What is the DFT of a constant?

[Mr.Tibbs]

The problem:

What is the discrete Fourier transform of a constant value?

Example DFT{2}

This is not my homework problem but will help me immensly in solving the actual problem.

DFT formula:

X$_{k}$ =$\sum$x[n] * e$^{\frac{-2\pi kn}{N}}$ from n = 0 to N-1

where N is the number of samples you can take in a 2$\pi$ period.

 

[cepheid]

So x[n] = 2 for all n.

That means you can take x[n] outside the summation, and you're left with a sum of exponential terms. Do you know how to work that out?

Naively I would expect it to reduce to an impulse (delta function) at k = 0, or perhaps regularly repeating impulses, due to the discrete nature of the DFT. I've have to sit down and think about it some more.

 

[Mr.Tibbs]

Ah, that helps out immensely. As it turns out it does turn into a delta function at k = 0. Thank you so much for clearing that up for me.