Low-complexity optical phase noise suppression in CO-OFDM system using recursive principal components elimination

Xiaojian Hong; Xuezhi Hong; Sailing He

doi:10.1364/OE.23.024077

1. Introduction

Coherent optical orthogonal frequency division multiplexing (CO-OFDM), which features high spectrum efficiency [1] and elastic bandwidth provisioning [2], is regarded as an important physical layer technology for flexible optical networks. However, its tolerance to optical phase noise is low due to the long symbol period [3], which hinders the application of CO-OFDM in systems that have large linewidth-symbol time products or employ high-order modulation formats, e.g., some metro/access networks [4–6 ].

Various digital algorithms have been proposed for optical phase noise suppression [7–13 ], which in principle mitigates the impact of common phase error (CPE) and inter-carrier-interference (ICI). Though the conventional CPE compensation (CPEC) algorithm [7,14 ] has the advantage of low computational complexity, the performance of the above systems after CPEC can still be severally degraded by the residual ICI. A linear interpolation (LI) algorithm [8] is proposed to partially reduce the ICI by emulating the time-varying optical phase noise inside one symbol through phase interpolation. Since ICI is caused by the convolution of transmitted data and the higher order components of phase noise [14], high performance ICI mitigation can be accomplished by estimating the higher order components and then eliminating their impact on the signal. In the orthogonal basis expansion (OBE) algorithm [9,10 ], eigenvalue decomposition of optical phase noise with a discrete Fourier transform (DFT) basis is employed to estimate multiple principal components, which requires matrix inversion and multiple times of domain conversion. Higher order components of optical phase noise can also be tracked by improving the temporal resolution of optical phase noise estimation by using sub-symbol processing [11,12 ]. Deconvolution by match filtering (MF) with a finite impulse response (FIR) filter has been proposed [13] to combat ICI, in which the FIR filter’s coefficients are chosen to be the DFT of optical phase noise and obtained by iterative pilot-aided decision-directed least mean squares estimation (DD-LMS). Nevertheless, compared with CPEC and LI, the computational complexities of the above high performance ICI mitigation algorithms [9–13 ] are substantially higher, which increases the difficulty of hardware implementation [15].

To achieve computationally efficient and high performance optical phase noise suppression in a CO-OFDM system, in this paper we propose a novel optical phase noise suppression approach based on recursive principal components elimination (R-PCE). In contrast to the DD-LMS-MF scheme [13], computationally efficient least squares (LS) estimation is adopted in R-PCE to reduce the complexity and latency. By employing LS estimation recursively in multiple principal components estimation, computationally intensive matrix inversion and domain transformation [9–11 ] are avoided in the R-PCE. The performance and optimization of R-PCE and its decision-aided version (DA-R-PCE) are investigated by Monte-Carlo simulations of the CO-OFDM system under different modulation formats (QPSK/16QAM) and laser linewidths, the results of which show that both high performance and low-complexity optical phase noise suppression are feasible with the proposed algorithm.

2. Theory of the proposed optical phase noise algorithm

The schematic diagram of the proposed R-PCE algorithm is depicted in Fig. 1(a) . The signal Y from the one-tap frequency domain channel equalizer [16] serves as the input to the proposed algorithm. For simplicity, single polarization is assumed in the following derivation. Y can be described as:

Y = {\hat{H}}^{- 1} H F Φ F^{H} X + W_{A S E}^{} \approx F Φ F^{H} X + W_{A S E}^{}

where the superscript (·)^H,

\hat{H}

and

H

denote the Hermitian transpose, estimated channel response matrix and real channel response matrix, respectively. In case that optical phase noise or amplified spontaneous emission noise (ASE) exists, the approximation on the right side of Eq. (1) introduces errors as channel equalization is not perfect with conventional channel estimation algorithms (e.g., ISFA [16]).

X = {[X (0), X (1), \dots, X (N - 1)]}^{T}

represents the transmitted OFDM symbol of length N in the frequency domain, where the superscript (·)^T denotes the transpose and

X (k)

is the data modulated on the kth subcarrier.

