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> \ | *<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 \ | *<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 | ||