I work with radar and I want to understand which spatial frequencies I can measure, i.e. the k-space coverage, given a set of coordinates of transmitter-receiver combinations. The ideal coverage for a reflective and transmissive mode of measurement can be found e.g. in this paper (figures 2 and 3), or here for a selection of measurement set-ups. Both of these papers give principal descriptions of these figures and I somewhat understand how to arrive at them, however, I want to understand how to calculate the sampling in k-space given an aperture of limited size at select positions and a signal of limited frequency bandwidth.

I understand that the transmissive measurement mode yields more of the low frequency content and therefore measures the "bulk" of the material, while the reflective mode measures high frequencies and therefore "edges", which corresponds to the shapes of the principal support in k-space. The limit of being able to measure a disk of radius $2*k$, with k being the wavenumber of the incident signal, also corresponds to the Nyquist criterion of a necessary antenna spacing of $\lambda/2$ to avoid aliasing. As far as I understand it, imaging (in the context of the Born approximation, where this k-space method is applicable) can be considered as a convolution of the PSF of the imaging system and the object function, which would be equivalent to the product of the Fourier Transform of the object function with the support of the system in k-space.

My current approach is quite simple: I implicitly assume a point scatterer at $(0,0)$. Given a number $n$ of transmitter coordinates $\vec{r}_t^n$, and corresponding receivers $\vec{r}_r^n$, as well as a wavenumber for the transmitted signal $k$, I then calculate the k-space coverage as: $$ \vec{\Omega} = k \cdot (\hat{r}_t^n - \hat{r}_r^n) $$ Where I normalised the vectors giving the antenna positions. This results in "sensible" images for the case of a pure linear aperture in a transmissive measurement mode, which look similar to Figure 4 in the Natterer paper. For a pure linear aperture in a reflective measurement mode, however, this results in nonsense, and the - has to be replaced with a + to make sense and have a similar result to the paper. I also tested some other set-ups, e.g. a bistatic set-up that gets rotated around $(0,0)$, and it looked sensible (reflective mode yielded a ring at high frequencies, up to the $2*k$ limit, while transmissive mode yielded a ring at low frequencies, close to the center), but I would like to make sure that what I am doing is actually correct (and why).

If this is in the wrong substack, please do let me know, I was wondering whether this, physics or mathematics would be correct.



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