Why Normalization Factors in Fourier Transform Pairs?

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Discussion Overview

The discussion revolves around the normalization factors in Fourier transform pairs, exploring why different conventions exist for these factors in relation to the mathematical expressions used in the transforms. The scope includes theoretical aspects of Fourier analysis and the implications of normalization in calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of different normalization factors (1/2*pi, 1/sqrt(2pi), or none) in Fourier transform pairs, linking it to the importance of normalization in coefficient calculations.
  • Another participant acknowledges the existence of several conventions for normalization, suggesting that these conventions are valid.
  • A participant raises a question about the normalization factors for the exponential forms exp(i*w*t) and exp(i*2*pi*f*t), indicating a potential inconsistency in their normalization.
  • Another reply suggests that the presence of the 2pi in the exponential function accounts for the difference in normalization factors.

Areas of Agreement / Disagreement

Participants express differing views on the normalization factors used in Fourier transforms, indicating that multiple conventions exist and that there is no consensus on a single approach.

Contextual Notes

The discussion does not resolve the underlying assumptions about the definitions of normalization factors or the implications of using different exponential forms in Fourier transforms.

fisico30
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hello forum,
a question about normalization factors in the Fourier transform pairs:

why is it that a factor 1/2*pi in only one, or 1/sqrt(2pi) in both, or no factor (in the case the complex basis functions have argument 2*pi*f) need to be in front of the Fourier transform pair?

I understand normalization and its importance in the dot product calculation of coefficients.
Sines, cosines, and complex exponentials have normalization factors equal to 1/sqrt(pi)...
in order to get modulus one...


thanks
fisico30
 
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it is there for normalization as you said, and there exists several conventions which works.
 
so,

exp (i*w*t) and exp (i*2*pi*f*t) don't have the same normalization factors? Why?
 
since you have 2pi already in the exponential
 

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