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4. Compute the coefficients <math>a_{k}</math> of the Fourier series of the signal x(t) with period T = 4 defined by
 
4. Compute the coefficients <math>a_{k}</math> of the Fourier series of the signal x(t) with period T = 4 defined by
  
<math>= \left\{ \begin{array}{ll} 0&, -2<t<-1\\ 1 &,  -1\end{array}\right. </math>
+
<math>= \left\{ \begin{array}{ll} 0&, -2<t<-1\\ 1 &,  -1\leq t\leq 1\end{array}\right. </math>
  
 
<math>\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &,  \text{ else}\end{array}\right. </math>
 
<math>\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &,  \text{ else}\end{array}\right. </math>

Revision as of 19:43, 8 October 2008

Test Problem 4

4. Compute the coefficients $ a_{k} $ of the Fourier series of the signal x(t) with period T = 4 defined by

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

$ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $

$ a_{k} = \frac{1}{T} \int_{0}^{T}x(t)e^{-jk\omega _{o}t}dt $

From the problem statement we know that T=4

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

Knowing that T=4 we can visualize the periodic signal in the range $ 0 \leq t \leq 4 $. x(t) = 1 for $ 0 \leq t \leq 1 $ and $ 3 \leq t \leq 4 $. Otherwise, x(t) = 0. Therefore:

$ = \frac{1}{4} \int_{0}^{1}e^{-jk\frac{\pi}{2}t}dt + \frac{1}{4} \int_{3}^{4}e^{-jk\frac{\pi}{2}t}dt $

$ = \frac{1}{4}\frac{j}{\frac{\pi}{2}k}e^{-jk\frac{\pi}{2}t}|_{0}^{1} + \frac{1}{4}\frac{j}{\frac{\pi}{2}k}e^{-jk\frac{\pi}{2}t}|_{3}^{4} $

$ =\frac{j}{2\pi k}(e^{-jk\frac{\pi}{2}} - 1 + e^{-jk2\pi} - e^{-jk\frac{3\pi}{2}}) $

$ =\frac{j}{2\pi k}(e^{-jk\frac{\pi}{2}} - 1 + 1 - e^{-jk\frac{4\pi}{2}}e^{jk\frac{\pi}{2}}) $

$ =\frac{j}{2\pi k}(e^{-jk\frac{\pi}{2}}-e^{jk\frac{\pi}{2}}) $

$ =\frac{1}{\pi k}(\frac{e^{jk\frac{\pi}{2}}-e^{-jk\frac{\pi}{2}}}{2j}) $

$ =\frac{1}{\pi k}sin(\frac{\pi}{2}k) $

But we are not done yet!

A quick glance at the above result reveals that $ a_{0} $ is undefined.

But $ a_{0} $ is simply equal to the average value of the function over 1 period.

$ a_{0}=\frac{2}{4} = \frac{1}{2} $

So in conclusion:

$ a_{k}=\frac{1}{\pi k}sin(\frac{\pi}{2}k) $ for all $ k \neq 0, \; a_{0}=\frac{1}{2} $

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Recent Math PhD now doing a post-doctorate at UC Riverside.

Kuei-Nuan Lin