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Normalized DFT

Ciro Santilli (@cirosantilli, 37) Mathematics Area of mathematics Calculus Fourier series Discrete Fourier transform
Updated 2025-07-16  0 By others on same topic  0 Discussions Create my own version
There are actually two possible definitions for the DFT:
  • 1/N, given as "the default" in many sources:
    xn​=N1​∑k=0N−1​Xk​ei2πNkn​
    (1)
  • 1/N​, known as the "normalized DFT" by some sources: www.dsprelated.com/freebooks/mdft/Normalized_DFT.html, definition which we adopt:
    xn​=N1​∑k=0N−1​Xk​ei2πNkn​
    (2)
The 1/N​ is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.
  • math.stackexchange.com/questions/3285758/scaling-magnitude-of-the-dft
  • dsp.stackexchange.com/questions/63001/why-should-i-scale-the-fft-using-1-n
  • www.dsprelated.com/freebooks/mdft/Normalized_DFT.html

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