Linearly interpolated sub-symbol optical phase noise suppression in CO-OFDM system

Xuezhi Hong; Xiaojian Hong; Sailing He

doi:10.1364/OE.23.004691

1. Introduction

Coherent optical orthogonal frequency division multiplexing (CO-OFDM) has been widely regarded as a promising technology for high-speed, i.e., Terabit/s and beyond, transmission systems [1–3]. However, the multicarrier based CO-OFDM system is more sensitive to optical phase noise compared to its single carrier counterpart [4,5]. Both common phase error (CPE) and inter-carrier-interference (ICI) are induced by the optical phase noise, and significantly degrade the CO-OFDM system’s performance. To suppress such distortions, several digital signal processing (DSP) algorithms based on either CPE compensation (CPEC) [6,7] or ICI mitigation have been proposed [8–14]. Symbol rotation can be corrected using a CPEC scheme in which the phase noise is assumed to be constant in one CO-OFDM symbol. However, the ICI induced by the residual time-varying phase noise after CPEC could still be a major issue in the CO-OFDM system with large linewidth-symbol time product or using high-order modulation formats, e.g. some coherent access systems which have a long symbol-time [15] and prefer cost-effective lasers [16], and some long-reach access/metro systems which employ high-order modulation formats [17], etc. Partial ICI suppression is achieved by temporal linear interpolation (LI) of CPE in the adjacent symbols [9]. Time domain orthogonal basis expansion (OBE) has been proposed to suppress ICI [10,11]. Since ICI is directly related to the time-varying feature of optical phase noise, the mitigation of ICI can be enhanced by increasing the temporal resolution of the phase estimation. High temporal resolution phase noise suppression methods, which in the time domain divide one OFDM symbol into several sub-blocks (in essence, sub-symbols), have been proposed for wireless [18,19] and optical systems [12–14]. A time domain blind ICI mitigation algorithm (BL-ICI) employing temporal partition is proposed for CO-OFDM [12]. We have proposed one ICI mitigation algorithm for CO-OFDM based on sub-symbol CPEC (SCPEC) [13], in which the CPE of every sub-symbol is estimated and then used for ICI mitigation. The phase noise suppression performance of such ICI mitigation algorithms is better than the CPEC algorithm at the expense of a higher complexity, e.g., a larger number of multiplications.

To further promote the CO-OFDM system’s tolerance to the optical phase noise without raising the DSP complexity level, we propose in this paper a novel phase noise suppression algorithm, LI-SCPEC, using phase linear interpolation and sub-symbol processing. By combining phase linear interpolation and sub-symbol processing, a better tracking of the carrier phase is achieved in our algorithm than several other ICI mitigation algorithms including SCPEC, OBE and BL-ICI. Monte-Carlo simulations of a 16QAM/32QAM CO-OFDM system show that the new algorithm significantly outperforms BL-ICI with the same number of sub-symbols in both back-to-back and 800 km SSMF transmission cases. Computationally intensive matrix inversion operations required in OBE and BL-ICI as well as cyclicity recovery required in [18] are avoided in the proposed algorithm. The complexity analysis shows that the required number of complex-valued multiplication in LI-SCPEC is independent of the number of sub-symbols.

2. Linearly interpolated sub-symbol phase noise suppression algorithm

The simulated CO-OFDM system model in the presence of optical phase noise is depicted in Fig. 1. The received signal is distorted by the imperfections in the channel (fiber), transmitter (laser 1), and receiver (laser 2), e.g., chromatic dispersion (CD), fiber nonlinearity (NL), laser phase noise and amplified spontaneous emission (ASE) noise from optical amplifiers. For simplicity, only a single polarization system is analyzed in this paper. Assuming that both timing synchronization and frequency tracking [20] are perfectly performed, the received mth frequency domain OFDM symbol $R_{m} = {[R_{m} (0), R_{m} (1), ..., R_{m} (N - 1)]}^{T}$ after cyclic prefix (CP) removal and discrete Fourier transform (DFT) can be described as:

R_{m} = F r_{m} = H F Φ_{m} F^{H} X_{m} + W_{m}

where

r_{m}

is the mth temporal sample, N is the number of the subcarriers, and superscripts (·)^T and (·)^H denote transpose and Hermitian transpose, respectively.

