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== | ||
Revision as of 18:57, 20 January 2023
The Fourier transform decomposes a signal (i.e. a time series) into multiple sine and cosine waves.
Background
Suppose we have signal\(\displaystyle f(x)\).
Then the Fourier transform \(\displaystyle \hat{f}(\xi)\)is defined as:
\[\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-i2 \pi \xi x} dx\]
Recall that Euler's formula states: \(\displaystyle e^{ix} = cos(x) + i \sin(x)\). Hence, \(\displaystyle e^{-i2 \pi \xi x} = cos(2 \pi \xi x) + i \sin(2 \pi \xi x)\) In other words, the Fourier transform is the integral (i.e. alignment) of signal times some sine and cosine waves.
The inverse of the fourier transform is: \[f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{i2 \pi \xi x} d\xi\]
Discrete Fourier Transform
A naive DFT would compute the matrix of \(\displaystyle e^{-i2 \pi \xi x}\) and multiply it with the signal. This would take \(\displaystyle \mathcal{O}(n^2)\) time.
However, most languages have an FFT library which can compute the DFT in \(\displaystyle \mathcal{O}(n \log n)\) time.
In most languages, FFT is defined as: \[A_k = \sum_{0}^{n-1} f(x) \exp\ \left\{ -2 \pi i \frac{mk}{n} \right\}\] and IFFT is defined as: \[a_m = \frac{1}{n} \sum_{0}^{n-1} \hat{f}(\xi) \exp \left\{ 2 \pi i \frac{mk}{n} \right\}\]
Properties
Note \(\displaystyle \bar{x}\) refers to the complex conjugate
- Linearity: \(\displaystyle FFT(\lambda f + g) = \lambda FFT(f) + FFT(g)\)
- \(\displaystyle IFFT(f) = (1/n) \bar{FFT}(\bar{f})\)
- \(\displaystyle FFT2(f) = FFT_{x}(FFT_{y}(f)) = FFT_{y}(FFT_{x}(f))\)
Short-time Fourier transform
The STFT applies FFT using a sliding window over the signal.
This produces a matrix of FFT values over time which allow you to see how the signal is changing.