Fourier transform: Difference between revisions
Created page with "The Fourier transform decomposes a signal (i.e. a time series) into multiple sine and cosine waves. ==Background== Suppose we have signal<math>f(x)</math>.<br> Then the Fourier transform <math>\hat{f}(\xi)</math>is defined as: <math display="block">\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-i2 \pi \xi x} dx</math> Recall that Euler's formula states: <math>e^{ix} = cos(x) + i \sin(x)</math>. Hence, <math>e^{-i2 \pi \xi x} = cos(2 \pi \xi x) + i \sin(2 \pi \xi x)<..." |
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<math display="block">A_k = \sum_{0}^{n-1} f(x) \exp\ \left\{ -2 \pi i \frac{mk}{n} \right\}</math> | <math display="block">A_k = \sum_{0}^{n-1} f(x) \exp\ \left\{ -2 \pi i \frac{mk}{n} \right\}</math> | ||
and IFFT is defined as: | and IFFT is defined as: | ||
<math display="block">a_m = | <math display="block">a_m = \frac{1}{n} \sum_{0}^{n-1} \hat{f}(\xi) \exp \left\{ 2 \pi i \frac{mk}{n} \right\}</math> | ||
==Properties== | ==Properties== | ||