F = [f_{0}, f_{1}, \dots, f_{N - 1}] / \sqrt{N}

with the kth row

f_{k} = {[1, e^{- j \frac{2 π k}{N}}, \dots, e^{- j \frac{2 π (N - 1) k}{N}}]}^{T}

and

F^{H}

are the size-N DFT matrix and inverse DFT (IDFT) matrix, respectively.

W_{A S E}

denotes the impact of the ASE noise after channel equalization.

Φ = d i a g {{[e_{}^{j ϕ (0)}, e_{}^{j ϕ (1)}, \dots, e_{}^{j ϕ (N - 1)}]}^{T}}

is a diagonal matrix representing the time-varying optical phase noise due to nonzero laser linewidth. The diagonal element

e_{}^{j ϕ (k)}

of

Φ

represents the impact of laser phase noise on the kth temporal sample of the CO-OFDM symbol. Laser phase noise can be modeled as a Wiener process [17], i.e.,

ϕ (k) = ϕ (k - 1) + u (k)

, where

u \sim N (0, σ_{u}^{2})

is the Gaussian distributed independent incremental movement of the laser phase noise with zero mean and a variance

σ_{u}^{2} = 2 π β T_{s}

. Note that T_s is the CO-OFDM sample period and β is the combined laser linewidth of lasers at the transmitter and receiver.

Fig. 1 (a) Schematic diagram of the proposed optical phase noise suppression algorithm R-PCE. The red dashed line denotes the decision-aided version of R-PCE. (b) The normalized spectrum |P(k)| of optical phase noise with perfect estimation (black) and the R-PCE algorithm (red, number of principal components q = 5). Only central five components are visible in the R-PCE (q = 5) case as all other components are assumed to be zero values. (c) BER versus OSNR with two different sets of i_k (1 ≤ k ≤ q = 6) in the 16QAM CO-OFDM system under three different laser linewidths (500 kHz, 1 MHz and 1.5 MHz). Better performance is obtained with set II in which the central frequency components are estimated first.

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Equation (1) can be rewritten as:

X \approx F Φ^{*} F^{H} Y + Δ δ

where

Δ δ

is the distortion induced by the ASE noise. Utilizing the property that

Φ^{*} = Φ^{- 1}

, the first term on the right side of Eq. (2) can be expanded as:

F Φ^{*} F^{H} Y = \frac{1}{N} [\begin{matrix} 1 & 1 & \dots & 1 \\ 1 & e^{- j \frac{2 π}{N}} & \dots & e^{- j \frac{2 π (N - 1)}{N}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & e^{- j \frac{2 π (N - 1)}{N}} & \dots & e^{- j \frac{2 π (N - 1) (N - 1)}{N}} \end{matrix}] [\begin{matrix} e^{- j ϕ (0)} & 0 & \dots & 0 \\ 0 & e^{- j ϕ (1)} & \dots & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 \\ 0 & \dots & 0 & e^{- j ϕ (N - 1)} \end{matrix}] [\begin{matrix} 1 & 1 & \dots & 1 \\ 1 & e^{j \frac{2 π}{N}} & \dots & e^{j \frac{2 π (N - 1)}{N}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & e^{j \frac{2 π (N - 1)}{N}} & \dots & e^{j \frac{2 π (N - 1) (N - 1)}{N}} \end{matrix}] [\begin{matrix} Y (0) \\ Y (1) \\ ⋮ \\ Y (N - 1) \end{matrix}]

Defining

P = {[P (0), P (1), ..., P (k), ..., P (N - 1)]}^{T}

as the spectral decomposition vector of the conjugated optical phase noise, whose kth element

P (k)

is:

P (k) = \frac{1}{\sqrt{N}} \sum_{n = 0}^{N - 1} e^{- j ϕ (n)} e^{- j \frac{2 π n k}{N}}

Equation (3) can be rewritten as:

