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hilbert_transform [2014/02/21 18:21] nikolaj |
hilbert_transform [2014/02/21 18:22] nikolaj |
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| ==== Partial Function ==== | ==== Partial Function ==== | ||
| | @#FFBB00: $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ | | | @#FFBB00: $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ | | ||
| - | | @#FFBB00: $H(f):=t\mapsto \frac{1}{\pi}\mathcal P\int_{-\infty}^\infty\frac{f(\tau)}{t-\tau}\,\mathrm dx$ | | + | | @#FFBB00: $H(f):=t\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(\tau)}{t-\tau}\,\mathrm dx$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| + | $H(H(f))=-f$ | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Hilbert_transform|Hilbert transform]] | Wikipedia: [[http://en.wikipedia.org/wiki/Hilbert_transform|Hilbert transform]] | ||
