In response to the exponential growth in traffic demands driven by high-speed optical access networks such as cloud services, ultrahigh-definition video, and virtual reality applications, passive optical networks (PONs) have emerged as a dominant architecture to deliver high data rate fiber access services^[1–3]. Future PON generations are expected to achieve capacities of 100 Gbps per wavelength (100G-PON) and beyond^[4]. However, standardized PON solutions based on intensity modulation and direct detection (IM/DD) are viewed as too challenging to support the growth of high-speed networks due to limited receiver sensitivity^[5]. To address this limitation, coherent systems, known for their superior receiver sensitivity and technological maturity, are now poised as promising solutions for future PONs^[6]. Coherent PONs (CPONs) with large dynamic ranges and high performance over 200G have been demonstrated in recent years^[7–9].

Despite its merits, the CPON faces several challenges that require attention, including sensitivity to laser wavelength drift and the implementation of coherent upstream burst-mode detection^[10,11]. The laser wavelength drift causes a frequency offset between the transmitter and receiver. This frequency offset makes the frequency offset estimation (FOE) a crucial part of the digital signal processing (DSP) in coherent optical fiber communication. Several studies have been reported that solve this issue. The traditional Viterbi–Viterbi algorithm, which relies on Mth power operations, is commonly used for M-ary phase-shift-keying (MPSK) signals^[12]. In contrast, the fast Fourier transform-based FOE (FFT-FOE) algorithm is suitable for higher-order modulation formats but suffers from computational complexity and increased power consumption^[13]. Given the cost sensitivity of PONs, achieving blind estimation algorithms with high computational complexity remains too challenging for the upstream link. Conversely, data-aided (DA) estimation algorithms, while requiring additional information for FOE, offer significantly lower computational complexity. A DA FOE algorithm employs training sequences instead of Mth power operations, enabling an estimation range of approximately 20% of the symbol rate^[14]. In Ref. [15], a method for estimating carrier frequency offset in orthogonal frequency-division multiplexing systems was introduced. This method utilized a pilot-tone-based maximum likelihood estimator (PBMLE) and employed a preamble with uniquely spaced pilot tones.

Another key issue here is achieving rapid convergence burst-mode DSP. In contrast to the continuous-mode downstream link, the central optical line terminal (OLT) in the upstream link receives signals burst-by-burst from the user-side optical network units (ONUs)^[16,17]. Therefore, the burst-mode DSP must quickly converge to recover the burst signal from the ONU promptly and then prepare for the subsequent upstream burst efficiently. Several studies have been conducted on fast-convergence DSP for CPON^[16,18–23]. For instance, in Ref. [16], a specially designed preamble was used for three burst-mode DSP functions, including frame synchronization, state-of-the polarization estimation, and FOE. In Ref. [18], a burst mode capable transmission system with an OFDM-based data-independent synchronization symbol was introduced for a fast adaption of the receiver. In Ref. [19], a fast-converged adaptive filter enabled blind packet recovery in less than 200 ns with three stages. A fast coarse estimation method of carrier frequency offset for orthogonal frequency division multiplexing systems using a burst mode transmission was demonstrated in Ref. [23].

In this paper, we introduce an adaptive dual-stage approach that includes coarse FOE and enhanced AEQ modules. After the coarse FOE in the first stage, a fine FOE tap is integrated into the AEQ to enhance FOE accuracy while simultaneously completing channel compensation. This integration reduces the number of pilot symbols required and achieves accurate FOE over a range of ∼±0.5 times the symbol rate. By the experiment, the effectiveness and feasibility of the proposed algorithm are demonstrated in 128-Gbit/s 16-ary quadrature amplitude modulation (16QAM) transmission systems with a single polarization.

In the following discussion, we focus on two critical modules: carrier frequency offset (CFO) and channel equalization. After completing down-conversion, match filtering, Gram–Schmidt orthogonalization procedure, timing recovery, and frame synchronization, three types of impairments remain in the signal: (1) carrier phase errors caused by the frequency mismatch between the transmitter laser and the local oscillator (LO); (2) in coherent optical communication systems, the received signal undergoes a downconversion operation followed by matched filtering to extract a baseband signal. However, due to carrier frequency shifts, the signal is not fully downconverted to the baseband. Consequently, after passing through the matched filter, the signal broadens in the time domain; (3) intersymbol interference due to chromatic dispersion (CD) and polarization mode dispersion (PMD). The CFO module compensates for phase errors, while the linear channel equalization module handles the latter two impairments.