F Φ^{*} F^{H} Y = \frac{1}{\sqrt{N}} [\begin{matrix} Y (0) & Y (N - 1) & Y (N - 2) & \dots & Y (1) \\ Y (1) & Y (0) & Y (N - 1) & \dots & Y (2) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ Y (N - 1) & Y (N - 2) & Y (N - 3) & \dots & Y (0) \end{matrix}] [\begin{matrix} P (0) \\ P (1) \\ ⋮ \\ P (N - 1) \end{matrix}]

Since the Wiener phase noise has a Lorentzian spectrum [18], the optical phase noise can be approximated with a few lower order harmonics, i.e., the principal components. Thus we artificially set the coefficients of higher order harmonics to zero, i.e., $P (k) = 0,$ where $L_{1} < k < N - L_{2},$ $0 \leq L_{1} (L_{2}) ≪ N - 1.$ Note that the coefficients of higher order harmonics are located in the middle of the vector P. We now define a principal components vector P _q of length $q = L_{1} + L_{2} + 1$ and a cyclic matrix T _q of size N × q as:

P_{q} = [\begin{matrix} P (0) \\ ⋮ \\ P (L_{1}) \\ P (N - L_{2}) \\ ⋮ \\ P (N - 1) \end{matrix}], T_{q} = [Y, Y_{1}, \dots, Y_{L_{1}}, Y_{N - L_{2}}, \dots, Y_{N - 2}, Y_{N - 1}], where L_{2} = ⌊ \frac{q}{2} ⌋, Y_{k} = \frac{1}{\sqrt{N}} [\begin{matrix} \begin{matrix} Y (N - k) \\ ⋮ \\ Y (N - 1) \end{matrix} \\ \begin{matrix} Y (0) \\ ⋮ \\ Y (N - k - 1) \end{matrix} \end{matrix}]

The operator

⌊ A ⌋

denotes the nearest integer less than A. Equation (2) becomes:

X \approx T_{q} P_{q} + Δ ζ

The modeling error induced by the truncation of P is included in the second term

Δ ζ

on the right side of Eq. (7). The principal components vector

P_{q}

is then estimated recursively using LS criteria. For q > 1, Eq. (7) is rearranged as:

X - T_{q - 1} P_{q - 1} \approx Y_{i_{q}} P (i_{q}) + Δ ζ, where i_{q} = {\begin{matrix} L_{1}, q ​ i s e v e n \\ N- L_{2}, q i s odd \end{matrix} ​ ​

Let

l_{m}

(

0 \leq m \leq N_{p} - 1

,

N_{p}

is the number of pilots) denote the index of pilots in each OFDM symbol, and the corresponding

N \times N

selection matrix is

S = {[e_{l_{0}}, e_{l_{1}}, \dots, e_{l_{N_{p} - 1}}]}^{T}

, where

e_{l_{m}}

is a

N \times 1

vector with the

l_{m}

element equals to 1 and the other elements equal to 0. With the above notations, the

P (i_{q})

in Eq. (8) can be estimated using the pilots as follows:

\hat{P} (i_{q}) = \frac{{(S Y_{i_{q}})}^{H} S X_{q}}{{(S Y_{i_{q}})}^{H} (S Y_{i_{q}})}, where X_{q} = X - T_{q - 1} {\hat{P}}_{q - 1} \approx X_{q - 1} - Y_{i_{q - 1}} \hat{P} (i_{q - 1})

For q = 1, the estimated first order component

\hat{P} (0)

can be obtained as:

\hat{P} (0) = \frac{{(S Y)}^{H} S X}{{(S Y)}^{H} (S Y)}

A refinement procedure is executed after Eq. (10) by eliminating the impact of amplitude noise on

\hat{P} (0)

. The updated

{\hat{P}}_{1} = \hat{P} (i_{1})

is expressed as:

\hat{P} (i_{1}) = \sqrt{N} \exp (j \cdot a n g l e (\hat{P} (0)))

Therefore, the principal components vector

{\hat{P}}_{q}

of q components can be estimated recursively using Eq. (9) with an initialized

{\hat{P}}_{1}

of one component from Eq. (11). The corresponding OFDM symbol after optical phase suppression and decision can be expressed as

\hat{X} = Θ (T_{q} {\hat{P}}_{q})

where the operator

Θ (\cdot)

stands for symbol decision operation.