X_{m} = {[X_{m} (0), X_{m} (1), ..., X_{m} (N - 1)]}^{T}

represents the mth transmitted OFDM symbol in the frequency domain and its kth element

X_{m}^{} (k)

is the data modulated on the kth subcarrier.

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

and F^H are the DFT and IDFT transformation matrix, respectively. The kth row of F is

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

. W_m denotes the ASE noise in the optical fiber link. H is the

N \times N

channel response matrix in the frequency domain, which is assumed to be constant over several CO-OFDM symbols and can be estimated periodically using training symbols [21].

Φ_{m} = d i a g {{[e_{}^{j ϕ_{m} (0)}, e_{}^{j ϕ_{m} (1)}, ..., e_{}^{j ϕ_{m} (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_{m} (k)}

of Φ_m represents the random phase modulation caused by laser phase noise on the kth temporal sample

x_{m} (k)

of the mth OFDM symbol. Laser phase noise is modeled as a Wiener process [22], i.e.,

ϕ_{m}^{} (k) = ϕ_{m}^{} (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 symbol period and β is the two-side 3-dB bandwidth of the laser phase noise, i.e., the combined laser linewidth of laser1 and laser 2.

Fig. 1 Simulation model of the CO-OFDM system in the presence of phase noise. CP: cyclic prefix, TS: training symbols, Mod.: modulator, CD: chromatic dispersion, NL: nonlinearity, Sync.: synchronization, FOC: frequency offset compensation, CP Remv.:CP removal, Channel Equ.: channel equalization.

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The principle of the proposed sub-symbol optical phase noise suppression algorithm is shown in Fig. 2(a). The proposed algorithm includes two stages: (I) preparation of “transmitted” temporal samples and (II) sub-symbol phase noise mitigation with reference samples from stage I.

Fig. 2 (a) Block diagram of the proposed phase noise suppression algorithm, and (b) one realization of the real carrier phase (black) and its estimations at stage I: CPEC (red) and LI (green), and stage II: LI-SCPEC (blue) (number of sub-symbols N_B = 4).

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In stage I, we first obtain the common phase error ${\bar{ϕ}}_{m}$ of the mth OFDM symbol using least-squares (LS) estimation [5]. Then the carrier phase is estimated via linear interpolation of the three adjacent symbols’ CPE ${\bar{ϕ}}_{m}$ as [9]:

{\hat{ϕ}}_{m} (k) = {\begin{matrix} (k - \frac{N - N_{cp}}{2}) \frac{{\bar{ϕ}}_{m} - {\bar{ϕ}}_{m - 1}}{N + N_{cp}} + {\bar{ϕ}}_{m}, 0 \leq k < \frac{N - N_{cp}}{2} \\ - (k - \frac{N - N_{cp}}{2}) \frac{{\bar{ϕ}}_{m} - {\bar{ϕ}}_{m + 1}}{N + N_{cp}} + {\bar{ϕ}}_{m}, \frac{N - N_{cp}}{2} \leq k < N \end{matrix}

The phase noise estimation

{\hat{Φ}}^{I}_{m}

for stage I can be expressed as

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

The sample

{\hat{X}}_{m}

after channel equalization and phase correction with the corresponding

{\hat{Φ}}^{I}_{m}

is:

{\hat{X}}_{m} = F {({\hat{Φ}}_{m}^{I})}^{*} F^{H} {\hat{H}}^{- 1} R_{m} = X_{m} + Δ ξ

where the superscript

{(\cdot)}^{*}

denotes the complex conjugation operation,

\hat{H}

is the estimated channel response, and

Δ ξ

is the residual error after channel equalization and phase correction. The estimated “transmitted” temporal sample

a_{m}

is obtained after a tentative decision on the corresponding

{\hat{X}}_{m}

and one IDFT operation:

a_{m} = {[a_{m} (0), a_{m} (1), ..., a_{m} (N - 1)]}^{T} = F^{H} Q ({\hat{X}}_{m})

where

Q (\cdot)

is the decision operation performed in a data slicer. Then

a_{m}

is passed to stage II and serves as the reference for the following sub-symbol phase estimation procedure.

