## 4 Time Gating and Demodulation Algorithms
### OAM Recovery Algorithm
Each linear combination of spectral replicas is determined by the spectral envelope with its unique fractional pseudo-Doppler shift as in Figure 13. An arrangement such as in Figure A6 can be used to combine all the replicas in such a way that only those of the desired OAM mode will add up to a non-zero complex amplitude while those of all other OAM modes will cancel to zero, much like in adaptive-array signal-processing which nulls interfering sources' signals. The coefficients should be low-pass so only the baseband spectral replicas are passed, as in down-conversion. The OAM recovery process can be understood in terms of using (29) truncated to \(M=K\) terms and transformed into time-domain, as \(x(t)\) in (A14), where \(K\) is at least the number of OAM modes. The coefficients of the down-converted signals at each stage will be different than in (A14); they will now be derived jointly for all \(K\) of the OAM modes. Note that each stage has as input to its coefficient one of the spectral replicas (index \"\(m\)\" in (29)) of all the OAM modes superposed with their amplitudes as received upon. The stage inputs to the coefficients can be arranged in an \(Mx1\) vector whose \(m\)-th row entry is  
\[x_{m}(t)=\sum_{k=1}^{K}U_{k,m}\underbrace{a_{k}s_{k}(t)e^{-j2\pi(m2F)t}}_{ Lowpass} \tag{32}\]  
and \(U_{k,m}\) is the spectral envelope coefficient for the \(k\)-th OAM mode at the \(m\)-th spectral replica (i.e., stage coefficient). The lowpass signal at each stage is the same (because the spectral replicas are in fact replicas of the same signal, composed of all the OAM modes in their arrival proportions \(\{a_{k}\}\)) so in vector-matrix form, (32) can be written as  
\[\left[\mathbf{U}_{K\times M}^{H}\mathbf{U}_{M\times K}\right]^{-1} \mathbf{U}_{K\times M}^{H}\mathbf{X}_{M\times 1}=\cdots\] \[\cdots=\left[\mathbf{U}_{K\times M}^{H}\mathbf{U}_{M\times K} \right]^{-1}\mathbf{U}_{K\times M}^{H}\mathbf{U}_{M\times K}\mathbf{S}_{K\times K }\mathbf{A}_{K\times 1}=\cdots \tag{37}\] \[\cdots=\mathbf{S}_{K\times K}\mathbf{A}_{K\times 1}\]  
Denoting the pseudo-inverse of matrix \(\mathbf{U}\), which is found on the LHS of (37), as \(\mathbf{U}^{\#}\) i.e.,  
\[\mathbf{U}_{K\times M}^{\#}=\left[\mathbf{U}_{K\times M}^{H}\mathbf{U}_{M \times K}\right]^{-1}\mathbf{U}_{K\times M}^{H} \tag{38}\]  
allows one to write (37) as is commonly done in least-squares optimization problems  
\[\mathbf{U}_{K\times M}^{\#}\mathbf{X}_{M\times 1}=\mathbf{S}_{K\times K} \mathbf{A}_{K\times 1} \tag{39}\]  
so by correspondence with (35), the joint OAM recovery coefficients matrix is  
\[\mathbf{C}_{K\times M}=\mathbf{U}_{K\times M}^{\#} \tag{40}\]  
Therefore, the recovered OAM modes are obtained from the input vector \(\mathbf{X}\) simply as  
\[\mathbf{Y}_{K\times 1}=\mathbf{U}_{K\times M}^{\#}\mathbf{X}_{M\times 1}= \mathbf{S}_{K\times K}\mathbf{A}_{K\times 1} \tag{41}\]  
which can be subsequently equalized and demodulated as in the \"back end\" (or DSP baseband section) of a \"conventional\" MIMO digital radio receiver. Although the matrix inverse found within the pseudo-inverse in (38) is not guaranteed to exist, it is more likely the more spectral replica are included, i.e., the larger the \(M\) is. It is dependent on the differences among the spectral envelopes of each of the OAM modes, which become more apparent as more spectral replica are included with increasing \(M\). Increasing \(M\) beyond \(K\) does not increase the dimensions of \(\mathbf{U}^{H}\mathbf{U}\), but can improve its condition number, thus enhancing its invertibility. In other words, including more spectral components beyond the minimum number \"\(K\)\" can lead to better least-squares estimates of the OAM signals, with smaller error-vector magnitudes (EVMs) due to crosstalk. Note also that this OAM recovery algorithm is deterministic because all the information contained in matrix \(\mathbf{U}\) is known at the receiver, in principle.