DFT Properties

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Periodicity

If you look back at the explanation of the DFT you will see that the finite input becomes periodic, as well the frequency response. This is something that has to be considered whenever you interpret the results of a DFT. This is represented mathematically as:

Linearity

If you take the DFT of two sequences and obtain their frequency responses, then the DFT of the combination of these sequences is the addition of their individual frequency responses. Mathematically this is:

Circular shift of DFT INPUT

For a sequence that exists for all n then this can be shifted by  When working with the DFT the sequences are only defined for 0 to N-1 therefore when the sequence is shifted, part of it would fall out of the area of interest. However the DFT is periodic before and after this area of interest. So when the sequence is shifted out if the window (0 to N-1), it can be thought of as wrapping around the part that falls off the end, back to the front of the sequence. Rather like the sequence is on a wheel with a fixed point for reference, and the wheel spins round whenever you shift the function.

Going back to the twiddle factor properties we can represent this idea as :

Circular shift of DFT OUTPUT

This same shifting property applies to the frequency domain as well as the time domain. Using the same notation we get:

This time it is the time domain signal that is multiplies by the twiddle factor. It should be noted that because the twiddle factor is complex that a shift in the frequency domain will usually result in a complex form of the time domain sequence even if it was originally real.

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