10/09 - Fourier Transform: Difference between revisions

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==Fourier Transform==
==Fourier Transform==
Remember from [[10/02 - Fourier Series]]
Remember from [[10/02 - Fourier Series]]
*<math> \alpha_m = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt</math>
*<math> \alpha_n = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j\,2\,\pi \,n\,t/T}\, dt</math>
*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_m e^{j2\pi m/T}</math>
*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_n e^{j\,2\pi \,n/T}</math>


If we let <math> T \rightarrow \infty</math>
If we let <math> T \rightarrow \infty</math>

Revision as of 17:31, 3 December 2008

Assuming the function is perodic with the period T

Fourier Transform

Remember from 10/02 - Fourier Series

If we let

Remember

Definitions

Examples

Sifting property of the delta function

The dirac delta function is defined as any function, denoted as , that works for all variables that makes the following equation true:

  • When dealing with , it behaves slightly different than dealing with . When dealing with , note that the delta function is . The is tacked onto the front. Thus, when dealing with , you will often need to multiply it by to cancel out the .

More properties of the delta function

Let and