## 4 Time Gating and Demodulation Algorithms
### OAM Recovery Algorithm  
As evident from the above simulation experiments, the OAM signals are present in different proportions in the various harmonic spectral replicas (more accurately, \"spectral shifts\") of the composite received signal at the output of the gating subsystem. It is expected, and will indeed be shown, that these proportions are not random but fixed and deterministic, as are the relative amplitudes of the spectral replicas themselves (e.g., ranging from \(-60\) to \(-90\) dB in Figure 10). They are in fact determined by the physical parameters of the link, which can be made known to the receiver a-priori, thus enabling it to recover the OAM modes more effectively than was done in the simulation experiment. Specifically, several shifted spectra can be shifted to baseband and linearly combined so as to cause the amplitudes of the desired OAM mode to add and those of the undesired OAM modes to cancel coherently, using an LMS adaptive FIR (Finite Impulse Response) filter type of algorithm for each OAM mode. Subsequently, or as part of the LMS algorithm, the desired OAM mode is adjusted in amplitude and phase so its dynamic range matches that of the decision or demodulating subsystem, compensating for the dynamic range of the wireless link. It is essential to recognize that the gating pseudo-Doppler modulations of the composite received signal constitute a \"time-limited fractional frequency shift\" operation on it in the discrete frequency domain. This can subsequently be recognized as the dual of a \"frequency-limited fractional time shift\", or band-limited fractional delay operation on a signal in discrete time domain as detailed in [19]. Specifically, the gating frequency, (which is twice the pseudo-Doppler modulation frequency), \(2F\), and the fraction thereof comprising the OAM spectral shift, \(kd\)/(\(2R\))\(F\) as evident in (10), correspond to the sampling interval and fraction thereof, respectively, in the band-limited fractional-delay problem treated in [19]. This can be expected on the basis of the duality relations that exist between time and frequency domains due to properties of the Fourier transform and its inverse that relates them. The property that sampling in time-domain at intervals \(T\) causes periodic extensions in frequency-domain by 1/\(T\) also helps to explain the received spectra observed in the simulations. Appendix C derives the frequency-domain effect of the time-limited fractional frequency-shift and its direct implementation along the same lines of reasoning as [19] for the impulse response and direct implementation of band-limited fractional delay in the time domain. The former can then be applied to (10) to plot the envelope of its spectrum and reveal the fractional pseudo-Doppler frequency shifts of the OAM modes. It also serves as the basis for an algorithm to recover the individual OAM modes from the gated output of the pseudo-Doppler modulator, as will be shown in the sequel. In order to discern the spectral shift by the fractional pseudo-Doppler frequency, the Fourier transform of (10) is derived, in stages. First, (10) is affected by the time-gating and frequency-shift  
\[ \begin{array}{l}h_{k}(t)=-\cos\bigl{(}2\pi Ft+\frac{\pi}{4}\bigr{)}\sin\bigl{(}2 \pi Ft-\frac{\pi}{4}\bigr{)}\times\cdots\\ \cdots\times\int\limits_{-\infty}^{\infty}u_{k}(\tau)\sum\limits_{n=-\infty}^{ \infty}\delta\bigl{(}t-\frac{\mu\pi}{2\pi F}-\tau\bigr{)}d\tau\end{array}  \tag{21}\]  
Using a trigonometric identity for the top line of (21) with \(a/2=2\pi Ft\) and \(b/2=\pi/4\) in  
\[2\cos\biggl{(}\frac{a+b}{2}\biggr{)}\sin\biggl{(}\frac{a-b}{2}\biggr{)}=\sin(a )-\sin(b) \tag{22}\]  
simplifies it to  
\[ \begin{array}{l}h_{k}(t)=(-1/2)[\sin(2\pi 2 Ft)-1]\times\cdots\\ \cdots\times\int\limits_{-\infty}^{\infty}u_{k}(\tau)\sum\limits_{n=-\infty}^{ \infty}\delta\bigl{(}t-\frac{\mu\pi}{2\pi F}-\tau\bigr{)}d\tau\end{array}