To illustrate the feasibility of the proposed algorithm, Fig. 1(b) shows the normalized spectrum $| P (k) |$ under a combined laser linewidth of 500 kHz and a symbol period of 28.8 ns with perfect estimation (black) and the corresponding estimation result (red) with the R-PCE (q = 5) algorithm. As most of the non-trivial components are around few lower order harmonics, a small q is expected to be sufficient for optical phase noise suppression. Due to the Lorentzian spectrum [18], the selection of i_k (1≤ k ≤ q) as in Eq. (8) is optimal in the sense of minimizing the approximation error in Eq. (9) during the recursion. To show the necessity of properly choosing i_k during recursion, Fig. 1(c) depicts the bit error ratio (BER) results versus optical signal-to-noise ratio (OSNR) under two different sets of i_k when calculating ${\hat{P}}_{6}$ , {i₁,i₂,i₃,i₄,i₅,i₆}_I = {0,1,2,3,N-2,N-1} and {i₁,i₂,i₃,i₄,i₅,i₆}_II = {0,1,N-1,2,N-2,3}. The algorithm employing set II (solid lines) obviously outperforms the one with set I (dashed lines) under all three different laser linewidths (500 kHz, 1 MHz and 1.5 MHz). Note that we use {i₁,i₂,i₃,i₄,i₅} = {0,1,N-1,2,N-2} in Fig. 1(b). The above R-PCE algorithm can be modified to a decision-aided (DA) algorithm, i.e., DA-R-PCE. As shown by the red dashed line in Fig. 1(a), the decision output from Eq. (12) is fed back to Eq. (9)-(11) , and the corresponding selection matrix S is set as the unit matrix of size-N. To make full use of the DA process, the size of the principal components vector (q ₂) in the DA estimation stage is chosen to be larger than that (q ₁) in the previous PA estimation stage.

3. Complexity analysis

The complexity of the proposed R-PCE algorithm in terms of required number of complex-valued multiplications (RNCM) is analyzed in Table 1 . For comparison, the RNCM of the CPEC, LI, OBE and DA-R-PCE algorithms are also listed. The computational complexity of R-PCE algorithm is mainly induced by calculating ${\hat{P}}_{q}$ [Eq. (9) and Eq. (10)] and $\hat{X}$ [Eq. (12)], whose RCNM are $O (2 q N_{p})$ and $O (q N)$ , respectively. For the DA-R-PCE, an additional RCNM of $O (3 q_{2} N)$ is needed for DA processing. Nevertheless, the RNCM of R-PCE and DA-R-PCE algorithms scale linearly with the number of principal components q. As indicated in Eq. (9) and Eq. (10), the proposed R-PCE algorithm with q = 1 is equivalent to the CPEC [14]. Compared with the computational efficient algorithm LI with Q of 8 [8], the proposed R-PCE algorithm with a moderate value of q shares a similar level of complexity. For instance, the RNCM with N = 16N_p for the R-PCE (q = 7) and LI (Q = 8) are about 7.88N and 8.17N, respectively.

Table 1. The required number of complex-valued multiplications (RNCM) in different algorithms

View Table

Unlike the OBE algorithm [9] which requires intensive computational effort for basis expansion (e.g. L times of size-N DFT are needed in calculating $C = F d i a g {η} B^{*}$ for L principal components estimation in the OBE, where $η$ is the channel equalized time domain vector of length N and $B^{*}$ is the $N \times L$ conjugate orthogonal basis matrix), the proposed R-PCE algorithm introduces a recursive approach to the estimation of multiple principal components, which do not need any multiplications for domain transformation or to obtain the cyclic matrix T _q. Consequently, the total complexity of the proposed R-PCE algorithm is substantially lower than the OBE algorithm with the same number of principal components (i.e., L = q). Extended from the computationally efficient R-PCE algorithm, the two-stage DA-R-PCE algorithm with a moderate q ₁ (q ₂) is found to have a computational complexity similar to OBE with small L. For instance, the RNCM with N = 16N_p for the R-PCE with q = 3, R-PCE with q = 4, DA-R-PCE (q ₁ = 5, q ₂ = 6) and OBE (L = 3) are about 3.38N, 4.5N, 23.63N and 23.42N, respectively. Therefore, complexity reduction with factors of about 7 and 5 are achieved with R-PCE with q = 3 and R-PCE with q = 4, respectively, when compared with the OBE (L = 3).

