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==Fourier Transform== |
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==Fourier Transform== |
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Remember from [[10/02 - Fourier Series]] |
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Remember from [[10/02 - Fourier Series]] |
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*<math> \alpha_m = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt</math> |
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*<math> \alpha_n = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j\,2\,\pi \,n\,t/T}\, dt</math> |
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*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_m e^{j2\pi m/T}</math> |
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*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_n e^{j\,2\pi \,n/T}</math> |
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If we let <math> T \rightarrow \infty</math> |
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If we let <math> T \rightarrow \infty</math> |
Revision as of 17:31, 3 December 2008
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Assuming the function is perodic with the period T
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Fourier Transform
Remember from 10/02 - Fourier Series
If we let
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Remember
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Definitions
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Examples
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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
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Let and
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