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, $k$ is the wavenumber, $n$ the refractive index of air and $p$ is the Fourier transform variable in spectral domain.

The idea is that we have an antenna which propagates in $(x,z)$ directon 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.

  • 2
    $\begingroup$ Nyquist Criterion is about sampling a continuous-time (or space) signal. And to avoid aliasing you shall use sufficiently dense samples which is determined by the bandwidth of the signal being sampled. Now Which signal is sampled here? And what's its bandwidth?And which spatial sampling grid is used? I don't want to read the paper, but you can put these information there for clarification. $\endgroup$ – Fat32 Jul 24 '20 at 15:08

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