## 4 Time Gating and Demodulation Algorithms
### Simulation of Basic Pseudo-Doppler Modulator and Gating Arrangement  
A preliminary numerical simulation to verify the proposed concept was conducted using MATLAB(r) R2012a (7.14.0.739) and Simulink(r) R2012a (7.9), with Communications System Toolbox (5.2), DSP System Toolbox (9.3) and Signal Processing Toolbox (6.17). For reference, the transmitted constellation from one of the sources is observed via a matched pulse-shaping filter, as shown in Figure 9a. Its spectrum is also observed; in fact, the spectra of all the sources are the same on average and are shown in Figure 9b as seen when superposed at one receive antenna (They would appear the same on average at the other receive antenna.) So, all the OAM modes (OAM 1 and OAM 8 in this case) occupy the same spectrum shown in Figure 9b, which is the only spectrum visible in the transmission medium and potentially subject to regulation. Yet, it will be shown that the OAM modes can be recovered separately from this composite signal in the receiver. Figure 9: (a), Constellation and (b) spectrum of transmitted signal on all OAM modes (OAM1 + OAM8). After modulation by the pseudo-Doppler waveforms, time-gating and down-conversion, the constellation and spectrum of the composite received signal appears as in Figure 10a,b respectively. The down-conversion frequency is 0 Hz in this case, but in general it will be some multiple of 2F, where \(F\) is the pseudo-Doppler modulation frequency, because the gating pulses occur 2 times per cycle of \(F\) and they have harmonics. Note that the spectrum of the receiver output composite signal consists of many harmonics of twice the pseudo-Doppler modulating frequency imposed at the receiver front end and is generally not symmetric about 0 Hz. Also note that the constellation does not look recoverable. The constellation is always obtained from the spectral replica positioned at 0 Hz after the down-converter. It is interesting to observe the received constellation and spectrum of each OAM mode separately. This is shown, still with 0 Hz frequency shift in the down-converter, in Figure 11 below. Note that, with a frequency shift of 0 Hz in the down-converter, OAM8 dominates over OAM1. With suitable scaling by a complex coefficient (amplitude and phase change), OAM8 could be recovered even in the presence of OAM 1, and after suitable conventional equalization and decoding, its QAM data symbols successfully demodulated. Figure 10: (**a**) Constellation, and (**b**) spectrum of the composite received signal of all OAM modes before demodulator (shifted by 0 Hz in this case). With a different frequency shift applied at the down-converter, other OAM modes can be recovered. For example, it turns out in this case that with a shift of 2F, i.e., twice the pseudo-Doppler modulation frequency, the complementary situation arises, as shown in Figure 12.  
Figure 11: Received constellations and spectra of OAM modes transmitted separately and 0 Hz down-conversion: (**a**) constellation of OAM 8, (**b**) spectrum of OAM8, (**c**) constellation of OAM1, (**d**) spectrum of OAM1. Figure 12: Separate OAM modes with shift by 2F: (**a**) constellation of OAM8, (**b**) spectrum of OAM8, (**c**) constellation of OAM1, (**d**) spectrum of OAM1. Now with frequency shift of \(2F\) in the down-converter, OAM1 dominates over OAM8, so OAM1 could be similarly recovered and demodulated in the presence of OAM8. Without any other signal processing, each OAM mode was recovered from the composite signal with an uncoded error rate in the order of BER \(\approx 10^{-1}\). These examples were chosen because the differences in OAM proportions happen to be very obvious, but this is not always the case. Moreover, the expected spectral shifts by fractions of the pseudo-Doppler modulation frequency appear to be absent in all of the output spectra. Were the above simulation results a lucky coincidence and did they disprove the theory of the pseudo-Doppler frequency shifting of the OAM modes? It will be shown in the next subsection that this is not the case.