4. Monte-Carlo simulation and performance analysis

To investigate the performance of the proposed algorithm in the CO-OFDM system, Monte-Carlo simulation is conducted using Matlab and VPItransmissionmaker. The total number of subcarriers is 256, of which 10 central subcarriers and 30 edge subcarriers are set as guard subcarriers. A cyclic prefix of length 32 is inserted into every symbol. ISFA channel estimation with 5 subcarriers for averaging is employed [16]. Electrical pre-distortion [19] at the transmitter is adopted to relieve the effect of the nonlinearity of the optical I/Q modulator. The sampling rate is chosen to be 10 GS/s to emulate a metro/access system [4,5 ], which typically has a large linewidth-symbol time product. Note that for the same linewidth-symbol time product $σ_{u}^{2}$ , a shorter symbol time (hence a higher sampling rate $1 / T_{s}$ for a constant oversampling ratio) implies a larger tolerable laser linewidth $β$ . 16 pilot subcarriers are uniformly inserted into the signal band. The total data rates are 13.89 Gb/s and 27.78 Gb/s for QPSK and 16QAM systems, respectively.

Figure 2 shows the BER result versus OSNR of the QPSK/16QAM CO-OFDM system with the R-PCE under certain laser linewidths (QPSK: 2 MHz, and 16QAM: 500 kHz). For systems with both modulation formats, the performances improved when q increased from 1 to 3. Compared to R-PCE with q = 1, which is equivalent to the conventional CPEC algorithm, the required OSNRs at a BER of 3.8 × 10⁻³ for the R-PCE algorithms with q = 3 and q = 2 decrease by 2.01 dB and 1.31 dB in the simulated QPSK system and 2.31 dB and 1.52 dB in the simulated 16QAM system, respectively. Further performance improvement from increasing q becomes trivial when q is sufficiently large (i.e., q ≥ 3 in Fig. 2).

Fig. 2 BER versus OSNR for the R-PCE algorithm in the QPSK/16QAM CO-OFDM system under combined laser linewidths of (a) 2 MHz and (b) 500 kHz.

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To quantize the system’s tolerance of optical phase noise, Fig. 3(a) depicts the required OSNR at a BER of 3.8 × 10⁻³ under various laser linewidths in the back-to-back 16QAM CO-OFDM system. For performance comparison, two conventional phase noise suppression algorithms (CPEC and LI) and one other high-performance ICI mitigation algorithm (OBE) are also listed in Fig. 3. The CPEC and R-PCE algorithms (q = 1) have identical performances as expected. The performance of R-PCE with q = 2 is found to be similar to LI (Q = 8). With an OSNR of 20 dB, the combined laser linewidth tolerances are about 584 kHz, 761 kHz, 1255 kHz, 1218 kHz and 1344 kHz for CPEC, LI (Q = 8), OBE (L = 3), R-PCE with q = 3 and R-PCE with q = 4, respectively. Thus, compared with the conventional CPEC and LI algorithms, the R-PCE with only a small q ( = 3) evidently promotes the system’s laser linewidth tolerance under a moderate OSNR. Note that R-PCE with q = 4 has a better performance than OBE (L = 3), yet its complexity is significantly lower than the latter one as analyzed in the previous section.

