Discrete Fourier transform in k and 1/k

gnulinger

Say you have some function that is periodic in a parameter k. The discrete Fourier transform from a sampling may be found in the usual way, giving the frequency spectrum in k. But what if I want to find the frequency spectrum in 1/k ?

I'm not really sure what this is called, and so I've had a hard time Google searching for it. Any links or help would be appreciated. Thanks.

chiro

Hey gnulinger.

When you say frequency spectrum are you talking about integer frequencies?

I do know that there are ways to get fractional frequencies that are based on fractional derivatives and subseqent integrals:

http://mathworld.wolfram.com/FractionalDerivative.html

But if you are talking about just having a transfer function to get something in F(1/k) instead of F(k), then I think this is going to be a bit more involved and you should probably outline the reason why you want the function in terms of 1/k as opposed to the linear transform space k.

gnulinger

I am talking about the latter, and yes, I think it will be fairly involved. I have a function that is periodic in 1/k, and I am wondering if there is some way of mapping the DFT in k to that in 1/k.

rbj

i know a lot about the DFT, it's definition, the theorems, how it is related to the continuous fourier transform. but i cannot decode at all what you're talking about. what do you mean that it is "periodic in 1/k" ? try tossing up equations to be clear.

BTW, even though i get in fights about this on comp.dsp, i maintain that the DFT is nothing other than the Discrete Fourier Series. the DFT maps one discrete and periodic sequence of length N to another discrete and periodic sequence of the same length. and the inverse DFT maps it back. dunno if that answers your question.

gnulinger

Part of the problem is that I too am unclear on this subject, so it is hard for me to ask the right questions. I was hoping that someone may have heard of something related to what I was asking about, and could have pointed me in the right direction.

In the De Haas-van Alphen effect, wikipedia link, the magnetic moment of a crystal oscillates with period related to 1/B, where B is the magnetic field. The DFT would ostensibly give you a frequency spectrum in 1/B.

This is similar to what I want to do.

rbj

so, are you sampling the magnetic moment function of time somehow? where do the numbers that go into the DFT get set to some value?

marcusl

The simplest way is to plot the results against 1/k on a nonlinear scale. You often see optical spectra plotted this way--the calculation is done for frequency but the plot is done against lambda.

gnulinger

Do plot your data or the DFT against 1/k?

mdo

You have a function f(t) that has a Fourier transform, F(ω), that is null or almost null for |ω| > Ω. f(t) is sampled at every multiple of a given interval h to obtain a discrete signal f(n) = f(nh), where h should be < π/Ω. When you compute the DFT, F(μ), you are working on a normalized domain 0≤μ<π, but you can express it in "real" ω by multiplying by Ω or by 2π/h.
It's indifferent.