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Fourier transform unitary, angular frequency |
Fourier transform unitary, ordinary frequency |
Remarks
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![{\displaystyle g(t)\!\equiv \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db4f64e9330c14a22b70228c614c089ef0b5a5d5)
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![{\displaystyle G(\omega )\!\equiv \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f7f916fbeb6f2f2e111a93bad95549534446793)
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![{\displaystyle G(f)\!\equiv }](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0b518e7b4dad9a4ebf2230ee04bdc4fca81ce4)
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10
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The rectangular pulse and the normalized sinc function
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11
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Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
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12
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tri is the triangular function
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13
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Dual of rule 12.
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14
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Shows that the Gaussian function is its own Fourier transform. For this to be integrable we must have .
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common in optics
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a>0
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the transform is the function itself
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J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
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it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
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![{\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33484b90b024cd7e6da8fb4f8cffaf6bd7121ef1)
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![{\displaystyle {\frac {2i}{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab20951f0a86bd96f3d11052fb603685ddf07db6)
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Un (t) is the Chebyshev polynomial of the second kind
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