Ambiguity about roots of unity in discrete Fourier transform

In summary, the discrete Fourier transform is a change of basis operator between the usual orthonormal basis and the "Fourier" basis, which consists of vectors with powers of the N roots of unity. However, in complex analysis, the root of a complex number is not unique. This raises the question of which solution is the actual first entry in the first Fourier basis vector. In the calculus of discrete Fourier transformations, we use finite fields instead of traditional complex analysis, which makes the solution unique. If temporary or final results do circle, additional information from the application is needed to determine the correct solution."
  • #1
CantorSet
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Hi everyone, I have a question on the discrete Fourier transform. I already know its a change of basis operator on [itex]C^N[/itex] between the usual orthonormal basis and the "Fourier" basis, which are vectors consisting of powers of the [itex]N[/itex] roots of unity.

But if i recall correctly from complex analysis, the root of a complex number is not unique. So for example, if we look at the first entry of the first Fourier basis vector, it is [itex] e^{\frac{2 \pi i }{N}} [/itex]. But there are N solutions here. Which one is the actual first entry in the first Fourier basis vector?
 
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  • #2
We do not use the usual complex analysis in the calculus of discrete Fourier transformations. Instead finite fields are used, that is we consider only one cycle on the circle and do not circulate, which makes it unique. If temporary or final results do circle, then additional information from the application is needed to determine which one.
 

FAQ: Ambiguity about roots of unity in discrete Fourier transform

1. What are roots of unity in discrete Fourier transform (DFT)?

Roots of unity refer to the complex numbers that satisfy the equation x^n = 1, where n is the number of data points in the DFT. In other words, they are the solutions to the equation that represent the frequencies of the signal in the time domain.

2. Why is there ambiguity about the roots of unity in DFT?

The ambiguity arises because there are multiple solutions to the equation x^n = 1, which can lead to different interpretations of the frequency components in a signal. This can cause confusion when interpreting the results of a DFT.

3. How does the choice of roots of unity affect the DFT results?

The choice of roots of unity can affect the resulting frequencies and amplitude values in the DFT. Different choices can lead to different interpretations of the signal and can also impact the accuracy of the DFT.

4. How can the ambiguity about roots of unity be resolved?

The ambiguity can be resolved by choosing a specific convention for the roots of unity, such as the principal root or the primitive root. This ensures consistency in the interpretation of the results and avoids confusion.

5. Are there any practical implications of the ambiguity about roots of unity in DFT?

Yes, the ambiguity can affect the accuracy and precision of the DFT results, especially in cases where the signal contains closely spaced frequencies. It is important to carefully consider the choice of roots of unity when analyzing signals using DFT to avoid misinterpretation of the results.

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