The time-domain channel equalized OFDM symbol $y_{m} = F^{H} {\hat{H}}^{- 1} R_{m}$ and $a_{m}$ from stage I are partitioned into N_B sub-symbols:

y_{m} = {[y_{m, 0}^{}, y_{m, 1}^{}, ..., y_{m, N_{B} - 1}]}^{T}, a_{m} = {[a_{m, 0}^{}, a_{m, 1}^{}, ..., a_{m, N_{B} - 1}^{}]}^{T}

The nth sub-symbol of

y_{m}

and

a_{m}

can be expressed as:

\begin{array}{l} y_{m, n}^{} = {[y_{m}^{} (n l + 0), y_{m}^{} (n l + 1), ..., y_{m}^{} (n l + l - 1)]}^{T} \\ a_{m, n}^{} = {[a_{m}^{} (n l + 0), a_{m}^{} (n l + 1), ..., a_{m}^{} (n l + l - 1)]}^{T} \end{array}

respectively, where n $\in$ [0,N_{B $-$}1], and l = N/N_B is the length of one sub-symbol. Let

{\bar{ϕ}}_{m, n}^{}

denote the CPE of the nth sub-symbol of the mth OFDM symbol, and it can be estimated by employing the LS criterion:

{\bar{ϕ}}_{m, n}^{} = a n g l e {\frac{y_{m, n}^{T} a_{m, n}^{*}}{a_{m, n}^{T} a_{m, n}^{*}}} = a n g l e {\frac{\sum_{k = 0}^{l - 1} y_{m}^{} (n l + k) {[a_{m}^{} (n l + k)]}^{*}}{\sum_{k = 0}^{l - 1} {| a_{m}^{} (n l + k) |}^{2}}}

The phase noise inside one sub-symbol is assumed to be constant. Therefore, the final phase noise estimation for the mth OFDM symbol can be expressed as

{\hat{Φ}}^{I I}_{m} = d i a g {{[e_{}^{j {\hat{ϕ}}_{m} (0)}, e_{}^{j {\hat{ϕ}}_{m} (1)}, \dots, e_{}^{j {\hat{ϕ}}_{m}^{} (n l + k)}, \dots, e_{}^{j {\hat{ϕ}}_{m} (N - 1)}]}^{T}}, {\hat{ϕ}}_{m}^{} (n l + k) ≅ {\bar{ϕ}}_{m, n}^{}

where k $\in$ [0,l $-$ 1] and n $\in$ [0,N_{B $-$}1]. The phase recovered frequency domain OFDM symbol is obtained via phase correction using the estimated phase noise:

T_{m} = F {({\hat{Φ}}^{I I}_{m})}_{}^{*} y_{m}

Figure 2(b) shows one realization of the real carrier phase (black) and its estimation in stage I (CPEC: red, LI: green) and stage II (LI-SCPEC: blue, number of sub-symbols N_B = 4). Though the LI procedure in stage I is similar to [9], the ICI mitigation in the proposed algorithm is done by sub-symbol processing in stage II, which is quite different from [9]. The sub-symbol processing provides a better approximation of the carrier phase with higher temporal resolution. According to sampling theory, the higher temporal resolution of phase estimation is beneficial to ICI mitigation as the higher order components of ICI can be tracked.

3. Monte-Carlo simulation and performance analysis

We conduct Monte-Carlo simulation to investigate the performance of the proposed algorithm using Matlab and VPItransmissionmaker. The laser phase noise is modeled as a Wiener process [22] in our simulation. The DFT/IDFT size is chosen to be 256. 10 guard subcarriers are added at the middle of the band, and 30 guard subcarriers are added and allocated at both sides of the band, which provides an oversampling ratio of 1.185 to relieve the design of the anti-aliasing low-pass filter. Pilot subcarriers are employed and uniformly distributed in the band. The CP of length 32 is inserted into every data symbol. ISFA channel estimation with 5 subcarriers for averaging is employed [21]. The sampling rate is 10 GS/s, which results in a symbol period of 28.8 ns. Optical phase noise of the laser is added at both the transmitter and receiver. Pre-distortion [23] is employed at the transmitter to minimize the effect of the nonlinearity of the optical I/Q modulator. More than 2000 symbols are simulated in every realization for BER calculation, which corresponds to ~2²⁰ (~2²¹) bits for the 16QAM (32QAM) system. The sampling rate is chosen to be 10 GS/s to emulate a metro/access system with low symbol rate. The sensitivity of a CO-OFDM system to the laser phase noise can be represented by the linewidth-symbol time product [4]. For the same linewidth-symbol time product, a shorter symbol-time (hence a higher sampling rate with a specific oversampling ratio) implies a larger tolerable laser linewidth.

