# Research Progress on Carrier-Free Phase-Retrieval Receivers
## 3 Modified CF-PR Receiver Scheme
### Improve the Performance of the CF-PR with a Reduced Number of Iterations and Better Steady Accuracy
#### 3.1.1 Parallel Alternating Projections  
The schematic diagram of the CF-PR receiver with parallel alternating projections [45] is shown in Figure 2a. This receiver scheme consists of multiple dispersive elements and PDs to increase the number of intensity measurements. Generally, each dispersive element owes a different amount of dispersion, which changes the received signal strength through its own linear model of dispersion. Figure 2b shows the schematic diagram of the algorithm flow with \(M\) iterations in each dispersion plane. Assuming that there are \(N\) dispersion planes in total, the total number of iterations is thus \(N\times M\). At each stage, the MGS algorithm iteratively refines the estimation of the complex-valued \(s(t)\) and its dispersed counterpart \(d_{i}(t)\), and finally recovers the optical field information of the signal by exploiting the intensity constraints. Because it provides more complementary intensity constraints, the parallel alternating projection scheme can improve the PR performance of the CF-PR receiver.  
The parallel alternating projection scheme computes the averaged amplitude error between \(a(t)\) and \(s(t)\) after each iteration, i.e., \(A_{err}(t)=\overline{\left|a(t)-\left|s(t)\right|^{2}\right|}\) as a metric to evaluate the phase recovery error accuracy. When the algorithm completes the iteration of one dispersion plane and moves to the next one, the \(A_{err}(t)\) will first show a peak and then fall sharply, which proves that the parallel alternating projection helps the PR algorithm solve the local minimum issue. Compared with a single alternative projection  
Figure 1: (**a**) Schematic of the transmission system using the CF-PR receiver; (**b**) the relationship of the signals at different planes.  
scheme, the parallel alternating projection scheme has faster convergence speed and higher phase recovery accuracy. We should also mention that the use of more dispersion planes is beneficial for improving the PR performance, but at the cost of more hardware complexity. Experiments have shown that in the case where the dispersion element is composed of 25, 35, and 55 km SSMFs, and the 30 Gbaud quadrature phase shift-keying (QPSK) signal is transmitted through a 55 km SSMF. After 2000 iterations, the bit-error ratio (BER) of the CF-PR receiver is stabilized below \(1\times 10^{-2}\) when the number of dispersion planes is three. When the number of dispersion planes is one, the BER curve of the CF-PR receiver cannot converge [45].  
It is worth mentioning that the space-time (ST) diversity scheme is similar to the parallel alternate projection scheme in that time decorrelation and spatial scrambling are employed to decorrelate the signals [62]. The specific implementation process is shown in Figure 3. In this scheme, the complex-valued signal is first split into two tributaries. One tributary is sent into a dispersive element. The other tributary is split equally into N copies, and the N redundant copies are temporally decorrelated by the relative time delays. The N copies are then mixed spatially using a symmetric N-by-N optical coupler. This scheme proves that the utilization of more ST diversity channels increases the convergence speed of the CF-PR algorithm and the receiver tolerance toward the receiver, but at the sacrifice of more hardware complexity.  
Figure 3: Diagram of **(a)** a CF-PR receiver with ST diversity; **(b)** field recovery process.  
Figure 2: **(a)** Illustration of a phase-retrieval receiver employing \(N\) parallel alternating projections; **(b)** schematic of the signal processing flow running the modified GS algorithm over multiple dispersed planes sequentially.