Figure 1 compares the DSP of conventional and proposed schemes. Traditionally, the coherent receiver (Rx) DSP module performs FOE followed by a conventional adaptive equalization (AEQ), as depicted in Fig. 1(d), which compensates for signal distortion induced by optical channels^[24,25]. Figure 1(b) illustrates the widely adopted Mth power-based FOE algorithm, which is computationally intensive due to the Fourier transform. In contrast, the DA-FOE algorithm reduces computational complexity by incorporating redundant information.

Figure 1(c) shows the block diagram of the traditional DA-FOE^[14], where t(n) represents the N pilot symbols transmitted, rp(n) denotes the data input to the FOE module, Ts is the sample interval, Δfest represents the estimated frequency offset, (·)* is the complex conjugate operation, θ is the carrier phase, and {γ(n)} are independent and identically distributed Gaussian noise: rp(n)=t(n)exp[j(2πΔfestnTs+θ)]+γ(n).(1)

By multiplying rp(n) and t*(n), the modulation phase in rp(n) is removed, z(n)=rp(n)t*(n)=exp[j(2πΔfestnTs+θ)]+γ(k)t*(n).(2)

Subsequently, as discussed in Ref. [14], we calculate the correlations R(m) shown in Eq. (3), R(m)=1N−m∑n=m+1Nz(n)z*(n−m),m∈[1,N/2].(3)

γ(k) are statistically equivalent Gaussian noises that can be neglected after averaging, we have R(m)≅exp[j(2πΔfestmTs)],m∈[1,N/2].(4)

Equation (4) establishes a relation between Δfest and arg{R(m)}, and we can calculate Δfest based on the increment of arg{R(m)}.

The traditional DA-FOE method uses pilot symbols for accurate FOE, followed by an adaptive equalizer to address channel impairments. Enhancing this approach, our proposed algorithm simplifies the frequency offset calculation and reduces the number of pilot symbols needed while still maintaining robust performance. This enhancement is achieved by integrating a fine FOE tap within the adaptive equalizer.

Figure 2 shows the principle of the proposed solution. Our algorithm consists of two stages: Initially, we estimate the frequency offset by analyzing a small set of symbols with identical phases. Next, we employ AEQ to compensate for both linear impairments and residual frequency offset (RFO). The input signal r(n) (one sample per symbol) to the proposed algorithm is composed of three components: the pilot symbols rp(n) that have the same phase information, the training symbols for AEQ, and the data symbols x(n) that contain the information to be transmitted. The structure of signal r(n) is illustrated in Fig. 2(a). It is important to note that rp(n) should consist of symbols with the same phase, while the amplitudes of rp(n) are not required for the process. In this paper, rp(n) comprise N ones (1+0i).

Figure 2(b) illustrates the first stage block diagram of the proposed algorithm. The phase rotation induced by frequency offset is calculated as 2πΔfestTs=1N∑Narg[rp(n)·rp(n−1)*],(5)where N is the length of rp(n). The signal x(n) is then compensated using the roughly estimated frequency offset and input to the second stage. An AEQ based on the least mean square (LMS) principle is used to compensate for both the linear distortion and the RFO error. Figure 2(c) illustrates the structure of the AEQ. The output y(n) of the equalizer is given by y(n)=∑i=−(M−1)/2(M−1)/2ωix(n−i)exp[j2πωf(n−i)Ts],(6)where ω represents the tap coefficients to compensate for linear impairments, ωf represents the coefficients to compensate for RFO, and M is the order of the adaptive equalizer. Then the error can be calculated as e(n)=d(n)−y(n), where d(n) is the training symbol. The cost function is calculated as J(n)=E[|e(n)|2]. The tap coefficients are then updated with step-sized parameters μ1 and μ2 as ωi←ωi+2μ1e(n)x*(n−i)exp[−j2πωf(n−i)Ts],(7)ωf←ωf+μ24πnTs·Im{d(n)·y*(n)}.(8)

The convergence parameters directly affect the update rate of tap coefficients and the stability of the system. Since there are differences in the convergence direction and target between ω and ωf, we set different step-sized parameters for ω and ωf, respectively.

