Difference between revisions of "HW4.1 William Schmidt ECE301Fall2008mboutin" - Rhea

Latest revision as of 11:06, 16 September 2013

Example of Computation of Fourier series of a CT SIGNAL

A practice problem on "Signals and Systems"

Problem

Let the signal $ x(t) = 5cos(3\pi t) + sin(\pi t)\, $
Find the Fourier Series coefficients

Analysis

First rewrite the signal as a sum of complex exponentials:

$ x(t) = 5(\frac{e^{3\pi jt} + e^{-3\pi jt}}{2}) + \frac{e^{\pi jt} - e^{-\pi jt}}{2j} $

Simplifying gives:

$ x(t) = \frac{5}{2} e^{3\pi jt} + \frac{5}{2} e^{-3\pi jt} + \frac{1}{2j} e^{\pi jt} - \frac{1}{2j} e^{-\pi jt} $

The fundamental period is:

$ \omega_o = \frac{2\pi}{T} = \pi $, with $ T = 2\, $ being the period of the original signal.

From the fundamental period, it is easily seen that the fourier series coefficients are:

$ a_{-3} = \frac{5}{2} $

$ a_{-1} = -\frac{1}{2j} $

$ a_{1} = \frac{1}{2j} $

$ a_{3} = \frac{5}{2} $

$ a_{k} = 0\, $, for all other k

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