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Mathematics

Area of mathematics

Calculus

Fourier series

Discrete Fourier transform

There are actually two possible definitions for the DFT:
* 1/N, given as "the default" in many sources:
  $$
  x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{i 2 \pi \frac{k n}{N}}
  $$
* $1/\sqrt{N}$, known as the "normalized DFT" by some sources: https://www.dsprelated.com/freebooks/mdft/Normalized_DFT.html[], definition which we adopt:
  $$
  x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{i 2 \pi \frac{k n}{N}}
  $$

The $1/\sqrt{N}$ is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.
* https://math.stackexchange.com/questions/3285758/scaling-magnitude-of-the-dft
* https://dsp.stackexchange.com/questions/63001/why-should-i-scale-the-fft-using-1-n
* https://www.dsprelated.com/freebooks/mdft/Normalized_DFT.html


Normalized DFT

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Normalized DFT

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