Basic Fourier Series
Academic Resource Center
Workshop for BME
by: Neha Bansal
Agenda
•
•
•
•
Fourier Series
Trigonometric Fourier Series
Compact Trigonometric Fourier Series
Examples
o Square Waves
o Sawtooth Waves
• References
Fourier Series
• A periodic function f(t) can be represented by an
infinite sum of sine and/or cosine functions that
are harmonically related. That is, the frequency
of any trigonometric term in the infinite series is
an integral multiple, or harmonic, of the
fundamental frequency of the periodic function.
Trigonometric Fourier
Series
• Given f(t) is periodic, f(t) can be represented as
follows:
where n is the integer sequence 1,2,3, ... , a0, an, and bn are called the Fourier
coefficients, and are calculated from f(t), 0 = 2 /To is the fundamental frequency
Compact Trigonometric Fourier Series
Exponential Fourier Series
Example: Square wave
Example: Sawtooth
Wave
Let us consider a sawtooth wave
For convenience, we shall shift our interval from
to
we have simply f(t)=t. Using Eqs. of Fourier series, we have
.
In this interval
Example: Sawtooth
wave
(7.15)
.
So, the expansion of f(t) reads
References
• WikiBooks Resources:
o http://en.wikibooks.org/wiki/Signals_and_Systems/Fourier
_Series
• Wolfram MathWorld Fourier Series:
o http://mathworld.wolfram.com/FourierSeries.html
• ARC Website:
o iit.edu/arc
• BME Schedule
o http://iit.edu/arc/tutoring_schedule/biomedical_engineerin
g.shtml