In this section, we conduct experimental evaluations to assess the effectiveness of the proposed scheme using 128-Gbit/s 16QAM transmission systems, as illustrated in Fig. 3. At the transmitter (Tx), 32-GBaud 16QAM signal is generated using a root-raised-cosine (RRC) filter with a roll-off factor of 0.01. A 32-GHz arbitrary waveform generator (AWG, Keysight M8196A) operating at 92 GS/s is used to load the digital signal. The output of the AWG is amplified by two linear electrical amplifiers (EAs) and modulated by a 22-GHz I/Q modulator (IQM). At the Rx, the signal is detected by a coherent front end with a balanced photodetector (BPD) having 3-dB bandwidth of ∼40GHz. The optical source from a tunable semiconductor laser (Santec TSL-710) with a 100-kHz linewidth serves as the LO. The polarization of the received signal and LO are aligned by a polarization controller (PC). The received power level is adjusted to −12dBm with a variable optical attenuator (VOA). The power of the local oscillator is 10 dBm. Finally, the signal is digitized by a 160 GS/s 59 GHz digital sampling oscilloscope (DSO) for offline signal processing.

The coarse FOE is first performed on the input signal r(n) using pilot symbols rp(n) with a length of N. We investigate the impact of the pilot symbol length on the accuracy of estimation results. The variation of normalized frequency variance (NFV, defined as E[|Δf^Ts−ΔfTs|2]) of coarse FOE with respect to N is demonstrated in Fig. 4. We conducted experiments under three different scenarios: N=20, N=50, and N=100. As N increases, the precision of coarse FOE improves accordingly. For example, with N=100, the NFV can reach a minimum of about 6.8×10−9, whereas with N=20, the NFV can only reach a minimum of 1.8×10−6. In addition, the stability of the algorithm is affected by N. As N decreases, the range of frequency offset that can maintain the stability of NFV narrows down. For instance, when N=100, NFV remains stable within the frequency offset range of −6 to 6 GHz. However, NFV deteriorates sharply when the frequency offset exceeds ±6GHz. When N=20, NFV is stable only if the frequency offset is within the range of −4 to 4 GHz. In conclusion, it is observed that the accuracy and stability of coarse FOEs are improved as N increases.

Figure 5 presents the effect of the AEQ convergence parameters on the performance of the algorithm in the second stage. After a coarse estimation of the frequency offset, an adaptive equalizer is employed in the subsequent stage to mitigate channel impairments and the RFO. Figure 5(a) illustrates the convergence curve of the AEQ for N=100 and M=11. The initial frequency offset is 13 GHz. After coarse FOE, the RFO is 86 MHz. Therefore, the tapping coefficient ωf, which represents the RFO deduced from the AEQ, needs to converge quickly from the initial value 0 to the RFO value. The black line in Fig. 5(a) depicts the convergence curve of ωf when performing fixed step size LMS (FSS-LMS) equalization. It begins to oscillate around 86 MHz as the number of equalization iterations increases.

To avoid oscillations and enhance the stability of AEQ, a variable step size LMS (VSS-LMS) equalization approach is employed, where the step size is written as a function of the error e(n) as shown in Ref. [26]. The analytical representation of the step size function is given by μi=βi[1−exp(−αi|e(n)|2)],i=1,2.(9)

In the given function, α determines the gradient, affecting the rate of change within the function’s output. β determines the function’s value domain, setting the boundary for μi. Figure 6 illustrates the relationship between step-sized parameters and e(n). When the AEQ error is large, a greater step size is employed to accelerate convergence. As the error diminishes, the step size is correspondingly reduced. In this experiment, we set α1=0.8, β1=0.001, α2=1, and β2=300 to have maximum convergence speed and minimum bit error rate in the equalizer process. As shown by the red line in Fig. 5(a), ωf converges stably to 86 MHz without further oscillations. Figure 5(b) illustrates the mean squared error (MSE) curve of the proposed AEQ, the MSE will oscillate with the iterative process when the FSS-LMS equalization is used. Conversely, the implementation of VSS-LMS equalization increases the performance of the algorithm.

We then evaluate the performance of the second stage algorithm, as shown in Fig. 7(a). After AEQ, the NFV is reduced by approximately 1 order of magnitude, and the FOE range is extended; however, the performance variation caused by different N in the first stage has not been eliminated. This is because the precision of the initial rough estimation has an impact on the spectral characteristics of the signal after matched filtering. Figure 7(b) compares the performance of the proposed algorithm with the FFT-based FOE and the DA-FOE. The proposed dual-stage scheme (N=100) offers an estimation range of [−15(GHz),15(GHz)]. In contrast, the FFT-based FOE has a range of [−Rs/4,Rs/4]=[−8(GHz),8(GHz)], and the DA-FOE (N=100) has a range of [–8(GHz),8(GHz)], where Rs is the symbol rate. Furthermore, increasing the length of pilot data enhances the estimation range of both the proposed scheme and the DA-FOE. Finally, with the same N, the NFV of our proposed dual-stage scheme is significantly lower than that of the traditional DA-FOE. This indicates that our scheme requires fewer pilot data to achieve the same estimation accuracy. For instance, our dual-stage solution with N=50 achieves a similar estimation accuracy as the DA-FOE with N=600.

Figure 8 illustrates the ability of the VSS-LMS AEQ to compensate for channel impairments when N=100, the frequency offset is 1 GHz, and M=11. Figure 8(a) shows the error convergence curves of the AEQ. The AEQ converges stably within 600 training symbols. Figures 8(c) and 8(e) depict the constellations of the input signal and the signal after rough frequency offset compensation with N=50 and N=100, respectively. From Fig. 8(e), it can be seen that the output signal still has some phase noises, primarily due to RFO and laser linewidth. The residual phase noise of the AEQ output signal is depicted in Fig. 8(b). To suppress the phase noise and recover the original signal, we perform a phase recovery process after the AEQ. Figure 8(f) displays the constellation diagrams of the signal after phase recovery.

Figure 9 presents a comparative analysis of the convergence velocity for three distinct schemes: (1) DA-FOE succeeded by cascaded multi-modulus algorithm non-data-aided (NDA) equalization, (2) DA-FOE succeeded by conventional DA equalization, and (3) the proposed scheme employing rough FOE followed by enhanced AEQ. Figure 9(a) illustrates the curve of NFV versus N in DA-FOE and the proposed scheme. The results show that the proposed scheme achieves a lower NFV with the same amount of pilot data. Consequently, the FOE pilot data length can be reduced when using the enhanced AEQ. From Fig. 7(b), it can be seen that the dual-stage solution with N=50 achieves a similar estimation accuracy as the DA-FOE with N=600. The MSE of the AEQ for three evaluated algorithms is presented in Fig. 9(b). The enhanced AEQ achieves convergence within 600 symbols. Its convergence rate is comparable to the conventional DA equalization scheme. In contrast, the blind NDA equalization scheme demonstrates the slowest convergence rate. In summary, the proposed scheme completes FOE and channel equalization using only 650 (50+600) training symbols. In contrast, the traditional approach requires 1200 (600+600) training symbols to achieve the same tasks. Notably, approaches like preamble-based channel estimation can be used to further accelerate the convergence speed of AEQ^[27,28].

Finally, we evaluate the computational complexity through a comparative analysis between the proposed algorithm and the conventional methods using DA-FOE followed by an AEQ. Given that multiplication operations are typically the most resource-intensive processes in hardware computation, we use the metric of complex multiplication operations required for algorithm execution to evaluate computational complexity. Traditional DA-FOE with N training symbols involves (3N2+14N+8)/8 multiplications, and then, the M-order adaptive equalize requires M+1 multiplications per iteration for tap updates and M multiplications for error function construction. In our proposed scheme, the computational complexity of the first stage rough estimation is N+1. The modified AEQ algorithm requires four additional multiplications for FOE, i.e., 2M+5 multiplications per iteration.

Figure 7(c) shows the multiplication count in the traditional and proposed schemes, considering M=11th order AEQ and Ntrain=600 iterations. To achieve an NFV lower than 10−8, the traditional DA-FOE requires 600 training symbols, whereas the proposed scheme utilizes only 50 pilot symbols. At this point, the numbers of multiplications required for traditional schemes and the proposed scheme are 136,051 and 16,251, respectively. Notably, the proposed scheme achieves an 88.06% reduction in computational complexity. Moreover, our approach significantly reduces the number of pilot symbols—only 650 compared to 1200 for traditional schemes —to achieve an NFV lower than 10−8 and channel equalization simultaneously.

An algorithm is proposed for a CPON that can estimate frequency offset over a wide range and mitigate channel impairments. We present the theoretical analysis of the scheme and validate its feasibility through 128-Gbit/s 16QAM signal transmission systems. The experimental results demonstrate that the proposed algorithm possesses an FOE range approaching ±0.5Rs, and it requires 88.06% fewer multiplications than the DA-FOE scheme to achieve comparable performance. Moreover, our approach reduces the length of pilot symbols compared to traditional schemes, thereby allowing the DSP to converge faster. Consequently, an efficient and rapid dual-stage scheme is presented in this paper for upstream burst-mode coherent detection that simultaneously achieves FOE and channel equalization.