Example of Computation of Fourier series of a CT SIGNAL

A practice problem on "Signals and Systems"


The function y(t) in this example is the periodic continuous-time signal cos(t) such that

$ y(t) = \ cos(t) $

where cos(t) can be expressed by the Maclaurin series expansion

$ \ cos(t) = \sum_{k=0}^\infty \left (-1 \right )^k \frac{t^{2k}}{ \left(2k \right )!} $

and its Fourier series coefficients are described by the equations below.

$ \ a_k = \frac{1}{T}\int_{0}^{T} y(t)e^{-jk\omega_0t}\, dt $

$ \omega_0 = \frac{2\pi}{T} = 1 $

With this information it is possible to determine the first fiew coefficients of the Fourier series of the signal, as seen below.

Also noting that

$ \cos(t) = \frac{e^{jt} - e^{-jt}}{2} $

then the solution for solving the coefficients becomes

$ \ a_k = \frac{1}{2\pi}\int_{0}^{2\pi} \frac{e^{jt} - e^{-jt}}{2}e^{-jkt}\, dt $


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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang