1. Introduction

Coherent optical orthogonal frequency division multiplexing (CO-OFDM) has been intensively investigated as an important technology for high-speed optical networks as it offers both high spectrum efficiency and high tolerance to the linear impairments of fiber [1,2 ], such as chromatic dispersion (CD) and polarization-mode dispersion (PMD). In addition to the advantages in fiber transmission, CO-OFDM is also considered a promising candidate for next generation flexible optical networks due to its seamless compatibility with elastic bandwidth allocation [2,3 ]. However, due to its multicarrier nature, CO-OFDM has a long symbol duration and is more susceptive to laser phase noise compared with its single carrier counterpart [4], which hinders its application in high spectrum efficiency systems that employ high-order modulation formats [4] and in systems that have large linewidth-symbol time products, e.g., some metro/access networks with moderate baud rates and low-cost lasers with large linewidths [5–7 ]. The performance of the CO-OFDM system under laser phase noise is degraded by both common phase error (CPE) and inter-carrier-interference (ICI) [4,8 ]. A conventional CPE compensation (CPEC) algorithm [8,9 ] corrects the rotation caused by CPE, while leaving the ICI untouched. This results in poor performance in the systems mentioned above. Algorithms that can effectively mitigate the impact of ICI are of great importance for such CO-OFDM systems.

The ICI can be mitigated in the frequency domain by estimating the higher order components of the optical phase noise and subsequently performing deconvolution. A computationally intensive deconvolution method has been proposed [10], in which the filter’s coefficients are obtained through iterative pilot-aided decision-directed least mean squares (DD-LMS) estimation. We have proposed a frequency domain method based on recursive principle components elimination (R-PCE) [11] that achieves a low-complexity but with some performance degradation (e.g., about 4% reduction of laser linewidth tolerance in the system studied in [11]). A digital coherent superposition (DCS) algorithm [12] has been proposed to self-cancel the effect of ICI by transmitting the subcarrier pairs with Hermitian symmetry. Though DCS has the advantage of low implementation complexity, it reduces the spectral efficiency by one half. Besides the frequency domain methods, the ICI induced by temporal multiplicative laser phase noise can also be compensated in the time domain. In the orthogonal basis expansion (OBE) algorithm [13,14 ], the eigenvalue decomposition of optical phase noise is employed to compensate both CPE and ICI. Sub-symbol optical phase suppression algorithms have also been proposed [15–17 ], in which a long CO-OFDM symbol is temporally partitioned into several sub-symbols. With high temporal resolution of carrier phase tracking, the ICI can be effectively suppressed in sub-symbol algorithms. Decision-aided sub-symbol algorithms, including the sub-symbol CPEC (SCPEC) algorithm [15] and linearly interpolated (LI-) SCPEC algorithm [16], have been proposed. The performances of decision-aided sub-symbol algorithms are severely affected by the propagation of decision errors when the laser linewidth is large. A non decision-aided sub-symbol blind ICI mitigation algorithm named BL-ICI has been proposed [17]. Though the laser phase noise tolerances of these ICI mitigation algorithms are much larger than that of the conventional CPEC algorithm, their complexities are generally substantially higher. For practical CO-OFDM applications, one should choose a certain optical phase noise suppression algorithm according to the amount of available hardware resources, the target performance (e.g., bit error rate) and the acceptable overhead ratio. As analyzed in [11,16 ], most of the high-performance ICI mitigation algorithms require multiple iterations of discrete Fourier transform (DFT) or inverse DFT (IDFT), which consume the majority of the hardware resources. For example, the LI-SCPEC algorithm requires domain conversion with DFT/IDFT to be executed four times, while the OBE and BL-ICI algorithms employ DFT/IDFT operations L (L is the order of basis expansion) and N_B (N_B is the number of sub-symbols) times in constructing the observation-based matrix, respectively.

