I am researching the split-step parabolic equation and its split step solution as in:

Ozgun, Ozlem & Apaydin, Gokhan & Kuzuoglu, Mustafa & Sevgi, Levent. (2011). PETOOL: MATLAB-based one-way and two-way split-step parabolic equation tool for radiowave propagation over variable terrain.

The idea is to solve equation:

enter image description here

where $F$ is a Fourier transform, $x$ is the horizontal range, $z$ is the vertical range, $k=2\pi/\lambda$ is the wavenumber, $n$ the refractive index of air and $p=k\text{sin}(\theta)$ is the Fourier transform variable in spectral domain, and $\theta$ is the angle propagation angle from horizontal

The idea is that we have an antenna which propagates in $(x,z)$ direction like shown in the image below

enter image description here

We need to choose the $\Delta z$ that is the altitude increment in a way that avoids alliasing. What I don't understand from the link I pasted is that authors say, that there is a "Nyquist Criterion" such that:

$$z_{max}\times p_{max}=\pi N$$

where $N$ is the Fourier transform size, $z_{max}$ is the maximum vertical height as shown above and $p_{max}$ is the maximum value of the spectral variable. Could anyone try and derive this for me? Tell me where it comes from? Thank you.


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