(New page: Question 4 Compute the coefficients <math>a_k</math> of the Fourier series of the signal <math>x(t)</math> periodic with period <math>T=4</math> defined by <math>\,x(t)=\left\{\begin{arra...) |
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<math> a_0= average of x(t) = /frac {1x2}{4}=/frac{1}{2}; | <math> a_0= average of x(t) = /frac {1x2}{4}=/frac{1}{2}; | ||
| − | T=4, <math>\omega _o = \frac{2 \pi} {4} = \frac{\pi}{2} | + | T=4, <math>\omega _o = \frac{2 \pi} {4} = \frac{\pi}{2}</math> |
for k<>0, | for k<>0, | ||
| − | <math>a_k=\frac{1}{T}\int_{0}^{T}x(t)e^{-jk\frac{2\pi}{T}t}dt | + | <math>a_k=\frac{1}{T}\int_{0}^{T}x(t)e^{-jk\frac{2\pi}{T}t}dt</math> |
| − | <math>a_k=\frac{1}{4}\int_{-2}^{2}x(t)e^{-jk\frac{\pi}{2}t}dt | + | <math>a_k=\frac{1}{4}\int_{-2}^{2}x(t)e^{-jk\frac{\pi}{2}t}dt</math> |
| − | <math>a_k=\frac{1}{4}\int_{-1}^{1}x(t)e^{-jk\frac{\pi}{2}t}dt | + | <math>a_k=\frac{1}{4}\int_{-1}^{1}x(t)e^{-jk\frac{\pi}{2}t}dt</math> |
| − | <math>a_k=\frac{1}{4} \left.\frac{ | + | <math>a_k=\frac{1}{4} \left.\frac{e^{-jk\frac{\pi}{2}t}}{-jk\frac{\pi}{2}}\right|_{-1}^{1}</math> |
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| + | <math>a_k=\frac{1}{4} (\frac{e^{-jk\frac{\pi}{2}t}}{-jk\frac{\pi}{2}}-\frac{e^{jk\frac{\pi}{2}t}}{jk\frac{\pi}{2}})<\math> | ||
| + | |||
| + | <math>a_k=\frac{1}{2jk\pi}(e^{jk\frac{\pi}{2}}-e^{-jk\frac{\pi}{2}})<\math> | ||
| + | |||
| + | <math>a_k=\frac{sin(k\frac{\pi}{2})}{k\pi}<\math> | ||
Revision as of 09:03, 15 October 2008
Question 4 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. \, $
(Simplify your answer as much as possible.)
Ans: $ a_0= average of x(t) = /frac {1x2}{4}=/frac{1}{2}; T=4, <math>\omega _o = \frac{2 \pi} {4} = \frac{\pi}{2} $
for k<>0,
$ a_k=\frac{1}{T}\int_{0}^{T}x(t)e^{-jk\frac{2\pi}{T}t}dt $
$ a_k=\frac{1}{4}\int_{-2}^{2}x(t)e^{-jk\frac{\pi}{2}t}dt $
$ a_k=\frac{1}{4}\int_{-1}^{1}x(t)e^{-jk\frac{\pi}{2}t}dt $
$ a_k=\frac{1}{4} \left.\frac{e^{-jk\frac{\pi}{2}t}}{-jk\frac{\pi}{2}}\right|_{-1}^{1} $
$ a_k=\frac{1}{4} (\frac{e^{-jk\frac{\pi}{2}t}}{-jk\frac{\pi}{2}}-\frac{e^{jk\frac{\pi}{2}t}}{jk\frac{\pi}{2}})<\math> <math>a_k=\frac{1}{2jk\pi}(e^{jk\frac{\pi}{2}}-e^{-jk\frac{\pi}{2}})<\math> <math>a_k=\frac{sin(k\frac{\pi}{2})}{k\pi}<\math> $