To address the conflict between low computational complexity and high optical phase noise tolerance in CO-OFDM systems, in this paper we propose two novel linewidth-tolerant algorithms, non-decision aided sub-symbol optical phase noise suppression (NDA-SPS) and partial-decision aided sub-symbol optical phase noise suppression (PDA-SPS), based on low-complexity sub-symbol processing without any DFT/IDFT operations. By arranging pilot-subcarriers at a specific spectral location, multiplier-free generation of the observation-based matrix is achieved in both algorithms, which drastically reduces the complexities. For the NDA-SPS algorithm, high accuracy low-complexity pilot-aided sub-symbol estimation is obtained without decision error propagation. To reduce the number of pilot subcarriers by half while maintaining a performance similar to that of the NDA-SPS algorithm, a partial-decision aided process is introduced in the PDA-SPS algorithm. The principles and computational complexities of the proposed sub-symbol algorithms are theoretically analyzed. The performances and the optimization of the proposed algorithms are investigated with Monte-Carlo simulations in the 16QAM CO-OFDM system under different laser linewidths. Numerical results show that compared with several other sub-symbol algorithms, the proposed algorithms offer larger laser linewidth tolerances yet with much lower complexities.

2. Theory of the proposed optical phase noise suppression algorithms

The schematic diagrams of the proposed sub-symbol phase noise suppression algorithms are depicted in Fig. 1(a) . For simplicity, a single polarization system is examined in the following derivation. After timing/frequency synchronization [18] and cyclic prefix removal, the received temporal samples $y={[y(0),y(1),\cdots ,y(N-1)]}^T$of one CO-OFDM symbol, which is distorted by laser phase noise, channel imperfections (e.g. CD and PMD) and amplified spontaneous emission (ASE) noise from optical amplifiers, can be connected to the frequency domain symbol $X={[X(0),X(1),\cdots ,X(N-1)]}^T$ at the transmitter by the following equation:

 

Fig. 1 (a) Schematic diagrams of the proposed sub-symbol optical phase noise suppression algorithms, i.e., NDA-SPS and PDA-SPS. The red dashed line represents a partial-decision aided process and only exists in the PDA-SPS algorithm. (b) Example of one carrier phase realization (black line), and the corresponding carrier phase estimated by the CPEC algorithm (red line) and the proposed NDA-SPS algorithm (green line, number of sub-symbols N_B = 3) under a combined laser linewidth of 500 kHz.

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Unlike the conventional CPEC algorithm, in which the optical phase noise is assumed to be a fixed value for the whole symbol period [red line in Fig. 1(b)], the proposed sub-symbol algorithms treat the optical phase noise as a process that fluctuates on a sub-symbol basis [green line in Fig. 1(b)] instead of on a symbol basis. Assuming that the received CO-OFDM symbol y is partitioned into N_B sub-symbols in the time domain, we then define $e^{j\overline{\varphi }b}$ as the average optical phase noise at the bth sub-symbol:

In order to estimate the length-N_B phase noise vector $O\left(N_{B1}^3\right)+O\left(N_{B2}^3\right)$, a total number of N_p pilot subcarriers are uniformly inserted into the signal band at the transmitter. Let $k_q$ $(\text{0}\le q\le N_p-1)$ denote the index of pilot subcarriers and the corresponding length-N selection vector $S={[c_0,\cdots ,c_{k_0-1},{\text{e}}_{k_0},c_{k_0+1},\cdots ,c_{k_1-1}{\text{,e}}_{k_1},c_{k_1+1},\cdots ,c_{k_{N_p-2}},e_{k_{N_p-1}},\cdots ,c_{N-1}]}^T$, where ${\text{e}}_{k_q}$equals 1 and $c_k$ equals 0. Note that when $k_0=0$, $c_0,c_{1,}\cdots ,c_{k_0-1}$ are null. From Eq. (1) and Eq. (3), we have:

When N_p is a divisor of N, and the first index of pilot subcarrier k ₀ is chosen to be:

An important observation about vector s in Eq. (8) is that all possible values of the non-zero elements a_n are on either in-phase coordinates or quadrature coordinates in the complex plane. Figure 2 shows the selection vectors S (left side) in the frequency domain and their corresponding normalized time domain vectors s (right side) with N = 256 and N_p = 8 for four possible choices of k ₀ in Eq. (7). Moreover, since N is often chosen to be a power of 2 to facilitate the implementation of DFT with hardware efficient radix-2 fast Fourier transform (FFT), $\frac{{\text{N}}_{\text{p}}}{\sqrt{\text{N}}}$ is also a power of 2. Therefore, the calculation of vector ${\Lambda }_b$ does not need any multiplications but only addition/subtraction operations, which is beneficial for low-complexity hardware implementation. In other words, the circular convolution operation with s in Eq. (9) can be realized using a “circular shift and add” method without any multiplications, and hence the generation of observation-based matrix C is multiplier-free.

