As per the title, I was wondering if there was a $\operatorname{sinc}$ based interoplation formula for reconstructing a signal in the frequency domain which has been sampled with respect the bipolar coordinate system?

For instance, we all know the $\operatorname{sinc}$ based Whitakker-Shannon interpolation formula for reconstructing signals from their samples in the time domain is:

$$x(t)=\sum_{n=-\infty}^{\infty}x[n]\cdot\operatorname{sinc}\left(\frac{t-n\Delta t}{\Delta t}\right)$$

provided that certain criterion are satisfied. This can quite easily be extended to higher dimensions so that if, for example, $\phi$ is now a function and $\boldsymbol{x}\in\mathbb{R}^n$ then

$$\phi(\boldsymbol{x})=\sum_{k\in\mathbb{Z}^n}\phi(k\Delta\boldsymbol{x})\cdot\operatorname{sinc}\left(\frac{\boldsymbol{x}-k\Delta\boldsymbol{x}}{\Delta\boldsymbol{x}}\right),$$ where we define $\operatorname{sinc}(\boldsymbol{x})=\operatorname{sinc}(x_1,...,x_n)=\operatorname{sinc}(x_1)\cdot...\cdot\operatorname{sinc}(x_n)$.

One can also find a similar $\operatorname{sinc}$ based interpolation method in the book 'Advanced Topics in Shannon Sampling and Interpolation Theory' which allows one to reconstruct a signal in the frequency domain from samples taken in the polar coordinate system $(\rho, \theta)$

Let $\phi(x,y)$ be space-limited to $2A$ and $\hat{\phi}(\rho,\theta)$ be angularly band-limited to $K$. Then $\hat{\phi}(\rho,\theta)$ can be reconstructed from its polar samples via

$$\hat{\phi}(\rho,\theta)=\sum_{n=-\infty}^{\infty}\sum_{k=0}^{N-1}\tilde{\hat{\phi}}\left(\frac{n}{2A},\frac{2\pi k}{N}\right)\operatorname{sinc}\left[\frac{2A(\rho-n)}{2A}\right]\cdot\sigma\left(\theta-\frac{2\pi k}{N}\right),$$ where $N$ is even, $$\sigma=\frac{\sin\left[\frac{1}{2}(N-1)\theta\right]}{N\sin\left(\frac{\theta}{2}\right)},$$ and $$\tilde{\hat{\phi}}\left(\frac{n}{2A},\frac{2\pi k}{N}\right)=\begin{cases} \hat{\phi}\left(\frac{n}{2A},\frac{2\pi k}{N}\right),& n\ge 0, \\ \hat{\phi}\left(-\frac{n}{2A},\frac{2\pi k}{N}+\pi\right),& n<0. \end{cases}$$

However, for if we take samples in the bipolar coordinate system $(\sigma,\tau)$ and we want to reconstruct $\hat{x}(\sigma,\tau)$ from $\hat{x}(n\Delta\sigma,m\Delta\tau)$ then I can find absolutely no such formula, despite scouring the internet for literature. Now that I mention it, for my intents and purposes, it would be more helpful to find such a formula for the coordinate system $(a,\tau)$, where $\tau$ are our isosurfaces and $(0,-a)$ and $(0,a)$ are their respective foci.

  • $\begingroup$ Your $x(t)$ is one-dimensional. How can it be, that your Fourier Transform can be expressed in a polar coordinate system (i.e. a 2-dimensional system)? $\endgroup$ – Maximilian Matthé Mar 8 '17 at 5:19
  • $\begingroup$ @MaximilianMatthé The FT for a 1 dimensional sequence has magnitude and phase components, correct? (Given the FT multiplies x(t) by a complex exponential function). $\endgroup$ – Dan Boschen Mar 8 '17 at 12:37
  • $\begingroup$ @MaximilianMatthé That was an example of $\operatorname{sinc}$ interpolation in one dimension. The formula can quite easily be extended to the multi-dimensional case. My $\hat{x}(\rho,\theta)$ is actually something very complicated, so I did not write it out. The $\operatorname{sinc}$ interpolation formula for reconstruction in the Fourier domain is also quite complicated. $\endgroup$ – Jason Born Mar 8 '17 at 13:02

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