## 2 Background and Pseudo-Doppler Principles
### Transmission of Multiple Overlapping OAM Radio Beams  
In numerous past applications, the axial beam patterns of phase modes, or OAM modes in modern parlance, have been derived and characterized as being proportional to [4]  
\[G_{k}\left(\theta,\phi\right)\ \approx(-j)^{k}e^{jk\theta}I_{k}((2\pi r\sin\phi)/\lambda) \tag{9}\]  
where \(I_{k}\) is a Bessel function of the first kind, order \(k\), \(r\) is the radius of the circular antenna array, \(\varphi\) is the elevation angle measure from the beam axis and \(\lambda\) is the wavelength of the RF carrier wave of the \(k\)-th OAM mode. The first few orders of this Bessel function are shown plotted in Figure 10 of Appendix B.  
Consequently, it is seen that OAM beams of higher orders have wider cone angles in the far field than those of lower orders, and beams of different OAM orders do not overlap much in space (except negative and positive modes of the same order). That is also evident from Figure 1, which was plotted in accordance with Equation (9) and Appendix B, where the peak positions along the \(x\)-axis correspond the peaks of the conical OAM beams at radii \"\(R\)\" from the beam axis. Clearly, the higher-order Bessel functions having peaks at larger values of \"\(x\)\" means that higher-order OAM modes have peaks at larger radii from the axis, hence larger cone angles, as dictated by (9). Also evident from (9) is the property that an OAM beam of order \(k\) generated from an array with a larger radius will have a smaller cone angle than the same OAM of order \(k\) generated by an array with a smaller radius. Because it is desired to transmit multiple data streams on the same RF carrier to one user using multiple OAM beams, their conical beam patterns must overlap at the user's location in the far field. Therefore, all OAM modes cannot be launched from the same circular array of antenna elements, but the lower-order ones should be launched from arrays having proportionally smaller radii and the higher-order ones from arrays having proportionally larger radii. These arrays may be concentrically stacked as shown for example in Figure 1a of [11], adapted below as Figure 5. Such a transmitting antenna arrangement is expected to be of the same physical size as in other, more \"conventional\" RF schemes for transmitting OAM beam and no attempt to improve the link SNR is inferred here; that is effected at the receiver--as will be shown in subsequent sections. Real-Time Implementation of Pseudo-Doppler Effect on Received OAM Beams by Dynamic Antenna Combining  
As noted in the Introduction, very few attempts at OAM multiplexing in electromagnetic-wave communication links did not rely on spatial rectification of the helical OAM phase fronts at the receiver, but a few should be mentioned before proceeding to describe the present scheme as being unique. Reference [12] even uses a time-based method for generating OAM modes in circular arrays, which saves some hardware but otherwise is not really necessary for overcoming difficulties in implementing an OAM communications link. A simple phase-gradient measurement is used in [13] to sequentially detect OAM modes, which are sequentially encoded with data symbols at the transmitter. The authors of [14] also use the phase gradient to resolve the OAM modes at the receiver, by switching between multiple pairs of receiving antenna elements. In [15], the authors use the phase gradient sensed by switching between two receiving elements to identify the transmitted OAM mode, which corresponds to an encoded data symbol. This generates harmonics of the switching frequency, which are used to detect the (sequentially) transmitted data symbol. No dynamic combining effects are employed, and the data symbols are transmitted and decoded sequentially. Another variation of the time-gated generation and detection of OAM modes appears in [16], using strategically-placed antenna elements covering only part of the circular aperture that an array such as the one in [11] would utilize.