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[[Category:problem solving]]
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[[Category:ECE301]]
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[[Category:ECE]]
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[[Category:Fourier transform]]
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[[Category:inverse Fourier transform]]
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[[Category:signals and systems]]
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== Example of Computation of inverse Fourier transform (CT signals) ==
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A [[CT_Fourier_transform_practice_problems_list|practice problem on CT Fourier transform]]
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----
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== The Signal ==
 
== The Signal ==
  
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<math>a_{-1} = e^{-j \frac{\pi}{3}}</math>
 
<math>a_{-1} = e^{-j \frac{\pi}{3}}</math>
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<math>= 2 \pi e^{j \frac{\pi}{3}} \delta(\omega - 4) + 2 \pi e^{-j \frac{\pi}{3}} \delta(\omega + 4)</math>
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duality applied
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<math>\frac{1}{2 \pi}( 2 \pi e^{j \frac{\pi}{3}} \delta(-t - 4) + 2 \pi e^{-j \frac{\pi}{3}} \delta(-t + 4))</math>
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<math>e^{j \frac{\pi}{3}} \delta(-t - 4) + e^{-j \frac{\pi}{3}} \delta(-t + 4)</math>
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----
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[[CT_Fourier_transform_practice_problems_list|Back to Practice Problems on CT Fourier transform]]

Latest revision as of 11:45, 16 September 2013

Example of Computation of inverse Fourier transform (CT signals)

A practice problem on CT Fourier transform



The Signal

$ X(j \omega) = \cos(4 \omega + \frac{\pi}{3}) $

Taken from 4.22.b from the course book, it looks interesting and I want to try it.


The Inverse Fourier Transform

$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j \omega)e^{j\omega t}d\omega $

For this problem I will not be using the above equation but in stead be using duality.


$ x(t) = \cos(4 t + \frac{\pi}{3}) $


note

$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{j k \omega_o t} $

and

$ \cos(4 t + \frac{\pi}{3}) = \frac{e^{j(4 t + \frac{\pi}{3})}}{2} + \frac{e^{-j(4 t + \frac{\pi}{3})}}{2} $

$ \omega_o = 4 $


$ a_1 = e^{j \frac{\pi}{3}} $


$ a_{-1} = e^{-j \frac{\pi}{3}} $


$ = 2 \pi e^{j \frac{\pi}{3}} \delta(\omega - 4) + 2 \pi e^{-j \frac{\pi}{3}} \delta(\omega + 4) $


duality applied

$ \frac{1}{2 \pi}( 2 \pi e^{j \frac{\pi}{3}} \delta(-t - 4) + 2 \pi e^{-j \frac{\pi}{3}} \delta(-t + 4)) $


$ e^{j \frac{\pi}{3}} \delta(-t - 4) + e^{-j \frac{\pi}{3}} \delta(-t + 4) $


Back to Practice Problems on CT Fourier transform

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

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