series, with respectively: N=10 and N=100 Plot the graphs in one
screen. As you see the series comes closer and closer to Sq(t), however, the
amount of overshoot nearby the discontinuities does not tend to zero with increasing
N. Calculate with Derive this overshoot for N=10 and N=100. Compare your result
with the limit which amounts to 8.95 % of the total jump. After this example
I will show the use of a Computer Algebra System in proofing the limit above!
Beside the Gibs phenomenon there are a lot of other examples in this field like,
the consequences of shifting a function for the amplitude spectrum and the phase
spectrum and filtering. In other parts of Signal Processing, like The Discrete
Fourier Transform, a Cas is very useful. Examination methods have been altered
as well. At the end of the course the student? progress is assessed in three
different ways: 1) by oral presentation on the outcome of the investigations.
2) a written report on the practical coursework. 3) a written examination covering
mainly theoretical knowledge. One of the major considerations leading to the
revised approach was whether the emphasis in coursework could be shifted from
acquiring skills to gaining insight. In my view, experience gained over recent
years working with CAS tells me we are on the right track.
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