## 2 Background and Pseudo-Doppler Principles
### Application to OAM Radio Waves in the Far Field
since, according to the discussion of Figure 1, it is clear that  
\[v_{n}=\frac{d}{dt}(R\theta_{n}) \tag{6}\]  
Thus, it has been shown that a spatial-domain property of an OAM beam of order \(k\), the phase gradient \(k/R\), can be converted to a frequency-domain property, namely a kind of transverse or rotational Doppler shift \(f_{n,Doppler}\), through the motion of the antenna element receiving the OAM beam. Note that the effect is real in the physical sense [9]; the subscript \"\(n\)\" may be omitted as there is only the one moving antenna. Note also, that this transverse Doppler shift is directly proportional to the OAM order, \(k\), and independent of RF carrier frequency. Next, invoke the pseudo-Doppler technique whereby the motion of a single antenna from position #1 to position #2 is emulated by switching among several antennas, as outlined at the end of subsection \(A\). Specifically, let the receiver employ two antennas separated by distance \"\(d\)\" tangentially to the footprint of the OAM beam, and instead of switching between their outputs, the receiver _combines_ their outputs in _time-varying proportions_ ranging from only output #1, to half of each output #1 and #2, to only output #2. This is in effect a form of gradual switching between the antennas, in _one_ direction; it is assumed to be repeated periodically at some rate in accordance with the principles employed in [10]. To understand how such time-variant combining emulates a transverse Doppler shift in the received carrier frequency, Appendix A reviews the principle of the Doppler effect in simple contexts. With that in mind, it is relatively easy to derive the **key relation** between the phases of the OAM beam as received at the two antennas at a given time in Figure 4.  
Figure 4: Applying pseudo-Doppler technique to OAM beams at a two-antenna receiver. Only the footprint of the OAM beam is shown; the color denotes the phase. At a given point in time, the phase of the RF wave arriving in the form of OAM mode \(k\) at RX antenna #1 is \(\psi_{1}\) and at the same time the phase at RX antenna #2 is \(\psi_{2}\). Because the phase delay advances \(k\) multiples of \(2\pi\) radians for one complete trip around the footprint, \(2\pi R\), at a given observation point in time, it advances by proportion \(kd/(2\pi R)\) for the portion of the footprint covered by the antenna separation \"\(d\)\". The phases at the two RX antennas are therefore related as  
\[\psi_{2}(t)=\psi_{1}(t)-k2\pi\bigg{(}\frac{d}{2\pi R}\bigg{)} \tag{7}\]  
Consequently, the signals at the inputs \(W_{1}\) and \(W_{2}\) of the time-varying combiner are modeled as being multiplied by the complex-exponential phase factors as  
\[ \begin{split} W_{1,k}(t)&=S_{k}(t)e^{i\psi_{1}(t)} \\ W_{2,k}(t)&=S_{k}(t)e^{i\psi_{2}(t)}=S_{k}(t)e^{i \psi_{1}(t)-jkd/R}\end{split} \] (8a,b)  
where \(S_{k}(t)\) is the signal of the \(k\)-th OAM beam received at the reference point in the far field. Before exposing the function of the time-varying combiner in Section 3, the next subsection describes briefly the transmitting end of the link where the multiple OAM beams are modulated with independent data streams on the same RF carrier and launched from the antenna structure. In this respect, the multiplexing of several data streams onto several OAM beams is still effectively performed in the spatial domain as in all other OAM transmission schemes in RF applications.