We first investigate the accuracy of phase noise estimation in different procedures of our algorithm. A noiseless and all-pass channel is used here. Figure 3(a) shows the possibility density functions (PDFs) of the residual phase error (RPE) in different procedures of the algorithm under a combined laser linewidth of 250 kHz for 25 different phase realizations. Compared to CPEC (red) and LI (green) in stage I, the PDF of RPE is clearly tightened in stage II (blue). Since the means of RPE are close to zero for all three distributions, the mean square error (MSE) of phase estimation can be approximated by the variance. Figure 3(b) shows the MSE of phase estimation at different stages for combined laser linewidths ranging from 0 kHz to 1 MHz. Due to the feed-forward structure, the MSE of the phase estimation drops when signals pass through from CPE estimation (red) to LI (green) and to sub-symbol processing (blue). The MSE reduction in stage II (i.e. from LI to sub-symbol processing) is larger than that in stage I (i.e. from CPE estimation to LI), and the difference increases for larger laser linewidths, which implies that stage II contributes more to the overall phase estimation accuracy improvement than stage I. To demonstrate the effectiveness of sub-symbol processing, we also include the MSE of decision-feedback (DF-) LI in Fig. 3(b). The DF-LI uses the same stage I procedure as the proposed algorithm while employing LI phase estimation other than sub-symbol processing in stage II. The MSE of LI-SCPEC is considerably lower than DF-LI, which verifies the superiority of sub-symbol processing.

Fig. 3 (a) Possibility density functions of residual phase error in different procedures of the algorithm (combined laser linewidth = 250 kHz) and (b) mean squared error (MSE) of phase estimation in different procedures for combined laser linewidth ranging from 0 kHz to 1 MHz.

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For a channel with ASE noise, the number of pilot subcarriers N_p must be increased in order to reduce the effect of ASE noise on phase estimation. It is important to find an appropriate N_p to achieve a satisfactory performance while maintaining a small spectrum overhead. Figure 4 shows the bit error rate (BER) performance of three different phase noise estimation algorithms (OBE, BL-ICI, LI-SCPEC) as a function of N_p at a fixed optical signal to noise ratio (OSNR) of 18 dB and combined laser linewidth of 500 kHz. Each BER result in Fig. 4 is obtained by measuring more than 2 × 10⁷ (i.e. ~2²⁴) bits. Since we are interested in systems with large laser linewidth and high-order modulation formats, the BL-ICI algorithm uses the average power method according to [12]. The OBE employs an expansion order L = 3 as suggested in [10]. The proposed LI-SCPEC (number of sub-symbols N_B = 2 or 4) has the least dependence on N_p and achieves a steady performance (variation of Log₁₀ (BER) < 0.05) for N_p > 5. The BL-ICI (N_B = 2 or 4) has a slower converging speed and performs steadily for N_p > 9. The OBE is much more sensitive to N_p than former ones. Though LI-SCPEC is more advantageous when N_p is small, to ensure a fair comparison, N_p is set to 10 in all simulations. Therefore, the effective data rate is 28.61 Gb/s (35.76 Gb/s) for the 16QAM (32QAM) system.

Fig. 4 BER performance versus number of pilot subcarriers.