 

Fig. 2 The selection vectors (S) (left side) in the frequency domain with different indices k ₀ of the first pilot subcarrier and their corresponding normalized time domain vectors s (right side) in a system with DFT/IDFT size of N = 256 and a number of pilot subcarriers N_p = 8. (a) k ₀ = 0, (b) k ₀ = 8, (c) k ₀ = 16, (d) k ₀ = 24. Since all possible values of the non-zero elements in the time domain vector s are on either in-phase coordinates or quadrature coordinates in the complex plane, the calculation of circular convolution with vector s becomes multiplier-free.

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The first N_p rows of the left-hand side of Eq. (5), i.e., the pilot-based vector R, can be rewritten as:

Combining Eq. (5), Eq. (6) and Eq. (11), we have:

To make Eq. (13) overdetermined, the number of pilots N_p should be no less than the number of sub-symbols N_B. A refinement procedure is executed by eliminating the impact of amplitude noise on ${\widehat{\Phi }}_{sub}$. The updated ${\widehat{\Phi }}_{sub}$ is rewritten as:

2.1 Non-decision aided (NDA) sub-symbol phase noise suppression

With ${\widehat{\Phi }}_{sub}$ from Eq. (14), the kth time domain sample z(k) after sub-symbol optical phase noise suppression is:

The signal $\text{z}={[z(0),z(1),\mathrm{...},z(\text{N}-1)]}^T$ is sent to a channel equalizer and a symbol slicer. The final output $\widehat{X}$ after decision is:

As no decision aided process is employed, the NDA sub-symbol algorithm is free of decision error propagation that largely limits the performance of decision aided algorithms such as our previous LI-SCPEC algorithm [16], when the optical phase noise level is very high (e.g., in a system with a CLW of several MHz).

2.2 Partial-decision aided (PDA) sub-symbol phase noise suppression

Although a large N_B is preferable to reduce the modeling error in Eq. (3), increasing N_B (i.e., the number of unknown variables to be estimated) degrades the accuracy of the LS estimation in Eq. (13) when N_p (i.e., the number of independent rows) is fixed. On the other hand, increasing N_p improves the accuracy of LS estimation, but leads to lower spectrum efficiency. A decision aided process can be employed in a sub-symbol algorithm [16] to mitigate the conflict between low overhead and high estimation accuracy. However, conventional decision aided processes are computationally intensive because they makes decisions on all subcarriers and uses the decisions for a second stage optical phase noise suppression.

To keep both complexity and overhead low, we devise a partial-decision aided (PDA) sub-symbol algorithm that is compatible with the multiplier-free process in Eq. (9) and yet offers outstanding performance comparable with that of the NDA sub-symbol algorithm, which has a doubled pilot-overhead. According to Eq. (7), for a certain number of pilot subcarriers N_p, there are four possible choices of selection vector S that enable multiplier-free generation of the observation-based matrix C with Eq. (9). We choose two of them, S ₁ and S ₂, to be the selection vectors for N_p pilot subcarriers in the pilot aided process and N_p data subcarriers in the partial-decision aided process, respectively. The PDA sub-symbol algorithm consists of the following two stages:

Stage 1: (a) Obtain a coarse estimation of optical phase noise with N_p pilot subcarriers. The received CO-OFDM symbol y is first divided into N_B1 sub-symbols for coarse optical phase suppression. With y and selection vector S ₁, the observation-based matrix C ₁ and pilot-based vector R ₁ are obtained using Eq. (9) and Eq. (11), respectively. Then the coarsely estimated sub-symbol optical phase noise vector ${\widehat{\Phi }}_{sub}$ is calculated using Eq. (14). (b) Obtain N_p data subcarriers through partial-decision with ${\widehat{\Phi }}_{sub}$ from the previous step. Similarly to the N_p pilot subcarriers, the unknown data vector ${X^{\prime }}_2$ carried on N_p data subcarriers corresponding with the selection vector S ₂ satisfies the following equation:

Stage 2: Fine sub-symbol optical phase noise suppression. The received CO-OFDM symbol is divided into N_B2 sub-symbols in this stage. ${\tilde{C}}_1$ and ${\tilde{C}}_2$ denote the new observation-based matrices corresponding with selection vectors S ₁ and S ₂, respectively. Note that $R_1=\frac{1}{\sqrt{N}}{\tilde{C}}_1{\Phi }_{new}+\Delta {\zeta }_1$ is the equation that relates to N_p pilot subcarriers, while $G_2^H({{\widehat{X}}^{\prime }}_2\ast {H^{\prime }}_2)=\frac{1}{\sqrt{N}}{\tilde{C}}_2{\Phi }_{new}+\Delta {\zeta }_2$ is the equation that relates to N_p partial-decision based data subcarriers. An updated sub-symbol optical phase noise vector ${\Phi }_{new}$ can be estimated through these 2N_p equations including both pilot subcarriers and partial-decision based data subcarriers:

Since two sub-symbol optical phase noise estimation stages are employed, the performance of the PDA sub-symbol algorithm depends on both N_B1 in the first stage and N_B2 in the second stage. With N = 256 and N_p = 8, Fig. 3(a) shows the required optical signal-to-noise ratio (OSNR) with different combinations of N_B1 and N_B2 at a BER of 3.8 × 10⁻³ in a back-to-back (B2B) 16QAM CO-OFDM system with the proposed PDA sub-symbol algorithm under a CLW of 1 MHz. It is clearly shown in the solid lines that singularities occur when N_B1 = N_B2, which can be attributed to the fact that the observation-based matrices used in stage 2 are the same as the ones used in stage 1 when N_B1 = N_B2, i.e., ${\tilde{C}}_1=C_1,{\tilde{C}}_2=C_2$. The independence of these two estimation stages is severely disrupted when these two estimation stages use the same observation-based matrices, which in turn results in poor performance in the decision aided estimation process. By contrast, the observation-based matrices are different when N_B1 ≠ N_B2, which provides diversity and hence significant improvement in the estimation accuracy in stage 2.

 

Fig. 3 (a) Required OSNR at a BER of 3.8 × 10⁻³ versus different numbers of sub-symbols N_B1 in stage 1 and N_B2 in stage 2 in the back-to-back 16QAM CO-OFDM system with the proposed PDA sub-symbol algorithm under a CLW of 1 MHz. (b) The OSNR gain at a BER of 3.8 × 10⁻³ versus different offset values with respect to the case with zero offset. (c) Two different symbol partition processes for the received time domain samples when N_B1 = N_B2. The upper part denotes the process without offset $(\text{i}\text{.e}\text{.},\Delta =0)$ in stage 1 and the lower part represents the process with offset $\Delta \ne 0$ in stage 2.

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To eliminate singularities when N_B1 = N_B2, we propose to include an offset when dividing a CO-OFDM symbol into sub-symbols in stage 2, as shown in Fig. 3(c). In stage 2, the end of the first sub-symbol is offset by $\Delta$ samples, and all other sub-symbols are shifted accordingly. Thus, the new sub-symbol matrix is $y^{″}=[{y^{\prime }}_1,{y^{\prime }}_2,\cdots ,{y^{\prime }}_{N_B}]$ with bth column vector as:

To illustrate the offset, the upper part of Fig. 3(c) depicts symbol partition in stage 1 without offset $(\Delta =0)$, and the lower part depicts the symbol partition with offset $(\Delta \ne 0)$ in stage 2. With such a procedure, the observation-based matrices in the two stages differ from each other. The effectiveness of the offset procedure is shown in the dotted lines in Fig. 3(a) with $\Delta =20$. To optimize the offset procedure, Fig. 3(b) shows the OSNR gain with respect to the case with $\Delta =0$ at a BER of 3.8 × 10⁻³ for various offset values when N = 256. It is found that $\Delta \approx \pm 20$ achieves the optimal performance in a system with $N_{B1}=N_{B2}\ge 4$. While for a system with $N_{B1}=N_{B2}\le 3$, the maximal gain is less than 0.05 dB, which is trivial for a system. Consequently, the offset values $\Delta$ are set to 20 and 0 for $N_{B1}=N_{B2}\ge 4$ and $N_{B1}=N_{B2}\le 3$, respectively.

