## 2 Background and Pseudo-Doppler Principles
### Implications in an OAM Radio Link  
As pointed out in [10], the resulting scaled pseudo-Doppler shift at the output of the combiner in the front end of the receiver should be greater than the bandwidth, \(B\), of the transmitted signal in any OAM mode (assuming each OAM mode carries an independent data stream at the same rate of \(B\) symbols/second). So one \"unit\" of frequency shift corresponds to \(k=1\) and satisfies  
\[2\pi Fd/(2R)\geq 2\pi B \tag{12}\]  
in keeping with radian units of frequency. In the example depicted in Figure 7, choose the parameters as in Table 1 below. The condition \(0.02k=kd/(2R)<<\pi/4\) is satisfied. From (22) and Table 1 the oscillator frequency for the pseudo-Doppler modulator of the receiver front end in Figure 7 is determined to be \(F=2\) GHz. This is a typical frequency in modern mobile radio hardware so the modulator and oscillator are easily achieved. The effective pseudo-Doppler shift of the \(k\)= 4 OAM mode will be 40 MHz. Note that the receiving antenna array in Figure 7 is not positioned on the beam axis as in \"conventional\" RF links employing OAM beams, but at the peak of the OAM beams which is at a radius \(R\) perpendicular to the axis. It is this arrangement which enables a higher SNR at the receiving end, owing to the much higher OAM signal amplitude there. Given the size of RF wavelengths, it would be grossly impractical to capture all of the OAM beam energy at radius \(R\) as the receive antenna array would need to be of the same order in size. As also stated in [10], it would be equally impractical to physically rotate such an antenna to impart a Doppler shift to its output, hence the motivation to use a virtually-moving, electronically interpolated \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter Symbol**} & \multicolumn{3}{c}{**OAM Radio Link Parameters**} \\ \cline{2-4}  & **Description** & **Value** & **Units** \\ \hline B & Signal bandwidth 1  & 10 & MHz \\ L & Link distance & 1 & km \\ R & Radius of overlapping OAM conical-beam footprints & 20 & m \\ d & Separation of receiving antennas & 20 & cm \\ \(\lambda\) & RF carrier wavelength & 5 & cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example Parameters of OAM Radio Link. antenna, to impart the much higher Doppler frequency shifts to the OAM modes using pseudo-Doppler techniques. It is not difficult to perceive that, despite having only two receiving antennas (necessarily) positioned off-axis, the SNR at the receiver can be much higher than that obtainable from a similar size of receiving radio antenna array necessarily positioned on-axis. Therefore, the subsequent signal processing of the pseudo-Doppler shifted OAM beam signals in the frequency domain is expected to yield much \"cleaner\" recovered OAM signals, at longer link distances in free-space RF applications than a conventional spatial OAM recovery technique relying on the reciprocity of launching OAM modes with a similarly-sized antenna array. (Note that even in such reciprocity-based spatial OAM recovery schemes, the receiving antenna is not large enough to capture all of the beam energy contained in the toroidal \"footprint\" with radius \(R\) of the OAM beams at link distance \(L\). The SNR penalty of those schemes is not due to the _size_ of their receiving antennas so much as it is due to their necessary _positioning on-axis_, where all (non-zero-order) OAM modes have a deep amplitude minimum and are spatially orthogonal.) Further possibilities for enhancing the effective SNR by frequency-domain signal-processing are described in Section 4.3. To determine the size of the transmitting antenna array, it is observed in Appendix B that the peak of the \(k\)-th order Bessel function in (9) occurs roughly where its argument is equal to \(k+1\) for orders below about 6. The receiver is situated off-axis, at the peak of the overlapping OAM beams. (The antennas are not drawn to the same scale as the beam pattern and link geometry.)