MathLiveDiscrete Fourier transform
The discrete Fourier transform converts a finite sequence of N samples into N complex coefficients describing the sequence's frequency content. It is used in signal processing, spectral analysis, and numerical computation of Fourier transforms.
X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} k n}What the symbols mean
| the k-th complex frequency coefficient of the transform | |
| the n-th input sample of the sequence | |
| the total number of samples | |
| the sample index, running from 0 to N-1 | |
| the frequency index, running from 0 to N-1 | |
| the imaginary unit, with i^2 = -1 | |
| Euler's number, the base of the natural logarithm |
About this formula
- FieldMathematics
- TopicFourier analysis
- LevelUniversity
- Also calledDFT formula, discrete Fourier transform equation, DFT, Fourier transform of a sequence, sampled signal frequency transform
- SourceWikidata Q2878
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