3. Performance evaluation

The performance of the proposed algorithms in a 16QAM CO-OFDM system is evaluated by Monte-Carlo simulations using Matlab and VPItransmissionmaker. We use the terms “NDA-SPS” and “PDA-SPS” to refer to the proposed algorithms in section 2.1 and section 2.2, respectively. The total number of subcarriers is 256, in which 8 central subcarriers and 14 edge subcarriers are set as guard subcarriers. Unless otherwise marked, 8 pilot subcarriers are uniformly distributed in the signal band as described in section 2. For PDA-SPS, the index of the first non-zero element of selection vector S ₁ (S ₂) is k ₀ = 8 (k ₀ = 24). A cyclic prefix of length 32 is used to accommodate the delay among subcarriers induced by chromatic dispersion. ISFA channel estimation is employed [21]. Electrical pre-distortion [22] is adopted at the transmitter to mitigate the nonlinearity in electrical-to-optical conversion with the optical I/Q modulator. The sampling rate is chosen to be 14 GS/s to emulate a 43.94 Gb/s metro/access system [5,6 ], which typically has a large linewidth-symbol time product. The photodiodes are modeled as 5th-order Bessel low-pass filters with 3 dB bandwidths of 14 GHz. Thermal noise and shot noise have also been included at the receiver. For comparison, the performances of two other sub-symbol algorithms, the BL-ICI algorithm with the average power method [17], the LI-SCPEC algorithm and the R-PCE algorithm, are also investigated. Unless otherwise indicated, all simulation results are obtained with homodyne detection using a practical synchronization method proposed in [18] (i.e., the impact of non-perfect time synchronization has already been taken into account in all numerical results). All BER results are obtained by measuring over 4.7 × 10⁷ bits under 25 different carrier phase realizations.

To demonstrate the effectiveness of the PDA process in pilot-overhead reduction, Fig. 4(a) shows the required OSNR at a BER of 3.8 × 10⁻³ with the NDA-SPS algorithm employing N_p = 16 and the PDA-SPS algorithm employing N_p = 8 in the B2B 16QAM CO-OFDM system under different CLWs. When N_p is fixed to 8, the PDA-SPS algorithm outperforms the NDA-SPS algorithm with the same number of sub-symbols. For a small number of sub-symbols, the performance of PDA-SPS (2, 2) with N_p = 8 is almost the same as that of NDA-SPS (N_B = 2) with N_p = 16. At an OSNR of 23 dB, the CLW tolerances are approximately 2.82 MHz, 2.56 MHz, 3.56 MHz and 3.22 MHz for NDA-SPS (N_B = 3) with N_p = 16, PDA-SPS (2, 3) with N_p = 8, NDA-SPS (N_B = 4) with N_p = 16 and PDA-SPS (3, 4) with N_p = 8. Therefore, with a moderate OSNR, the optical phase noise tolerance of the system with the PDA-SPS algorithm, which employs only half the number of pilot subcarriers, is only slightly smaller than that of the system with the NDA-SPS algorithm. We then focus on the performance of the PDA-SPS algorithm in the following investigation.

 

Fig. 4 (a) Required OSNR at a target BER of 3.8 × 10⁻³ in the back-to-back system with the proposed PDA-SPS algorithm and proposed NDA-SPS algorithm versus different CLWs. (b) Required OSNR at a BER of 3.8 × 10⁻³ in the back-to-back system with various phase noise suppression algorithms versus different CLWs. Unless specifically stated, the number of pilots N_p = 8.

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Figure 4(b) shows the required OSNR at a BER target of 3.8 × 10⁻³ with respect to CLW using different optical phase noise suppression algorithms in the B2B 16QAM CO-OFDM system. When the CLW exceeds 1 MHz, the system with a conventional CPEC algorithm fails to achieve the target BER in the OSNR range of interest. At an OSNR of 23 dB, the system’s CLW tolerances with CPEC, R-PCE (q = 3), BL-ICI (N_B = 4), LI-SCPEC (N_B = 4) and PDA-SPS (3, 4) are about 835 kHz, 1.44 MHz, 2.41 MHz, 2 MHz and 3.22 MHz, respectively. Though the ICI mitigation procedure in the BL-ICI algorithm is not affected by decision error, its performance with N_B = 4 is found to be worse than that of PDA-SPS (3, 4) when CLW exceeds 750 kHz. The performance variance of these two sub-symbol algorithms is attributed to the additional modeling error in the BL-ICI algorithm with approximation $\sin \theta \approx \theta$ [17]. The benefit of higher estimation accuracy in stage 1 of the PDA-SPS algorithm than that of the LI-SCPEC algorithm with the same number of sub-symbols is clearly demonstrated by comparing their performances when CLW is larger than 1.2 MHz. Therefore, the system’s optical phase noise tolerance is evidently improved by the PDA-SPS algorithm.

