Basic Fourier Series Academic Resource Center Workshop for BME by: - PowerPoint PPT Presentation

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 to . In this interval For convenience, we shall shift our interval from we have simply f(t)=t. Using Eqs. of Fourier series, we have

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