Blind phase search algorithm based on threshold simplification

A simplified scheme for the blind phase search (BPS) algorithm is proposed. By comparing the loss function with a threshold value, the number of test phases is reduced. Simulation and experimental results show that compared with the traditional BPS algorithm, this algorithm only needs half of the test phases under the same performance.


Introduction
Phase noise seriously affects the performance of high-order QAM coherent optical communication systems and laser linewidth is the main source of phase noise. In optical transmission, laser phase noise is usually described as the Wiener process (Foschini 1988). Optical carrier phase recovery (CPR) is an important technology in coherent optical transmission systems. Many CPR algorithms have been proven to compensate for laser phase noise in digital coherent receivers (Kikuchi 2006;Zhou 2010, 4). In the QPSK modulation format, the Viterbi-Viterbi algorithm (Bosco 2018) is usually used for carrier phase recovery. In the high-order QAM modulation format, a combination of QPSK partition and Viterbi-Viterbi algorithm (VVPE) is used (Fatadin et al. 2010). Another commonly used phase recovery algorithm for high-order modulation formats is the blind phase search algorithm (Pfau et al. 2009). The accuracy and linewidth tolerance of the blind phase recovery algorithm is very high, but the cost is very high in complexity. Various approaches have been proposed to reduce the complexity of BPS (Tang et al. 2019;Lu et al. 2016;Sun et al. 2014;Shi et al. 2022;Rozental et al. 2017). Tang et al. (2019), Lu et al. (2016), Shi et al. (2022) reduce the algorithm's overall complexity by reducing each stage's computational complexity. However, the number of test phases has not decreased. Sun et al. (2014) uses interpolation to reduce the number of test stages and thus reduce the complexity of 16QAM (Hereafter referred to as ECOC-BPS). However, the algorithm in ref. 10 has a lower effect on the advanced QAM format than 16QAM. This paper proposes to reduce the number of test phases of the algorithm by comparing the phase compensation loss function with a threshold (Hereafter referred to as T-BPS). Compared with the traditional BPS algorithm, the number of test phases of the proposed algorithm is reduced by half.

Experiment and simulation setup
A dual-polarization 64QAM simulation system was built with a symbol rate of 83 Gbaud. The power of the transmitting laser and the local oscillator laser is 6 dBm and 16 dBm, respectively, and the linewidth is set to 100 kHz. PM-64QAM transmission system adopts an advanced vector modulation format and DSP technology. The coherent optical communication system is shown in Fig. 1, in which the physical device is connected with the corresponding simulation function through the dotted line. In the simulation system, the bandwidths of the Digital-to-Analog Converter, Driver, Mach-Zehnder modulator, optical front-end (Optical Hybrid + Photo-Diode + Trans-Impedance Amplifier) and Analog-to-Digital Converter are set to 35 GHz, 40 GHz, 40 GHz, 40 GHz and 35 GHz (Rafique et al. 2017). Pre-emphasis technology is used to compensate for the device bandwidth to make the system work normally with limited bandwidth.
An experimental system was built in which the arbitrary waveform generator is KEY-SIGHT M9502A. Due to the limitation of experimental equipment, the baud rate of the system is set to 10 GBaud. Therefore, the experimental results are only used to verify the feasibility of the scheme without performance analysis. The emitting laser is an external cavity laser with a linewidth of less than 100 kHz and an operating wavelength of 1550 nm. The sampling rate of the DAC is 65 GS/s and the bandwidth is 20 GHz. The system adopts an IQ modulator from coherent solutions. The channel Gaussian white noise is generated by ASE. The sampling rate of ADC is 80 GS/s, and the bandwidth is 36 GHz. The receiver used in the system comes from Lecroy. Figure 1 clearly shows polarization and phase diversity architecture. By CW, two optical 90 degrees hybrids to down convert the modulated optical signal output from the channel. Four couple balanced photoelectric detectors (BPD) achieve photoelectric conversion.

