10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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|<math>=\frac{1}{2}X(f-f_0)+\frac{1}{2}X(f+f_0)</math> |
|<math>=\frac{1}{2}X(f-f_0)+\frac{1}{2}X(f+f_0)</math> |
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|<math>x(t)\,\!</math> |
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|<math>=F^{-1}\left[X(f)\right]</math> |
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|<math>F\left[\frac{dx}{dt}\right]</math> |
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|<math>=F\left[\frac{d}{dt}F^{-1}\left[X(f)\right]\right]</math> |
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|<math>=F\left[\frac{d}{dt}\int_{-\infty}^{\infty}X(f)\,e^{j\,2\pi f\,t}\,df\right]</math> |
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|<math>=F\left[\int_{-\infty}^{\infty}j\,2\pi fX(f)e^{j\,2\pi f\,t}\,df\right]</math> |
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|<math>=F\left[j\,2\pi f\,F^{-1}[X(f)]\right]</math> |
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|<math>=j\,2\pi f\,X(f)</math> |
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|Thus <math>\frac{dx}{dt}</math> is a linear filter with transfer function <math>j\,2\pi f</math> |
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Revision as of 18:57, 21 November 2008
Properties of the Fourier Transform
Linearity
Time Invariance (Delay)
Let and | ||
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Thus is a linear filter with transfer function |