## 4 Time Gating and Demodulation Algorithms
### OAM Recovery Algorithm
 \tag{23}\]  
Now, with the help of (23), the final output spectrum (30) can be expressed as  
\[\boxed{Z_{G,k}(f)=\Pi_{CS}(f)*\left[U_{k}(f)\sum\limits_{m=-\infty}^{\infty}S_ {k}(f-m2F)\right]} \tag{24}\]  
with the understanding that  
\[ \begin{array}{l}\Pi_{CS}(f)=\Bigl{(}\frac{-1}{2}\Bigr{)}\int\limits_{-\infty }^{\infty}[\sin(2\pi 2 Ft)-1]e^{-j2\pi/t}dt=\cdots\\ \cdots=\Bigl{(}\frac{1}{2}\Bigr{)}\int\limits_{-\infty}^{\infty}\Bigl{[}e^{-j2 \pi/t}-\frac{e^{j-2\pi(f-2 Ft)}-e^{-j2\pi(f+2 Ft)}}{j2}\Bigr{]}dt=\cdots\\ \cdots=\delta(f)/2+j\delta(f-2F)/4-j\delta(f+2F)/4\end{array}  \tag{25}\]  
This means that the spectrum of the gated output is a periodic extension of the transmitted spectrum of the signal with repetition interval equal to twice the pseudo-Doppler modulation frequency, \(2F\), multiplied by the spectral envelope \(U_{k}(f)\). The next step is to evaluate \(U_{k}(f)\), which is the envelope of the spectrum, and manifests the fractional pseudo-Doppler frequency shift expected according to the order, \(k\), of the OAM mode. This shift is traceable to the complex \"pulse shape\" function \(u_{k}(t)\) defined in (15). Then the convolution with (25) is performed at the end. The spectrum of the envelope is  
\[U_{k}(f)=\int\limits_{\frac{-\pi/2}{2\pi F}}^{0}e^{j2\pi(\frac{ik}{2R})Ft}e^{- j2\pi/t}dt=\int\limits_{\frac{-1}{4F}}^{0}e^{-j2\pi(f-\mu_{k})t}dt \tag{26}\]  
where the fractional pseudo-Doppler shift is \(\mu_{k}=F(kd/(2R))\) in accordance with Appendix C. It is straight-forward but tedious to evaluate, producing  
\[U_{k}(f)=\frac{e^{j\frac{\pi(f-\mu_{k})}{4F}}}{4F}\left[\frac{\sin\Bigl{(} \frac{\pi(f-\mu_{k})}{4F}\Bigr{)}}{\Bigl{(}\frac{\pi(f-\mu_{k})}{4F}\Bigr{)} }\right] \tag{27}\]  
Then according to (24), the convolution of (27) with (25) gives the complete spectral envelope  
\[ \begin{array}{l}U_{k,CS}(f)=\Pi_{CS}(f)*U_{k}(f)=\cdots\\ \cdots=\frac{1}{2}U_{k}(f)+\frac{i}{4}[U_{k}(f-2F)-U_{k}(f+2F)]\end{array}  \tag{28}\]Therefore, a sufficiently representative expression for the final output spectrum can be obtained by using (28) in (24) to obtain the output spectrum  
\[Z_{G,k,1}(f)=U_{k,CS}(f)\sum_{m=-\infty}^{\infty}S_{k}(f-m2F) \tag{29}\]  
It remains to evaluate (28) and plot its amplitude, so as to visualize the fractional pseudo-Doppler frequency shift at the output of this receiving subsystem. The magnitude of (28) is plotted in the subsequent Figure for the parameters used in the simulation with \(k=8\) in \(\mu_{k}=F(kd/(2R))\). What is actually plotted is the amplitude  
\[ \begin{array}{l}E_{k}(f)=\left(\frac{1}{2}\right)Sinc\left(\frac{f-\mu_{k}} {4F}\right)+\cdots\\ \cdots+\left(\frac{1}{4}\right)Sinc\left(\frac{f-2F-\mu_{k}}{4F}\right)+ \cdots\\ \cdots+\left(\frac{1}{4}\right)Sinc\left(\frac{f+2F-\mu_{k}}{4F}\right)\end{array}  \tag{30}\]  
with the common factor  
\[P_{k}(f)=\frac{1}{4F}e^{j\frac{\pi(f-\mu_{k})}{4F}} \tag{31}\]  
being omitted for clarity. The spectral replica according to (29) are superposed at intervals of \(2F\). Equation (29) is representative of the final output of the gating subsystem of the pseudo-Doppler modulated OAM receiver. Clearly, the spectral replica are spaced at twice the pseudo-Doppler frequency, \(2F\), while the peak of the envelope is at the fractional pseudo-Doppler frequency, \(\mu_{k}=F(kd/(2R))\), as expected and as observed in the simulation results. More accurate representation of the spectral envelope can be obtained by including higher harmonics of \(F\) when simplifying (16). Now that it has been established that the gated output (29) consists of different proportions of spectral replica of each OAM mode, an algorithm for recovering them may be proposed. First it will be necessary to truncate the series of spectral replicas to a minimum equal to the number of OAM modes to be recovered, because at least that many different linear combinations of them will be needed.