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The performance of a 16QAM/32QAM CO-OFDM system under varying combined laser linewidths (200 kHz and 500 kHz) with the proposed LI-SCPEC algorithm (N_B = 2 or 4) in terms of BER versus OSNR is shown in Fig. 5. The simulation results of several previously proposed phase noise suppression algorithms including CPEC, LI, DF-CPEC, OBE (L = 3) and BL-ICI (N_B = 2 or 4) are listed for comparison. With the same number of sub-symbols N_B, LI-SCPEC outperforms BL-ICI in all cases. For a target BER (e.g. 3.8 × 10⁻³), the OSNR gain of the proposed algorithm depends on the modulation format and laser linewidth. Consider that a commercial external cavity laser can has a laser linewidth less than 100 kHz, Fig. 5(a) and Fig. 5(c) show the BER versus OSNR of the 16QAM/32QAM system under a combined laser linewidth of 200 kHz. Under such laser linewidth, LI-SCPEC with N_B = 4 and LI-SCPEC with N_B = 2 achieve the best and second best performance, respectively, for both the 16QAM system and 32QAM system. Compared with CPEC and LI, the OSNR gain obtained by the proposed algorithm with N_B = 4 are 0.96 dB and 0.69 dB (2.24 dB and 1.40 dB) in the 16QAM (32QAM) system, respectively. For the combined laser linewidth of 500 kHz, the LI-SCPEC with N_B = 4 still outperforms all other counterparts irrespective of the modulation formats. The performance of LI-SCPEC with N_B = 2 is similar to BL-ICI (N_B = 4) and better than OBE (L = 3) for the 16QAM system, while the former one performs worse than the later ones for the 32QAM system.

Fig. 5 BER versus OSNR for different phase noise suppression algorithms in a 16QAM/32QAM CO-OFDM system. (a) & (c) combined laser linewidth of 200kHz, (b) & (d) combined laser linewidth of 500kHz. The proposed algorithm with four sub-symbols outpforms all other listed algorithms.

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To quantize the 16QAM system’s tolerance of optical phase noise, Fig. 6(a) depicts the required OSNR at BER = 1.0 × 10⁻³ under various laser linewidths in the back-to-back system. For a combined laser linewidth of 300 kHz, the OSNR gain at such BER from LI-SCPEC (N_B = 2) over CPEC, LI, OBE (L = 3), and BL-ICI (N_B = 2) are about 2.77 dB, 1.27 dB, 0.86 dB and 0.66 dB, respectively. When the combined laser linewidth exceeds 600 kHz, the CPEC and LI fails to achieve BER = 1.0 × 10⁻³ within the OSNR range of interest here. For a combined laser linewidth of 600 kHz, the OSNR gain from LI-SCPEC (N_B = 4) over OBE (L = 3) and BL-ICI (N_B = 4) are 2.16 dB and 1.16 dB, respectively. With OSNR = 19 dB, the combined laser linewidth tolerance are about 886 kHz, 732 kHz, 634 kHz, 524 kHz and 575 kHz for LI-SCPEC with N_B = 4, BL-ICI with N_B = 4, LI-SCPEC with N_B = 2, BL-ICI with N_B = 2 and OBE (L = 3), respectively. Therefore, compared with other ICI mitigation algorithms, LI-SCPEC with N_B = 4 evidently promotes system tolerance. We also notice that the advantage of the LI-SCPEC algorithm over the previous SCPEC algorithm [13] tends to be larger when the laser linewidth increases. This attributes to a more accurate reference achieved with LI in stage I of LI-SCPEC than CPEC in the first stage of SCPEC.

Fig. 6 (a) Required OSNR at BER = 1.0 × 10⁻³ in the back-to-back system with various phase noise suppression algorithms; (b) BER versus optical launch power after 800 km SSMF transmission with combined laser linewidth of 500 kHz. Considerably larger optical phase noise tolerance is achieved with the proposed algorithm (N_B = 4).

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In addition to the back-to-back scenario, the system’s BER performance after 10 spans of 80 km SSMF transmission (attenuation: 0.2 dB/km, chromatic dispersion: 17 ps/nm/km, nonlinear coefficient: 1.3 W⁻¹km⁻¹) versus optical launch power has also been evaluated, as shown in Fig. 6(b). Each span employs one EDFA (noise figure of 6 dB, gain of 16 dB) to fully compensate the fiber loss but with no optical dispersion compensation. The lasers at both the transmitter and receiver have a linewidth of 250 kHz, i.e., the combined laser linewidth is 500 kHz. The LI-SCPEC with N_B = 4 consistently performs better than other algorithms for all simulated optical launch powers.

