Fourier Transforms - what's the constant?

In summary, the constant in front of the Fourier Transform and its inverse can vary depending on convention. As long as the two factors multiplied together equal 1/2π, it does not affect the overall result. However, using the same constant for both the FT and IFT can make the equations more similar and aid in interpreting Fourier coefficients in physical units.
  • #1
RESolo
7
0
Hi

At university, in a previous module the constant in front of the Fourier Transform was given as (1/2π), and the constant in front of the inverse F.T. was 1.

However in a current module the lectrurer gives the constant as (1/√2∏), for both the F.T. and its inverse. Why is this?

Thanks
 
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  • #2
Different people use different conventions. So long as both factors multiplied together give (1/2π), it doesn't matter.

The specific convention only matters if you want to interpret the Fourier coefficients in physical units, e.g. as a power spectrum.

There is some logic to making both factors the same, because it makes the equations for the FT and IFT more similar.
 

Related to Fourier Transforms - what's the constant?

1. What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to break down a complex function into simpler parts, specifically into a combination of sinusoidal functions. It is commonly used in signal processing, image processing, and other areas of science and engineering.

2. How does a Fourier Transform work?

A Fourier Transform works by decomposing a function into an infinite sum of sinusoidal functions with different frequencies, amplitudes, and phases. This allows us to analyze the frequency components of a signal or function.

3. What is the purpose of the constant in a Fourier Transform?

The constant in a Fourier Transform, also known as the scaling factor, is used to ensure that the amplitude of the Fourier Transform is consistent with the original function. It helps to preserve the energy of the function and ensures that the inverse Fourier Transform will accurately reconstruct the original function.

4. How do you calculate the constant in a Fourier Transform?

The constant in a Fourier Transform is calculated by dividing the function by the square root of its period. This ensures that the amplitude of the Fourier Transform is consistent with the original function and that the total energy is preserved.

5. Can the constant in a Fourier Transform be negative?

Yes, the constant in a Fourier Transform can be negative. This happens when the function has a negative amplitude or when it is shifted by a phase angle of pi. In most cases, the constant is positive, but it is important to consider both positive and negative values when analyzing a Fourier Transform.

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