Why Normalization Factors in Fourier Transform Pairs?

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In summary, there are different normalization factors that can be used in Fourier transform pairs, including 1/2*pi, 1/sqrt(2pi), or no factor at all. These factors are necessary to ensure that the basis functions, such as sines, cosines, and complex exponentials, have a magnitude of 1. The choice of normalization factor depends on the specific convention being used.
  • #1
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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  • #2
it is there for normalization as you said, and there exists several conventions which works.
 
  • #3
so,

exp (i*w*t) and exp (i*2*pi*f*t) don't have the same normalization factors? Why?
 
  • #4
since you have 2pi already in the exponential
 

1. Why is normalization necessary in Fourier Transform pairs?

In Fourier Transform, the amplitude of the signal in the frequency domain is directly proportional to the amplitude of the signal in the time domain. However, the amplitude of the signal also depends on the number of samples used in the calculation. Therefore, normalization is necessary to ensure that the amplitude of the signal is not affected by the number of samples used.

2. What is the purpose of normalization in Fourier Transform?

The main purpose of normalization in Fourier Transform is to ensure that the amplitude of the signal in the frequency domain is independent of the number of samples used in the calculation. This allows for easier comparison and analysis of different signals.

3. How is normalization factor calculated in Fourier Transform pairs?

The normalization factor in Fourier Transform is calculated by dividing the amplitude of the signal in the frequency domain by the number of samples used in the calculation. This ensures that the amplitude of the signal is independent of the number of samples used.

4. Can the normalization factor be different for different types of signals?

Yes, the normalization factor can vary for different types of signals. This is because the amplitude of the signal in the frequency domain is dependent on the amplitude of the signal in the time domain, which can be different for different types of signals.

5. What happens if normalization is not applied in Fourier Transform pairs?

If normalization is not applied in Fourier Transform, the amplitude of the signal in the frequency domain will be affected by the number of samples used in the calculation. This can lead to incorrect analysis and comparison of signals, making it difficult to interpret the results.

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