Results and Discussions In a triple-carrier 128 GBaud polarization multiplexing hexadecimal quadrature amplitude modulation (PM-16QAM) FTN wavelength division multiplexing (WDM) system, a back-to-back and fiber transmission simulation platform is constructed to comprehensively investigate the performance of the proposed scheme.The simulation results demonstrate that the proposed scheme can effectively perform high-precision fine FOE based on the rough FOE results (Fig. 5). When the CZT search range is f1-100MHz-f1+100MHz, the FOE range of the proposed scheme is about -18-18 GHz, which means the estimation accuracy is significantly improved compared with the 1024-point 4th-FFT algorithm [Fig. 6(a)]. When the acceleration factors are 0.95, 0.90, and 0.85, the residual frequency offsets of the proposed method are about 2, 2.5, and 3 MHz, respectively [Fig. 6(b)], and the optical signal-to-noise ratio (OSNR) tolerances of the system using the proposed scheme are 23.5 dB, 23.8 dB, and 24.5 dB, respectively under the soft-decision forward error correction (SD-FEC) threshold at the bit error rate (BER) of 2×10^-2 (Fig. 7).The experimental results show that the absolute value of the maximum FOE error of the proposed method is about 2 MHz and 3 MHz when the acceleration factors are 1 and 0.9 [Fig. 9(a)]. Additionally, the proposed scheme can achieve a stable FOE in the -8-8 GHz frequency offset range, and the estimation accuracy is higher than that of the 512-point 4th-FFT algorithm [Fig. 9(b)]. When the acceleration factors are 1 and 0.9, the OSNR tolerance of the system using the proposed scheme is 19 dB and 20 dB respectively under the SD-FEC threshold at BER of 2×10^-2 (Fig. 10).Compared with the traditional blind FOE algorithm based on 4th-FFT, the overall computational complexity of the proposed scheme is significantly reduced, which is about 8% of the conventional algorithm, under similar estimation accuracy (Table 1).

In virtue of the computing power provided by advanced digital signal processing (DSP) technology, the faster-than-Nyquist (FTN) optical transmission technology can compensate for impairments, which has recently been considered a potential approach in the field of large-capacity coherent optical transmission. However, the severe inter-symbol interference (ISI) caused by tight FTN filtering will lead to the heavy deterioration of accuracy and stability in frequency offset estimation (FOE), which needs to be combated to guarantee the system performance. At present, the 4th fast Fourier transform (4th-FFT) algorithm is usually used for FOE in FTN coherent optical systems. To obtain a more ideal estimation accuracy, the 4th-FFT algorithm requires a large number of estimated samples to obtain a high-resolution FFT spectrum, which will bring greater computational complexity. Thus, the conventional FOE algorithm in the DSP module of the receiver is faced with conflicting effectiveness and complexity, which significantly degrades the stability and accuracy of FOE. Aiming at the above problems, this paper proposes a two-stage FOE algorithm based on phase difference of training sequence and chirp Z-transform (CZT) for dual-polarization (DP) 16QAM FTN-WDM systems.

This paper puts forward a two-stage FOE algorithm. In the first stage of the proposed scheme, the periodic multi-symbol structure of the training sequence is adopted to process multiple inter- and intra-period averaging to alleviate the influence of the noise on FOE, which can achieve stably rough FOE with low overhead. In the second stage, according to the rough frequency offset value f1 in the first stage, the CZT spectrum search range is set and the starting and ending frequency points (the lower and upper limit values of the FOE range) are determined. Specifically, with the rough frequency offset value f1 as the center point, the spectrum search of equal length is carried out to both sides respectively. According to the starting and ending frequency points and the preset frequency resolution (i.e., the accuracy of CZT-based FOE), the number of output sequence points of CZT is determined. Then, CZT is performed based on the number of the points on the received signal after the 4th power operation to obtain the CZT spectral function, and the maximum spectral line value of the CZT spectral function is extracted. Through the maximum spectral line value and the CZT spectrum search range, the fine frequency offset value f2 can be obtained. Thus, the high-precision FOE can be completed with low computational complexity while ensuring estimation accuracy.

Aiming at the requirement of the FTN system for low complexity, high precision, and high-reliability FOE algorithm, this paper proposes a two-stage FOE algorithm based on training sequence and CZT. The simulation results of the 128 GBaud PM-16QAM FTN-WDM system show that when the acceleration factors are 0.95, 0.90, and 0.85, the residual frequency offsets of the proposed scheme are about 2, 2.5, and 3 MHz in the carrier frequency offset range of -1.6-1.6 GHz. Under the typical 1 GHz frequency offset, the OSNR tolerances at BER of 2×10^-2 are 23.5 dB, 23.8 dB, and 24.5 dB respectively. The computational complexity of the proposed scheme is reduced by 92% compared with the 4th-FFT algorithm of the same precision. The 40 GBaud PM-16QAM FTN-WDM offline experimental results show that when the acceleration factor is 0.9, the absolute value of the maximum FOE error in the frequency offset range of -1.6-1.6 GHz is about 3 MHz. Thus, the excellent performance and outstanding advantages of the proposed scheme make it a preferable candidate for the FOE of DP-16QAM signal in practical FTN-WDM systems.