# 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
#### 3.1.2 Enhanced CF-PR Algorithms  
CF-PR Scheme with Selective Phase Resetting (PR-PR)  
The CF-PR scheme with parallel alternating projections improves the PR performance by adding more hardware such as more PDs, which would significantly increase the implementation cost. In contrast, the PR-PR scheme [47], which uses only one dispersive projection, and adopts the MGS algorithm with a selective phase reset, can ensure a faster and accurate phase retrieval performance, as shown in Figure 4. After \(k\) iterations, the 10-symbol moving average of \(A_{err}(t)\), which is represented by \(A_{err}^{MV}(t)\), is calculated. If \(A_{err}^{MV}(t)>\varepsilon\), the phase reset procedure is applied to reinitialize the phase information of the signal. The parameter \(\varepsilon\) is the acceptable GS error level. The use of \(A_{err}^{MV}(t)\) helps eliminate error spikes that disturb with convergence, thus avoiding the local minima. Experiment results have shown that after 30 Gbaud, QPSK signals are transmitted through a 55 km SSMF and the PR-PR receiver can improve the convergence speed using a phase reset by adaptively updating the acceptable GS error level. Consequently, the PR-PR scheme can realize fast algorithm convergence with less than 200 iterations. Joint Optimization of CF-PR and Forward Error Correction (FEC) Overheads  
Generally speaking, the use of pilot symbols is essential for the PR process to impose additional pilot constraint. The more pilot symbols we utilize, the more accurate the retrieved phase we can obtain when enough iterations for the CF-PR algorithm are used to guarantee the convergency. However, using more pilot symbols could have worse performance than the case with less pilot symbols when not having enough iterations, considering limited complexity and a fixed FEC overhead [63]. It has been reported that the joint optimization of the CF-PR and FEC overheads can enable a higher data rate and significantly reduced number of PR iterations [64]. Figure 5 shows the distribution of CF-PR pilot symbols and FEC overhead. The M here represents the number of bits for the insertion of pilot symbols for the CF-PR and is related to the modulation format. The overall overhead (OH) can be expressed as \(OH=1/R-1\), where \(R\) is the overall rate as follows:  
\[R=\frac{1}{1+OH_{FEC}}\cdot\frac{1}{1+OH_{PR}}=\frac{K}{N-M}\cdot\frac{N-M}{N}= \frac{K}{N} \tag{6}\]  
where \(M=\frac{OH_{PR}}{1+OH_{PR}}R_{S}\log_{2}m\), \(R_{S}\) is the symbol rate, and \(m\) is the cardinality size of signal constellation. A larger CF-PR overhead indicates a stronger phase information acquisition capability, which ensures the CF-PR algorithm converge fasters and the bit-error rate (BER) after convergence is lower. On the other hand, in order to maintain the same net data rate, an increase in CF-PR pilot overhead also implies a decrease in the FEC overhead, leading to reduced error correction capability. Therefore, at a fixed data rate, overhead allocation between CF-PR and FEC becomes critical to optimize performance. The CF-PR and FEC overhead joint optimization scheme is carried out by optimizing the combination of FEC-OH, PR iteration, and PR pilot OH, at a total OH of 44%. Under the condition  
Figure 4: Schematic representation of the PR-PR process. of BER \(<1\times 10^{-6}\), the optical field recovery of 30 GBaud QPSK signal after 55 km SSMF transmission is completed, and the total number of PR iterations can be reduced by 66% [64]. CF-PR Scheme with Adaptive Intensity Transformation (AIT-PR)  
In order to further improve the phase retrieval performance of the PR-PR receiver, an AIT-PR scheme is proposed by imposing an AIT constraint [65,66]. The principle of the AIT-PR receiver is shown in Figure 6. The two measured intensities are transformed adaptively with the current number of iterations. Regarding the QPSK signal, the constant modulus constraint is also included so that the AIT-PR algorithm can converge faster and more accurately. The AIT operation can be generally expressed as  
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