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I'd like to describe a problem that I've been struggling with for a while. I want to apologize in advance due to the long text. I just want to be as clear as possible in my first post.

Consider the equation below that provides a relationship between a variable $u(t)$, function of time $t$ ($u$ is given in meters) and the variable $B(z)$, function of a position $z$ in the space ($B$ is given in Tesla).

$$u(t) = \alpha \int_{-\infty}^{\infty} \frac{e^{i\kappa c \beta}-1}{\kappa^2c^2} \overline{B}(\kappa)e^{-i\kappa ct} d\kappa \, \, \, \, \, (1)$$

In this equation, $\kappa$ is the wavenumber (rad/m), $c$ is the wave speed (m/s), $\overline{B}(\kappa)$ is the Fourier Transform of $B(z)$, and $\alpha$ and $\beta$ are constants. It is clear to me that this equation is a representation of an inverse Fourier transform. (Maybe that becomes more clear if we define the angular frequency $\omega = \kappa c$).

Now, consider two things:

  1. I need to represent this equation in the discrete world. In other words, I only have discrete samples of $B$ (i.e., $B(n\Delta z) = B[n]$ and, consequently, samples for $t$, $\kappa$ and $u$. If $c$ is constant (let's say $c_0$), I would try something like

$$u[m] = \alpha \frac{2}{N} \operatorname{Re}\left\{\sum_{n=2}^{N/2} \frac{e^{i\kappa [n]c_0}-1}{(\kappa [n]c_0)^2} \overline{B}[n] e^{-i\kappa [n]c_0t[m]}\right\} \, \, \, \, \, (2)$$

where $N$ is the number of samples, $\kappa[n] = 2\pi(n-1)/(\Delta z N)$ (evenly spaced samples) and $t[m] = m\Delta t$. This representation seems to work pretty well. Notice that $n$ starts from 2 because I am not interested in the DC component. Besides, for $n=1$, I'd have a division by 0.

  1. Consider now that $c$ is not a constant, but instead, a function of $\kappa$, given by

$$c(\kappa) = c_0 \sqrt{1+a\kappa^2} \, \, \, \, \, (3)$$

or

$$c[n] = c_0 \sqrt{1+a(\kappa[n])^2} \, \, \, \, \, (4)$$

for the discrete samples, where $c_0$ and $a$ are constants. Can I still use the same representation of equation (2)? I've tried so, but I am afraid it would not be correct. The reason is because the exponential argument $k[n]c[n]$ (which corresponds to an angular frequency $\omega[n]$) is no longer evenly spaced. I do not know if I have to use something related to an inverse non-uniform discrete Fourier transform. In other words, if you have samples of $B[n]$ instead of the continuous function $B(z)$ and $c$ is given by equation (3), how would you represent equation (1) in the discrete domain?

Thank you. Take care. =)

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  • $\begingroup$ Using uneven sampling results in distorted frequency domain, with the periodicity affected. Using irrational numbers (z(x)=1+x/2**.5) seems to result in aperiodic spectrum. There most probably is a mathematical explanation, too. I think it's related to the way two or more sines combine, similar to $\sin{a}+\sin{b}=2\sin{\frac{a+b}{2}}\cos{\frac{a-b}{2}}$, the lowest argument dictates the overall periodicity. Here, it probably involves a gcd. I'll let others explain, I'm curious, too. $\endgroup$ – a concerned citizen Oct 13 '20 at 19:34

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