Why Normalization Factors in Fourier Transform Pairs?

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SUMMARY

The discussion centers on the normalization factors in Fourier transform pairs, specifically addressing the different conventions used, such as 1/2π, 1/√(2π), or no factor at all. Participants clarify that these factors are essential for ensuring proper normalization in the dot product calculations of coefficients. The normalization factors for sine, cosine, and complex exponential functions are highlighted as being equal to 1/√(π) to achieve a modulus of one. The conversation also touches on the implications of using different arguments in the exponential functions, such as exp(iωt) versus exp(i2πft).

PREREQUISITES
  • Understanding of Fourier transforms and their mathematical properties
  • Familiarity with normalization in mathematical functions
  • Knowledge of complex exponentials and their applications
  • Basic grasp of trigonometric functions, specifically sine and cosine
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  • Research the different conventions of normalization in Fourier transforms
  • Study the implications of normalization factors on signal processing
  • Learn about the mathematical derivation of Fourier transform pairs
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Mathematicians, signal processing engineers, and students studying Fourier analysis will benefit from this discussion, particularly those interested in the nuances of normalization 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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