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

Let us consider a sawtooth wave

For convenience, we shall shift our interval from to . In this interval

we have simply f(t)=t. Using Eqs. of Fourier series, we have

Example: Sawtooth Wave

Example: Sawtooth wave

So, the expansion of f(t) reads

(7.15) .

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_engineering.shtml