Fig. 3 (a) Required OSNR at a BER of 3.8 × 10⁻³ in the back-to-back 16QAM CO-OFDM system with various phase noise suppression algorithms. Unless otherwise marked, all BER results are obtained with ISFA channel estimation (the number of training symbols N_t = 2). (b) The contour plots of the receiver sensitivity at a BER of 3.8 × 10⁻³ for the R-PCE algorithms with respect to different combined laser linewidths and different numbers of principal components.

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To show the impact of imperfect channel equalization in system with the R-PCE optical phase noise suppression algorithm, the required OSNR at a BER of 3.8 × 10⁻³ for the R-PCE (q = 4) under three different channel estimation cases, i.e., the ideal channel equalization (CE), ISFA-based CE with the number of training symbols N_t = 2 and N_t = 12, are shown in Fig. 3(a). The results show that, compared with the ideal CE case, the performance degradation due to imperfect channel estimation with ISFA (N_t = 2) is small when the combined laser linewidth is moderate (e.g., ~0.39 dB OSNR penalty in system under a combined laser linewidth of 500 kHz). The performance of system employing R-PCE algorithm is improved with a higher channel estimation accuracy at the expense of a larger overhead for training symbols (e.g., when the number of training symbols N_t increases from 2 to 12).

To investigate the dependence of R-PCE’s performance on q, Fig. 3(b) shows the contour plots of receiver sensitivity (OSNR in dB) at a BER of 3.8 × 10⁻³ for the R-PCE with respect to different combined laser linewidths and different numbers of principal components q in the 16QAM CO-OFDM system. As shown in Fig. 3(b), the minimum required OSNR is obtained with an optimal value of q (q _optimal) for a certain laser linewidth (i.e. each single curve). Figure 3(b) also shows that q _optimal becomes larger with an increased laser linewidth. The choose of q affects the error in two processes: one is the modeling error in Eq. (7), where the coefficients of higher order components are artificially set to zero values, the other one is the cascaded estimation error in Eq. (9). To illustrate the dependence of modeling error on q in the 16QAM system, Fig. 4(a) shows the mean squared modeling error versus the number of principal components q. The modeling error keeps dropping as the number of principal components q increases, which implies that a larger q can reduce the modeling error. However, the mean squared cascaded estimation error using Eq. (9) increases with a larger q, as shown in Fig. 4(b). Therefore, the specific value of q _optimal is jointly determined by the above two processes. In practice, the number of steps in recursion, which equals to q, can be set to a moderate value with a small performance degradation. For example, in Fig. 3(b), the maximal OSNR penalty when q is set to 4 for system under a combined laser linewidth between 200 kHz and 1 MHz is about 0.35 dB compared with the case with an optimal q.

Fig. 4 (a) Mean squared modeling error versus the number of principal components q. (b) Mean squared cascaded estimation error versus the number of principal components q.

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Figure 5 shows the performance of DA-R-PCE in the back-to-back 16QAM CO-OFDM system. Since two estimation stages are employed, the performance of DA-R-PCE depends on both values of q in these two stages, i.e., q ₁ of the PA estimation stage and q ₂ of the DA estimation stage. Similarly to R-PCE, the optimal q in both stages of DA-R-PCE increases when the laser linewidth increases. Nevertheless, optimal performance can still be achieved with a moderate q ₁ and q ₂. The required OSNR for the above system under a combined laser linewidth of 500 kHz with the DA-R-PCE when q ₁ = 3 and q ₂ = 5 is 15 dB, which is about 1 dB lower than the R-PCE with q = 3. At an OSNR of 17.5 dB, the 16QAM CO-OFDM system with the DA-R-PCE when q ₁ = 7 and q ₂ = 10 achieves a 2 MHz laser linewidth tolerance which is about one time larger than the R-PCE with q _optimal of 5.

Fig. 5 The contour plots of receiver sensitivity at a BER of 3.8 × 10⁻³ with respect to different q ₁ in the PA estimation stage and different q ₂ in the DA estimation stage of DA-R-PCE under combined laser linewidths of (a) 500 kHz and (b) 2 MHz.

