# 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
After \(N\) iteration, WD is carried out on the I/Q components of the complex signal, respectively. Afterward, the pilot constraint is applied to update the sampling phase at the predetermined position. After being up-sampled to two SPSs, the signal is then shaped by another RRC filter. After the signal is propagated to the Rx dispersed plane, its strength is constrained by the transformed intensity measurement. Finally, the updated signal is transmitted to the Rx undispersed plane, and the phase of the signal is preserved until the next iteration. ```
1. Initialize \(a(t),b(t),R,V,h_{CD},h_{D},M,N\)
2.\(\angle(t)=\angle rand(t)\) \(\Diamond\) Initialize phase
3.forfromfromfrom\(1\)to\(M\) \(a(t)\gets a(t)^{P},b(t)\gets b(t)^{P}\); \(\Diamond\) AIT \(\Diamond\) \(\Diamond\) \(\Diamond\) \(\Diamond\) \(\Diamond\) \(\Diamond\) \(\Diamond\) Reconstruct the filed
4.\(\hat{s}(t)=h_{CD}^{-1}(t)\otimes\hat{s}(t)\) \(\Diamond\) Propagate back to Tx
5.\(\hat{s}_{k}(t)\gets h_{RRC}(t)\otimes\hat{s}_{k}(t)\) \(\Diamond\) RRC shaping
6.\(\hat{s}_{k}(t^{\prime})\leftarrow\hat{s}_{k}(t)\) \(\Diamond\) Down-sample to 1 SPS
7.\(\hat{s}(t^{\prime})\leftarrow\frac{s(t^{\prime})}{\sqrt{\left(mean\left(\{s(t )^{\prime}\}^{2}\right)\right)}}\) \(\Diamond\) Normalization
8. if\(i>N\) \(\Diamond



be improved accordingly with the help of this optical carrier [69; 70], but this is out of the scope of this submission, as we focus on carrier-free phase retrieval. Leveraging Digital Up-Sampling

The CF-PR receiver can reconstruct the full field information of the signal using only the intensity measurements, as is shown in Figure 7. B represents the bandwidth of modulated signals. The bandwidth of the signal \(|s(t)|^{2}\), \(|d(t)|^{2}\) after PD detection becomes twice that of \(s(t)\), \(d(t)\). Additionally, the amplitude information of \(a(t)=|s(t)|\), \(b(t)=|d(t)|\) is calculated on the measured intensities by the square root operation, which is a nonlinear operation and broadens the signal spectrum. In addition, the filter roll-off factor used for RRC shaping filtering is non-zero. Therefore, capturing \(|s(t)|^{2}\) and \(|d(t)|^{2}\) without frequency clipping requires a sampling rate larger than two SPSs, otherwise the signal will be distorted. The above problems can be solved by increasing the sampling rate. An iterative scheme was proposed to average the interleaved up-sampling symbols to maintain two samples per symbol of the estimated complex signal to achieve faster algorithm convergence [71]. The principle of the scheme is shown in Figure 8, where \(T\) is the symbol duration of the signal. Signals \(a(t)\) and \(b(t)\) are up-sampled four times, and the sampled symbols are divided into even and odd subsequences \(a_{even}(t)\), \(b_{even}(t)\), \(a_{odd}(t)\), \(b_{odd}(t)\). On both projection planes, \(\overset{\wedge}{S}(t)\) is duplicated and intensity constraints are applied in an interleaved fashion in order to estimate \(\overset{\wedge}{S}(t)\) with two SPSs per iteration. Finally, under the condition that OSNR is 30 dB and RRC filter roll-off factor is 0.01, the optical field recovery of the 30 GBaud QPSK signal after transmission over a 55 km SSMF is completed, and the CF-PR receiver using interleaved up-sampling symbols needs only 110 and 250 iterations to reach the 20% and 7% FEC thresholds, respectively. ### Reduce the Complexity of Each PR Iteration

CD compensation (CDC) and simulation (CDE) will introduce high computational complexity for each PR iteration. Compared with traditional single-carrier receivers,

Figure 8: CF-PR receiver using interleaved up-sampling symbols. Figure 7: (**a**) Concept of the CF-PR process. (**b**) Bandwidth of signal \(s(t)\) and its intensity \(|s(t)|^{2}\). a digital subcarrier multiplexing (DSM)-enabled CF-PR is proposed and beneficial for reducing the implementation complexity of the CF-PR process [72], as the lower symbol rate of each subcarrier helps reduce the implementation complexity of multiple CDCs and CDEs during PR iteration [73; 74].