As implied in section 2.2, the performance of PDA-SPS depends on the number of sub-symbols in both stages. To optimize the performance of the PDA-SPS algorithm, Fig. 5 shows the contour plots of receiver sensitivity (OSNR in dB) at a target BER of 3.8 × 10⁻³ with respect to different numbers of sub-symbols N_B1 in stage 1 and N_B2 in stage 2 under a CLW of 1 MHz [Fig. 5(a)] or 2 MHz [Fig. 5(b)]. For each single stage, the modeling error in Eq. (3) becomes smaller with a larger number of sub-symbols. However, with a limited number of equations [i.e., Eq. (12)] available, the estimation accuracy drops for larger numbers of unknown variables (i.e., more averaged optical phase noise of sub-symbols to be estimated). In both studied cases, optimal performance is achieved with moderate N_B1 and N_B2, i.e., N_B1 ≤ 4 and N_B2 ≤ 6. Comparing these two cases, the optimal number of sub-symbols in both stages becomes larger when the CLW increases. Since the number of equations doubles in stage 2, the optimal N_B1 in stage 1 is also found to be smaller than N_B2 in stage 2.

 

Fig. 5 The contour plots of receiver sensitivity at a BER of 3.8 × 10⁻³ with respect to different numbers of sub-symbols N_B1 in stage 1 and N_B2 in stage 2 of the PDA-SPS algorithm under CLWs of (a) 1 MHz and (b) 2 MHz. The optimal performances are achieved with moderate N_B1 and N_B2.

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In addition to the back-to-back scenario, the Q² factor of a 16QAM CO-OFDM system under a CLW of 1 MHz after 4 × 80 km standard single mode fiber (CD: 17 ps/nm/km, nonlinear coefficient: 1.3 W⁻¹km⁻¹, loss: 0.2 dB/km) transmission without inline dispersion compensation is also investigated, as shown in Fig. 6(a) . Each fiber span employs one erbium-doped fiber amplifier (EDFA) with a noise figure of 6 dB and a gain of 16 dB. The optical launch power is set to −6 dBm for optimal transmission performance. To show the impact of non-perfect channel estimation with ISFA under the impact of optical phase noise and ASE noise, the performance of PDA-SPS (3, 4) with ideal channel estimation is also included in Fig. 6(a). The results show that the performance degradation of PDA-SPS (3, 4) due to the non-perfect channel estimation is small (Q² factor penalty < 0.35 dB) in the simulated system. With the same number of sub-symbols and pilot-overhead, the PDA-SPS outperforms other listed sub-symbol algorithms (i.e., BL-ICI, LI-SCPEC) and the R-PCE algorithm in the transmission scenario. The performance of the PDA-SPS algorithm is only slightly worse than that of the NDA-SPS algorithm with a doubled pilot-overhead and the same number of sub-symbols. Compared with the BL-ICI algorithm and LI-SCPEC algorithm with the same number of sub-symbols, which show a similar performance in the system, the Q² factor increases approximately by 0.5 dB (1 dB) with the proposed algorithms when N_B = 2 (N_B = 4). By comparing the signal constellations before the symbol slicer with CPEC [point A in Fig. 6(a)] and PDA-SPS (3, 4) [point B in Fig. 6(a)], it is clear that the ICI induced by optical phase noise is effectively mitigated by the sub-symbol algorithm PDA-SPS. To show the impact of non-perfect frequency synchronization in case of intradyne detection, Fig. 6(c) compares the performance of different algorithms under carrier frequency offset from 0 GHz to 5 GHz. The measured mean residual frequency offset is about 3.08 MHz regardless of the actual carrier frequency offset before synchronization. Compared with the homodyne detection scenario, about 0.5 dB Q² factor penalty is observed for PDA-SPS (3, 4) under non-perfect frequency synchronization. From this figure one sees that even with such a residual frequency offset the Q² factor is improved by at least 1.22 dB with PDA-SPS (3, 4) when compared with BL-ICI (N_B = 4).

