HW6.3 Caleb Tan - Question 4 ECE301Fall2008mboutin - Rhea

Compute the coefficients $ a_k $ of the Fourier series of the signal x(t) periodic with period T = 4 defined by

$ \,x(t)=\left\{\begin{array}{cc} 0, & -2<t<-1 \\ 1, & -1\leq t\leq 1 \\ 0, & 1<t\leq 2 \end{array} \right. \, $

Answer:

$ T = \frac{2\pi}{w_0} = 4 $

$ w_0 = \frac{\pi}{2} $

$ a_k = \frac{1}{T} \int_{0}^{T} x(t) e^{-jkw_0t} dt $

File:X(t) ECE301Fall2008mboutin.jpg

From diagram, $ a_0 = \frac{1}{2} $ because it is the average of the signal over one period.

Then, using the formula:

$ a_k = \frac{1}{4} \int_{0}^{4} x(t) e^{-jk\frac{\pi}{2}t} dt $