As shown in Fig. 6(a), the required OSNR is significantly reduced in the whole range of the simulated linewidth for LI-SCPEC when N_B increases from 2 to 4. To investigate the dependence of performance on N_B in the 16QAM system, Fig. 7(a) and Fig. 7(b) show the contour plots of the receiver sensitivity (OSNR in dB) at BER = 1.0 × 10⁻³ and 2.0 × 10⁻² for LI-SCPEC with respect to different combined laser linewidths and different numbers of sub-symbols N_B, respectively. For a certain laser linewidth, the optimal number of sub-symbols N_B are approximately the same in the two cases. We also found that the lower BER case has a wider range of N_B around the optimal value with negligible performance loss than the other case. Note that unlike BL-ICI, an odd N_B is feasible in our algorithm.

Fig. 7 Receiver sensitivity for the different combined laser linewidths and various number of sub-symbols N_B at (a) BER = 1.0 × 10⁻³ and (b) BER = 2.0 × 10⁻²; (c) BER results with different N_B for combined laser linewidths of 300 kHz, 450 kHz, and 600 kHz under OSNR = 15.8 dB.

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Two observations are made: (1) for each line (i.e. a fixed receiver sensitivity), the slope changes from positive to negative as N_B increases (i.e. from left to right), and (2) line density becomes higher as the laser linewidth increases (i.e. from bottom to top). The first observation means that an optimum N_B exists under a certain receiver sensitivity (OSNR) for achieving the largest phase noise tolerance. Observation (2) implies that for a fixed N_B and the same amount of extra OSNR_, the increment of the laser linewidth tolerance tends to decrease when the laser linewidth increases. Figure 7(c) shows the BER performance with different N_B for combined laser linewidths of 300 kHz, 450 kHz and 600 kHz under OSNR = 15.8 dB. For each case shown in Fig. 7(c), a certain value of N_B is required to achieve the lowest BER.

An insight of the optimization of N_B is given. The performance of the proposed algorithm is constrained by three factors as shown in Eq. (8): (i) error in modeling, where the continuously varying carrier phase is “artificially” assumed to be constant inside one sub-symbol, (ii) error in phase estimation caused by non-zero additive ASE noise in y_m, and (iii) error in phase estimation caused by an inaccurate reference sample a_m due to the decision errors in the first stage of the algorithm. To illustrate the dependence of modeling error on N_B in the 16QAM system, Fig. 8(a) shows the MSE of phase estimation versus N_B without decision error in (iii) and with or without ASE noise (ii). In the ASE noise-free case, the MSE keeps dropping as the temporal resolution of the phase estimation increases. It implies that a large N_B is preferred to reduce the modeling error. However, in the case where ASE noise exists, the MSE is found to rise again when N_B exceeds a certain value, which implies that the size of the sub-symbol (i.e. l = N/N_B) need not be too small to average the effect of ASE noise in (ii). Figure 8(b) shows the mean squared phase difference between the outputs of Eq. (8) using either the ideal samples (decision error-free) or the samples affected by decision error in stage I as reference a_m. When decision error in (iii) exists, the derivation of phase estimation from the ideal case increases as N_B increases. Thus, by considering all three above factors in a system affected by ASE noise and non-zero optical phase noise (hence with decision error), the overall best phase estimation performance can only be achieved with a moderate N_B.

Fig. 8 (a) MSE of phase estimation versus number of sub-symbols in the case of no decision error in the first stage of the algorithm; (b) mean squared phase difference between outputs of Eq. (8) with either ideal samples or samples with decision error as the reference for phase estimation. An optimal and moderate N_B is required to achieve the best phase nosie suppression performance.