Principle of optimized algorithm
We assume that the received signal undergoes perfect timing synchronization and frequency offset compensation, and there are no other channel dispersion effects that cause inter-symbol interference (Li et al. 2019). Therefore, the received signal is only interfered by additive white gaussian noise (AWGN) and carrier phase noise (such as laser phase noise in optical communication systems). Therefore, the received signal can be expressed as: where A(k) is the k-th complex M-QAM symbol transmitted. A(k) = I(k) + jQ(k) , the value range of I(k) and Q(k) are ±1 , ±3 , … , √ M − 1 .The n(k) is a series of independent and identically distributed complex Gaussian random variables. The mean of each variable is zero and the variance is 2 n . (k) represents the carrier phase, which is usually simulated as a Wiener process in coherent optical communication.It can be derived from the following equation: (k) is a series of independent and identically distributed Gaussian random variables with a mean value of zero and a variance of : T s and Δ indicate the symbol duration of the transmitter and local oscillator laser and the total 3 dB laser linewidth. Figure 2 shows the block diagram of the proposed BPS carrier recovery algorithm. The digitized signal (one sample per symbol) entering the carrier phase recovery module is expressed as X k . The received samples X k are rotated by a number of M uniformly distributed (1) r(k) = A(k) exp[j (k)] + n(k).
(2) (k) − (k − 1) = (k). Then all rotated symbols are fed into a decision circuit and the squared distance to the closest constellation point is calculated. In order to eliminate the influence of additive noise, the distances of 2N+1 consecutive test symbols rotated by the same carrier phase angle are added together. In this paper, the N of the experiment and simulation is set to 20.
When the remaining parameters of the BPS algorithm are the same, Fig. 3 shows the loss function when the number of test phases is 16 and 32. Use M,J to represent the phase  Figure 3a shows 16,1 = 32,1 . Figure 3b shows 32,1 = ( 16,1 + 16,2 )∕2 . Figure 3 shows that when the two minimum values of the loss function are closest, the optimal test phase is actually the average of the two test phases. The difference between the minimum value of the error function and the second minimum value is denoted as e. We use the threshold T to describe the distance between the minimum and sub-minimum of the error function. When the test phase corresponding to the minimum value and subminimum value of the loss function is not adjacent, the optimal phase corresponds to the loss function's minimum value.. Then the optimal phase is calculated as follows: Figure 4 shows the difference between the second smallest value and the smallest value of the loss function. The blue point indicates 16,1 = 32,1 . The red point indicates 32,1 = ( 16,1 + 16,2 )∕2 . As can be seen from Fig. 4, the real threshold should be a curve. But our goal is to simplify the process of calculating the optimal phase. Therefore, we use a fixed threshold, so we choose the threshold that separates the red dot from the blue dot as much as possible according to the statistical characteristics. The cutoff value between the red and blue dots is about 0.15. Therefore, 32,1 can be calculated by finding the loss function when the number of the test phase is 16.
In equation 6, the optimal phase is one of the 32 test phases. Therefore, the possible value of the optimal phase can also be calculated in advance. A total of 15 comparators are required to find sub-minimum values and compare threshold values. The ECOC-BPS algorithm requires 5 adders and 2 multipliers. The power consumption of DSP is mainly reflected in the multiplier (Desset and Fort 2003). Therefore, the proposed algorithm saves the number of multipliers and has lower complexity than ECOC-BPS.  Figure 5c, d are the constellation diagrams after compensation using traditional BPS and the T-BPS algorithm proposed , respectively. The number of test phases for traditional BPS is 32, and the number of test phases for T-BPS is 16. The number of test phases of the proposed algorithm is half that of the traditional BPS algorithm.
We compared the performance of T-BPS, ECOC-BPS and traditional BPS. The number of test phases of T-BPS and ECOC-BPS is 16. The number of traditional BPS test phases is 32 and 64. Figure 6 also included a curve showing the lowest BER theoretically without phase noise. Figure 6 shows that the T-BPS algorithm performs better than ECOC-BPS at the FEC threshold. As mentioned above, the T-BPS algorithm reduces the number of test phases by half.