(New page: Compute the Fourier Transform of x(t): <math>\,x(t)=2e^{-3t}u(t)+3[u(t+3)-u(t-3)]</math> Using the Formula for Fourier Transforms: <math> \mathcal{F}(x(t))= \mathcal{X}(\omega)= \int_{-...)
 
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Compute the Fourier Transform of x(t):
 
Compute the Fourier Transform of x(t):
  
<math>\,x(t)=2e^{-3t}u(t)+3[u(t+3)-u(t-3)]</math>
+
<math>\,x(t)=2e^{-3t}u(t)+4[u(t+3)-u(t-3)]</math>
  
 
Using the Formula for Fourier Transforms:
 
Using the Formula for Fourier Transforms:
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So the calculation follows as:
 
So the calculation follows as:
 +
 
<math>
 
<math>
 
\mathcal{X}(\omega)=
 
\mathcal{X}(\omega)=
\int_{-\infty}^{\infty}(2e^{-3t}u(t)+3[u(t+3)-u(t-3)])e^{-j\omega t}  \,dt
+
\int_{-\infty}^{\infty}(2e^{-3t}u(t)+4[u(t+3)-u(t-3)])e^{-j\omega t}  \,dt
 +
</math>
  
 +
<math>
 +
=2\int_{0}^{\infty}e^{-t(3+j\omega)}\,dt + 4\int_{-3}^{3}e^{-j\omega t}\, dt
 +
</math>
  
 +
<math>
 +
=2\frac{-e^{-t}}{3+j\omega}\bigg|^{\infty}_{0}+4\frac{-e^{-j\omega t}}{j\omega}\bigg|^{3}_{-3}
 +
</math>
 +
 +
<math>
 +
=\frac{1}{3+j\omega}+4*\frac{2}{\omega}*\frac{-e^{-3j\omega} + e^{3j\omega}}{2j}
 +
</math>
 +
 +
<math>
 +
=\frac{1}{3+j\omega}+\frac{8sin(3\omega)}{\omega}
 
</math>
 
</math>

Revision as of 14:21, 8 October 2008

Compute the Fourier Transform of x(t):

$ \,x(t)=2e^{-3t}u(t)+4[u(t+3)-u(t-3)] $

Using the Formula for Fourier Transforms:

$ \mathcal{F}(x(t))= \mathcal{X}(\omega)= \int_{-\infty}^{\infty}x(t)e^{-j\omega t} \,dt $

So the calculation follows as:

$ \mathcal{X}(\omega)= \int_{-\infty}^{\infty}(2e^{-3t}u(t)+4[u(t+3)-u(t-3)])e^{-j\omega t} \,dt $

$ =2\int_{0}^{\infty}e^{-t(3+j\omega)}\,dt + 4\int_{-3}^{3}e^{-j\omega t}\, dt $

$ =2\frac{-e^{-t}}{3+j\omega}\bigg|^{\infty}_{0}+4\frac{-e^{-j\omega t}}{j\omega}\bigg|^{3}_{-3} $

$ =\frac{1}{3+j\omega}+4*\frac{2}{\omega}*\frac{-e^{-3j\omega} + e^{3j\omega}}{2j} $

$ =\frac{1}{3+j\omega}+\frac{8sin(3\omega)}{\omega} $

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva