How to recover a discrete set of samples from a variable evenly spaced in time but unevenly spaced in frequency?

Hi Everyone,

I've been struggling for a while with a mathematical problem, which I would like to explain here. I have used Matlab to model this phenomenon, so I will include my question in this forum.

Let me describe the origin of my challenge. First, consider the two main variables of my problem, B (magnetic field, in T) and u (amplitude of the movement of a wire, in m). The relationship between them (under a specific situation) is given by

where is the Fourier Transform of B, κ is the wavenumber (in rad/m), c is the wavespeed (in m/s), t is the time (in s), and and T are just constants. In the equation above, c is not necessarily constant, but instead, it is a function of κ(we say it models a dispersive wire), given by

where is a constant and is the non-dispersive wave speed.

For my real problem, I do not have continuous functions. Instead, I would have discrete (data) samples. In this case, would be the DFT of . Since I would be working on the discrete world,

where N is the number of samples and is the sampling step (in m). Similarly,

Alright. If I know , , , , T and , I would calculate the samples of the movement of the wire by applying a discrete version of Eq. (1), i.e.,

Notice that I took the samples from n=2, since n=1 is the DC component, which I am not interested in. I also summed up until N/2, and multiplied by 2 before the sum operator. The real part is taken because complex does not add anything to my representation (remember that u is movement in meters).

Now I can describe my problem: Imagine that I want to use this phenomenon to measure from the movement of the wire over time . How would you do that? In other words, if I measure the movement of the wire , how could I try to recover the magnetic field (or )?

The main issue here is that the argument is not evenly spaced, so the Fourier analysis becomes harder. For example, I have tryied to calculate the DFT of and compare it with the right side of Eq. (2). But again, it does not work very well, because has evenly spaced samples in time, while the frequency is not evenly spaced.

Another thing that I've tried: I set Eq. 2 as a linear system. I treated and as column matrices, and created a matrix M such that

In this way, , so I may try something like . Nevertheless, M seems to have rank problems, and frankly, I do not know how to deal with it.

So, do you have any ideas/suggestions on how to measure B from u, assuming I know the parameters , , T, and ?

Thank you all.

[Mathieu NOE]

hello

can you sned a simple extract of your code with the numerical values of u, M, B

if I can try to solve that equation...