Kerr nonlinearity of optical fibers is considered as one of the main factors limiting the capacity enhancement of long-haul optical fiber communications [1]. Some researchers have proposed to utilize the nonlinear Fourier transform (NFT) to establish the mapping relationship between time domain (TD) signal and nonlinear Fourier spectrum (NFS) [2], [3], [4]. This breakthrough theory overcomes the Kerr nonlinearity by encoding and multiplexing information on NFS. It converts the propagation of TD signals following the nonlinear Schrödinger equation (NLSE) into an independent linear propagation of NFS in the fiber channel [5]. Nowadays, nonlinear frequency division multiplexing (NFDM) transmission has been considered as a promising technique to overcome Kerr nonlinear limitations [6].

Nowadays, according to the different NFS modulation regions used, the current studies on NFDM system can be divided into three categories, namely discrete spectrum (DS) modulation [7], continuous spectrum (CS) modulation [8] and full spectrum modulation [9]. The proposed representative techniques include modulation of nonlinear b-coefficients [10], constellation shaping techniques [11], [12], novel signal detection designs [13], multi-symbol coding processing method [14], and neural network-based receivers which can reduce the effects of noise [15], [16], [17], etc. Even so, the NFDM transmission still faces many challenges, such as amplifier spontaneous emission (ASE) noise in fiber link [11], processing noise from NFT calculations [18], frequency deviation, and laser phase noise [19], etc. Among them, one conundrum that needs to be solved urgently is the center frequency offset (FO) problem between the transmitter laser and the local oscillator (LO) laser. It causes inaccurate locations of the received eigenvalues and miscalculation of the corresponding NFS, which has been deemed as a non-negligible impairment in NFDM systems.

Concretely, for the FO problem of NFDM systems, current proposed schemes can be roughly classified into two main categories. The first option is to minimize the FO effect as much as possible. For example, some existing works have considered a use of self-coherent configuration to avoid laser FO [11], [20], [21], [22], whereas after the power of homologous laser is evenly divided, the sensitivity of coherent reception can be significantly reduced. Thus, this type of solutions can only be applied in laboratory or short distance scenarios, and it is not suitable for practical long-distance transmission. The second category is to perform frequency offset estimation (FOE) in linear Fourier domain (LFD) or nonlinear Fourier domain (NFD). Concretely, the LFD schemes usually inserts pilots at the transmitter in NFDM system, and carry out FOE before NFT within the DSP of coherent receiver. Generally, the LFD scheme can be applied to DS-NFDM system [23] or CS-NFDM system [24], [25]. Nevertheless, the FOE accuracy and complexity of LFD schemes are closely related with the number of samples of linear fast Fourier transform, and its FOE performance needs to be improved [26]. Next, we mainly focus on the FOE scheme in DS-NFDM system. Furthermore, the NFD FOE scheme has been proposed in DS-NFDM system [27], which is based on the idea that the FO value can be calculated by the eigenvalue offset. However, this scheme is only applicable to the eigenvalue modulation scenario based on on-off encoding. In addition, the performance of classical FOE schemes without data assistance are often limited by its estimation range in NFD. For example, the FOE range of 4th-power feedforward scheme [28] is [−Rs/8, ＋Rs/8], where Rs represents the baud rate. The limited FOE range makes such schemes inappropriate for DS-NFDM transmission systems of several GBaud [7], since the frequency shift of used ECL lasers can be as high as 1.5 GHz. Besides that, Ref. [26] has proposed a NFD scheme with an idea similar to blind phase search (BPS) algorithm, which is named the angle search FOE (AS-FOE) scheme. The AS-FOE scheme is implemented after NFT, and its estimation accuracy and stability are better than that of the LFD scheme. However, a large number of search angles must be applied for this scheme (typically 4096), which directly leads to high computational complexity. In addition, transmitter imperfections, fiber channel impairments and sampling numbers affect the performance of AS-FOE scheme. Importantly, this scheme not fully consider the impact of ASE noise. Actually, the accumulation of ASE noise in fiber link would affect the accuracy of angle search operation, thereby reducing the FOE accuracy and stability of the AS-FOE scheme. Accordingly, it is still necessary to carefully analyze the influence of ASE noise on FO in NFD, establish a corresponding theoretical model, and propose a FOE scheme with high ASE noise tolerance and low computational complexity.

In this work, firstly, we have established an FO impairment model with ASE noise for DS-NFDM transmission. This model reveals that the accumulation of ASE noise directly deteriorates the FOE process in NFD, and this effect is mainly related to the transmission distance and eigenvalues shift. Afterwards, a low-complexity training sequence-assisted FOE (TS-FOE) scheme has been proposed, which is based on the derived NFD-FO model. This scheme could effectively reduce the impact of ASE noise on FOE operation using three sequential steps, which are removing modulation information, reducing the influence of phase noise, and calculating the average FOE value. The simulation results have shown that the proposed TS-FOE scheme achieves an FOE error within 1 MHz using a TS of length 64. Moreover, the estimated range of this scheme is [−Rs/2, ＋Rs/2], where Rs is the baud rate of the DS-NFDM signal. Furthermore, the effectiveness of the TS-FOE scheme has been verified by a 2 GBaud single-eigenvalue DS-NFDM QPSK experimental system. For different FOs within 1 GHz, the experimental results manifest that when OSNR is fixed at 13 dB, the TS-FOE scheme presents a stable FO compensation performance, and the corresponding bit error rate (BER) achieves below the 7% hard-decision forward error correction (HD-FEC) threshold (BER=3.8e−3). More importantly, the overall computational complexity of this TS-FOE scheme is related to the length L of the TS, which is on the order of OL. It is only as one thousandth as that of AS-FOE scheme.

The remainder of this paper is organized as follows. Firstly, the NFD-FO impairment model in presence of ASE noise is detailedly analyzed in Section 2. Subsequently, based on the NFD-FO model, the principle of the TS-FOE scheme is demonstrated in detail. Next, in Sections 3 Simulation results, 4 Experiment results, the effectiveness of the scheme has been verified by simulation and experimental platform, respectively. In Section 5, we have analyzed the computational complexity of the scheme and compare it with that of AS-FOE scheme. Finally, the conclusions are given in Section 6.