 

Fig. 6 (a) Q² factor of a 16QAM CO-OFDM system under a CLW of 1 MHz after 320 km SSMF transmission versus different optical launch powers with different optical phase noise suppression algorithms. (b) The constellation diagrams for CPEC algorithm and PDA-SPS (3, 4) algorithm under an optimal optical launch power of −6 dBm. (c) The Q² factor of intradyne detection system (solid lines) using the synchronization method in [18] with PDA-SPS (3, 4) and BL-ICI (N_B = 4) under different carrier frequency offsets. All listed algorithms except the NDA-SPS algorithm are tested in a system with 8 pilot subcarriers. For the NDA-SPS algorithm, the number of pilot subcarriers is 16.

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4. Complexity analysis

The complexity of the proposed sub-symbol algorithms in terms of the required number of complex-valued multiplications (RNCM) is analyzed in Table 1 . The RNCM of the CPEC [8], LI [23], BL-ICI [17] and previously proposed LI-SCPEC algorithms [16] are also listed for comparison. The complexity of the proposed NDA-SPS algorithm is mainly induced by calculating the phase noise vector ${\widehat{\Phi }}_{sub}$ [Eq. (13)] and phase noise suppression [Eq. (15)], whose RCNM are $O\left((N_B^2+N_B)N_P+N_B^2\right)$ and $O(N),$ respectively. For the PDA-SPS algorithm, an additional RNCM of $\left[O\left((N_{B1}+2N_{B2}+2+2N_p+2N_{B2}^2)N_P+N_{B2}^2\right)+O\left(N_{B2}^3\right)\right]$ is required for the partial-decision aided process.

Table 1. Number of operations required for various phase noise suppression algorithms

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As indicated in Eq. (3), the proposed NDA-SPS algorithm with N_B = 1 is equivalent to the conventional CPEC. Unlike the BL-ICI algorithm and LI-SCPEC algorithm that require multiple repetitions of size-N DFT/IDFT, the proposed sub-symbol algorithms do not need any DFT/IDFT in the process of phase noise estimation. Therefore, the total RNCM of NDA-SPS and PDA-SPS are substantially lower than that of the former two algorithms. For instance, as shown in Table 2 , the RNCM with N = 32N_p for LI (Q = 4), R-PCE (q = 3), NDA-SPS (N_B = 2), NDA-SPS (N_B = 4), PDA-SPS (2, 2), PDA-SPS (3, 4), BL-ICI (N_B = 2), BL-ICI (N_B = 4) and LI-SCPEC (N_B = 2 or 4) are about 4.08N, 3.19N, 1.23N, 1.94N, 2.28N, 3.73N, 15.09N, 29.87N and 20.31N, respectively. When the number of pilot subcarriers is set to ${\text{N}}_{\text{p}}\text{=}N/16$, the RNCM are about 1.42N and 2.56N for NDA-SPS (N_B = 2) and NDA-SPS (N_B = 4), respectively. Thus, with moderate numbers of sub-symbols, the PDA-SPS algorithm with half pilot-overhead has a complexity comparable to that of the NDA-SPS algorithm that exhibits a similar performance.

Table 2. Typical RNCM with N = 32N_p for various phase noise suppression algorithms

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5. Conclusion

In this paper, we propose two novel low-complexity linewidth-tolerant sub-symbol optical phase noise suppression algorithms, NDA-SPS and PDA-SPS, for CO-OFDM systems. In the NDA-SPS algorithm, high performance time domain sub-symbol optical phase suppression is achieved without decision error propagation. To improve the spectrum efficiency, a partial-decision aided process that helps to reduce the pilot-overhead by half is proposed in the PDA-SPS algorithm. To emulate an optical access/metro system, in our simulation we choose similar parameters as those in literature [10,13 ] (e.g., a moderate band rate of 10 to 20 GBaud/s, a moderate DFT/IDFT length of 256 and the number of pilot subcarriers of 10 to 20). Numerical results show that the PDA-SPS algorithm with a 50% pilot-overhead offers a laser linewidth tolerance similar to that of NDA-PSP algorithm at a moderate OSNR, which is considerably larger than several other sub-symbol algorithms. The optimization of the PDA-SPS algorithm verifies that the optimal performance can be achieved with moderate numbers of sub-symbols for a CLW of several MHz. By employing specially designed comp-type pilot subcarriers, multiplier-free observation-based matrix generation for sub-symbol optical phase estimation is achieved in the proposed algorithms. Since DFT/IDFT operations that are usually required in other high performance ICI mitigation algorithms are avoided in the proposed algorithms, the complexities of the proposed algorithms with superior performance are significantly lower than several other sub-symbol algorithms, which is beneficial for real-time hardware implementation.