# 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
\begin{array}{l}a(t)\gets a(t)^{P}\,b(t)\gets b(t)^{P}\\ P=(R-1)\cdot\exp(-i/V)+1\end{array}  \tag{7}\]  
Specifically, the measured signal strength is powered by the factor \(P\), and \(P\) is adaptively adjusted at each iteration. \(P\) is determined by the maximum power factor \(R\) and attenuation factor \(V\), which need optimization to ensure the best PR performance in terms of convergency speed and steady BER performance. The AIT operation can be efficiently implemented using a look-up table (LUT). The flow of the AIT-PR algorithm is shown in Algorithm 1. By utilizing intensity transformation, the PR algorithm can avoid falling into suboptimal solutions, leading to a better convergence performance. By performing CD compensation on the reconstructed signal, it propagates back to Tx. The reconstructed signal is then down-sampled to one sample per symbol (SPS) after being shaped by a root-raised cosine (RRC) filter, and pilot constrained to update the sample phase at the pre-determined locations. Due to the constant modulus of the QPSK format, another modulus constraint is applied to force the signal strength to be constant. After being up-sampled by two SPSs, the signal is shaped by another RRC filter. After the signal propagates to the Rx dispersed plane, its strength is constrained by the transformation. Finally, the updated signal is transmitted to the Rx undispersed plane, and the phase of the signal is preserved until the next iteration. Figure 5: CF-PR pilots overhead vs. FEC overhead. Figure 6: Illustration of (**a**) the carrier-less PR receiver; (**b**) the intensity transformation process. It has been verified that the AIT-PR receiver scheme only requires 50 iterations to numerically reconstruct the 56 GBaud QPSK signal after transmitting through an 80 km standard single-mode fiber. And in terms of 20% soft-decision forward error correction (SD-FEC) and 7% hard-decision forward error correction (HD-FEC), the required optical signal-to-noise ratio (OSNR) of the AIT-PR receiver is reduced by 1.95 and 1.89 dB when compared to the PR-PR receiver, respectively. Weighted Decision-Enhanced PR Receiver with Adaptive Intensity Transformation (WD-AIT-PR)  
Hard decisions (HD) can be employed before the pilot symbol insertion to enhance the PR performance [55]. On the other hand, the HD operation can only be applied after a large number of iterations with better pre-convergency to guarantee the correct decisions. In addition, HD-induced errors inevitably limit the PR performance. In this context, the WD-AIT-PR scheme is proposed to improve the PR performance [67; 68]. By providing weighted decision feedback, this scheme increases the \"pilot\" proportion without additional occupation of spectrum efficiency. WD is carried out separately for in-phase and quadrature (I/Q) components of the complex QAM signal. The principle of WD can be expressed as  
\[\widehat{S^{l}}(t)=\Re\Big{(}\widehat{S}(t)\Big{)},\widehat{S^{ Q}}(t)=\Im\Big{(}\widehat{S}(t)\Big{)}\] (8) \[\widehat{S^{l}_{WD}}(t)=\widehat{S^{l}_{WD}}(t)+f\big{(}\gamma^{ L}(t)\cdot E^{l}(t)\big{)},E^{l}(t)=\widehat{S  
The use of reliability can minimize the impact of decision errors. In addition, \(f(\cdot)\) is a compressed sigmoid function to increase the effect of reliability value when it is a high value, as well as decrease this effect when it is a low value, as shown below.  
\[f(x)=\frac{1}{2}\bigg{(}\frac{1-\exp[-\alpha(x/\beta-1)]}{1+\exp[-\alpha(x/ \beta-1)]}+1\bigg{)} \tag{10}\]  
where \(\alpha\) is an integer and \(\beta\) is a compression factor in the range of \([0,1]\). These two parameters need to be optimized to ensure the best performance of the algorithm. When the condition of \(f(x)=1\) is satisfied, WD can be regarded as a hard decision. The algorithm flow of a WD-AIT-PR receiver is shown in Algorithm 2. We first perform CD compensation on the signal after intensity transformation, and in this way, the reconstructed signal is considered as being transmitted back to Tx. After RRC filtering, the signal is down-sampled to one SPS and normalized.