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4. Complexity analysis

The complexity of the proposed algorithm is analyzed in Table 1. For comparison, the computational complexities of the CPEC, OBE, BL-ICI and previously proposed frequency domain compensated algorithm FC-LI-SCPEC [14] are also considered. In general, optical phase noise suppression algorithms with high performance ICI mitigation (e.g. OBE, BL-ICI, FC-LI-SCPEC and the proposed algorithm) are more complicated than the CPEC or LI. If minor performance degradation is allowed, the complexity of the proposed algorithm can be further reduced by changing the LI procedure from the time domain with a DFT to the frequency domain with a small order Q as used in [9]. For hardware implementation, one can conduct further optimization of the algorithm to minimize resource usage (see e.g [24].).

Table 1. Complexity of different algorithms in terms of the required number of complex-valued multiplications

View Table

As shown in Table 1, the computationally intensive matrix inversion is completely avoided in our algorithm. By implementing the phase estimation and compensation in the time domain other than the frequency domain, the new algorithm needs less DFT/IDFT operations than FC-LI-SCPEC. Table 1 also summarizes the required number of complex-valued multiplications (RNCM) per symbol for optical phase noise suppression. The IDFT/DFT operation is considered to be radix-2 IFFT/FFT. Since both the number of matrix inversion operations and the number of DFT/IDFT operations in BL-ICI are dependent on N_B, the RNCM of BL-ICI increases rapidly as N_B increases. On the contrary, the RNCM of LI-SCPEC is independent of the number of sub-symbols N_B, which implies that choosing a large enough N_B to achieve a satisfactory performance is feasible. Since the majority of additional procedures compared to LI/CPEC are implemented with low complexity IFFT/FFT, the RNCM of our algorithm scales gracefully with the number of the subcarriers N and the number of pilot subcarriers N_p. Consequently, the proposed algorithm provides a better performance and yet its complexity is similar to OBE and lower than BL-ICI or FC-LI-SCPEC.

5. Conclusion

We propose a new optical phase noise suppression algorithm, LI-SCPEC, based on phase linear interpolation and sub-symbol processing. The Monte-Carlo simulation results show that the performance of the proposed algorithm with four sub-symbols is superior to several other ICI mitigation algorithms including LI, OBE, BL-ICI, and SCPEC. Performance analysis verifies that the optimal performance is achieved with a moderate N_B. Complexity analysis shows that the required number of complex-valued multiplications is independent of the number of sub-symbols N_B, which implies that a large enough N_B is feasible to achieve optimal performance without raising the multiplication complexity.

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 [5]	-	-	-	$O {(2N}_{p} +N)$
OBE [10]	Matrix Inversion	L*L	1	$O (L^{3})$
	DFT/IDFT	$N$	L + 1	$O (\frac{N (L + 1)}{2} \log_{2} N)$
	Other Operations	-	-	$O {(2NL+N}_{p} {L+2N}_{p} L^{2})$
BL-ICI [12]	Matrix Inversion	(N_B-1)*(N_B-1)	1	$O (\frac{{(N_{B} - 1)}^{3}}{4})$
	DFT/IDFT	$N$	$N_{B} +1$	$O (\frac{N}{2} {log}_{2} N)+ O (\frac{N_{B} N}{2} \log_{2} \frac{N}{N_{B}})$
	Other Operations	-	-	$\begin{array}{l} O {(3N}_{p} + 4 N) \\ + O (\frac{N}{8} {(7N}_{B}^{2} - {3N}_{B} +10)) \end{array}$
FC-LI-SCPEC [14]	Matrix Inversion	-	-	-
	DFT/IDFT	$N$	4	$O (2 N \log_{2} N)$
	DFT/IDFT	$\frac{N}{N_{B}}$	${2N}_{B}$	$O (N \log_{2} \frac{N}{N_{B}})$
	Other Operations	-	-	$O {(2N}_{p} + \frac{N}{4} +4N)$
LI-SCPEC	Matrix Inversion	-	-	-
	DFT/IDFT	N	4	$O (2 N \log_{2} N)$
	Other Operations	-	-	$O {(2N}_{p} + \frac{N}{4} +4N)$

Linearly interpolated sub-symbol optical phase noise suppression in CO-OFDM system

Abstract

1. Introduction

2. Linearly interpolated sub-symbol phase noise suppression algorithm

3. Monte-Carlo simulation and performance analysis

4. Complexity analysis

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (10)

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