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In addition to the back-to-back scenario, the system’s BER performance after 4 spans of 80 km standard single model fiber (SSMF) transmission is also investigated (attenuation: 0.2 dB/km, chromatic dispersion: 17 ps/nm/km, nonlinear coefficient: 1.3 W⁻¹km⁻¹, optimized optical launch power: −7 dBm). Each span employs one EDFA (noise figure: 6dB, gain: 16 dB) to fully compensate for the fiber loss but with no optical dispersion compensation.

After 320 km SSMF transmission, the system’s BER under different laser linewidths with different algorithms are shown in Fig. 6 . Similarly to the back-to-back scenario, the R-PCE algorithm promotes the system’s laser linewidth tolerance significantly more compared to CPEC and LI. Though the R-PCE with q = 3 and q = 4 have much lower complexities than the OBE (L = 3), their laser linewidth tolerances are found to be similar (~1.5 MHz). The largest laser linewidth tolerance of 3 MHz is achieved with DA-R-PCE when q ₁ = 7 and q ₂ = 10. For the DA-R-PCE with q ₁ = 5 and q ₂ = 6, which has a complexity level similar to the OBE (L = 3), its laser linewidth tolerance at a BER of 3.8 × 10⁻³ is 2.6 MHz, which is 1.7 times that of the latter algorithm.

Fig. 6 BER performance of the 16QAM CO-OFDM system after 320 km SSMF transmission with various phase noise suppression algorithms versus different combined laser linewidths.

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5. Conclusion

We propose and theoretically derive a novel optical phase noise suppression algorithm, R-PCE, based on recursive principal components elimination. The Monte-Carlo simulation results show that the performance of the R-PCE algorithm with only a small q ( = 3) is evidently superior to conventional algorithms such as CPEC and LI. Complexity analysis shows that the required number of complex-valued multiplications of R-PCE increases linearly with the number of principal components q. Compared with the OBE (L = 3), the R-PCE algorithms with q = 3 and q = 4 are shown to have comparable laser linewidth tolerance, but with reduction of RNCM by factors of about 7 and 5, respectively. Decision-aided R-PCE is also investigated and the results verify that a 70% increase in laser linewidth tolerance is achieved after 320 km SSMF transmission by DA-R-PCE with q ₁ = 5 and q ₂ = 6 compared with the OBE (L = 3) that has a similar complexity. Numerical results show that optimal performance is achieved with a moderate q for both the R-PCE algorithm and its decision-aided version, which is beneficial for low-complexity hardware implementation.

Acknowledgments

This work was supported in part by the China Postdoctoral Science Foundation under Grant 2013M531868, in part by the Young Faculty Research Fund of SCNU under Grant 13KJ04, and in part by the Guangdong Innovative Research Team Program under Grant 201001D0104799318.

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Algorithm	Operation Category	Size of Operation	Number of Operation	Required Number of Complex-valued Multiplications(RNCM)
CPEC [14]	-	-	-	$O (2 N_{p} + N)$
LI [8]	DFT/IDFT	$Q$	1	$O (\frac{Q}{2} l o g_{2} Q)$
LI [8]	Other Operations	-	-	$O (2 N_{p} + Q N)$
OBE [9]	Matrix Inversion	$L * L$	1	$O (L^{3})$
	DFT/IDFT	$N$	$L + 1$	$O (\frac{N (L + 1)}{2} \log_{2} N)$
	Other Operations	-	-	$O (L N_{p} +2 L^{2} N_{p} +2 L N)$
R-PCE	Matrix Inversion	-	-	-
	DFT/IDFT	-	-	-
	Other Operations	-	-	$O (2 q N_{p} + q N)$
DA-R-PCE	Matrix Inversion	-	-	-
	DFT/IDFT	-	-	-
	Other Operations	-	-	$O (2 q_{1} N_{p} + q_{1} N+3 q_{2} N)$

Low-complexity optical phase noise suppression in CO-OFDM system using recursive principal components elimination

Abstract

1. Introduction

2. Theory of the proposed optical phase noise algorithm

3. Complexity analysis

4. Monte-Carlo simulation and performance analysis

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (12)

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