# 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
The use of DSM means dividing a high baud rate single-carrier signal into several low-baud-rate subcarriers for transmission, and the generation process of the DSM signal is shown in Figure 9. The QAM symbol sequence is generated using the delayed de-correlation sequence, which is divided into \(M\) streams (\(M\) is the number of subcarriers). Each subcarrier signal is first up-sampled to two SPSs. After the signal is filtered through an RRC filter with a roll-off factor of 0.01, each subcarrier signal is up-sampled M times (essentially zero filling in the frequency domain) and shifted to corresponding intermediate frequencies (subcarrier positioning). The DSM signal is finally obtained by multiplexing all subcarrier signals. For single-carrier modulation, the QAM symbol sequence is generated using the delay decorrelation sequence. After the signal is up-sampled to two SPSs, the single-carrier signal is passed through an RRC filter with a roll-off factor of 0.01. In fact, the single-carrier signal can be regarded as a special DSM signal when \(M=1\). The specific value for reducing the complexity of each PR iteration using a DSM-enabled CF-PR receiver is as follows. The AIT-PR algorithm is implemented in a block-wise manner, with block sizes of \(N\) and \(N/M\) for single-carrier and DSM-enabled receivers, respectively. Assuming that the received signal after analog-to-digital converters (ADC) has two SPS, it requires \(6N\cdot\left[\log_{2}(N/M)+\log_{2}\bigl{(}N/M^{2}\bigr{)}+3\right]\) real multiplications and \(3N\cdot\left[\log_{2}(N/M)+\log_{2}\bigl{(}N/M^{2}\bigr{)}+2\right]\) real adder for each iteration. As shown in Figure 10, as the number of carriers increases, the number of multiplications and additions of CF-PR decreases accordingly. Therefore, the DSM-enabled CF-PR receiver can greatly reduce the algorithm complexity. By numerically retrieving the optical field information of 56 GBaud DSM signals after transmitting through a 80 km SSMF using QPSK/16 QAM/32 QAM modulation, it can be found that compared to traditional single-carrier receivers, the DSM-enabled CF-PR receiver has the advantage of reducing the complexity of the CF-PR iteration process. In addition, a DSM-enabled CF-PR receiver has been verified to be the most robust to transmission defects such as laser linewidth and wavelength drift at the transmitter side, time skew and amplitude imbalance between two intensity tributaries at the receiver side. Finally, the superiority of the DSM-enabled CF-PR receiver was experimentally verified by restoring the optical field information of the 25 GBaud 16 QAM signal after transmission over a 40 km SSMF. Figure 9: Generation of DSM signals. ### CF-PR Receiver with Polarization and Spatial Mode Diversity

#### 3.3.1 CF-PR Receiver with Polarization Mode Diversity

Polarization division multiplexing can double the SE in comparison with its single-polarization counterpart. Figure 11 shows the diagram of the receiver architecture for the recovery of a polarization multiplexed QAM signal with intensity measurements only. A PBS separates the polarization-mixed received signal after fiber transmission onto the two polarizations with respect to the PBS axis. Thereafter, the single-polarization CF-PR receiver can reconstruct the full field of the single-polarization QAM signal. To guarantee the full field of polarization multiplexed QAM signal with high accuracy, the estimation of the 2 \(\times\) 2 fiber transfer matrix is essential [43]. However, the acquisition of the fiber transfer matrix is challenging as the conventional polarization demultiplexing algorithm only works well when the full field of the QAM signals is accurately retrieved.