## Abstract

A unified framework for phase noise suppression is proposed in this paper, which could be applied in any coherent optical block transmission systems, including coherent optical orthogonal frequency-division multiplexing (CO-OFDM), coherent optical single-carrier frequency-domain equalization block transmission (CO-SCFDE), etc. Based on adaptive modeling of phase noise, unified observation equations for different coherent optical block transmission systems are constructed, which lead to unified phase noise estimation and suppression. Numerical results demonstrate that the proposal is powerful in mitigating laser phase noise.

© 2011 OSA

## 1. Introduction

Intersymbol interference (ISI) caused by chromatic dispersion (CD) and polarization-mode dispersion (PMD) limits the transmission performance of high-speed long-haul optical communication systems. As very efficient options to deal with this problem, coherent optical orthogonal frequency-division multiplexing (CO-OFDM) [1, 2] and coherent optical single-carrier frequency-domain equalization block transmission (CO-SCFDE) [3, 4] have attracted wide interests recently which are powerful in mitigating ISI thanks to the block transmission structure where cyclic prefix (CP) is inserted in each block. Although CO-OFDM or CO-SCFDE could be totaly free of ISI if the length of CP is longer than that of channel spread, laser phase noise which induces time-variant rotation of signals, has been shown as an important impairment to the two coherent optical transmission schemes [5–8].

An effective phase noise suppression algorithm for the discrete Fourier transform (DFT) based CO-OFDM system is proposed in [9], which bases on expanding the phase noise by orthogonal basis. Due to the extra inter-carrier interference (ICI) suppression, the proposed approach outperforms traditional methods that are only designed for common phase error (CPE) compensation. As mentioned in [9], in order to derive the proposed algorithm, frequency domain transfer matrix is calculated by utilizing the close property of DFT under the Hadamard product. So except for DFT based CO-OFDM system, the phase noise suppression algorithm can be hardly adopted in other coherent optical block transmission systems e.g. the discrete cosine transform (DCT) based CO-OFDM system [10] and the CO-SCFDE system. Besides the orthogonal basis expansion based method in [9], a RF-pilot based method [11] and a matched filtering approach [12] have also been proposed to mitigate the phase noise induced ICI effect, however these methods are also designed for DFT based CO-OFDM systems.

In this paper, to circumvent the difficulty of deriving frequency domain transfer matrix for non-DFT based CO-OFDM systems and CO-SCFDE systems, we propose a new framework of time-domain phase noise suppression which can be applied in any coherent optical block transmission systems while maintains the same treatment of phase noise mitigation. A family of phase noise suppression algorithms have been derived under this framework which are proved to be very effective.

## 2. Phase noise suppression

#### 2.1. System model

Two different transmission schemes: DFT based CO-OFDM and CO-SCFDE whose schematic diagrams are shown in Fig. 1, are employed to describe the time-domain phase noise suppression framework. Extension to other block transmission systems is straightforward.

At the transmitter, the *i*th transmitted block **a**
* _{i}* = [

*a*(0)

_{i}*a*(1) …

_{i}*a*(

_{i}*N*– 1)]

*is encoded into the baseband CO-OFDM or CO-SCFDE symbol after pilot, CP and preamble insertions while for CO-OFDM, an inverse discrete Fourier transform (IDFT) is needed to transform the time-domain data into frequency-domain ones. The signal structure in which several transmitted blocks form a frame and the first block in each frame is employed as the preamble is adopted. Consider all the linear effects in the optical fiber channel, the phase noise induced by lasers in both the transmitter and the receiver, as well as the ASE noise induced by the optical amplifiers periodically placed in the optical link. Assuming a perfect time and frequency synchronization at the receiver, the*

^{T}*i*th received block

**r**

*= [*

_{i}*r*(0)

_{i}*r*(1) …

_{i}*r*(

_{i}*N*– 1)]

*, where the superscript*

^{T}*T*denotes transpose, can be obtained as

*ϕ*(

_{i}*m*) is the phase noise introduced at the transmitter and the receiver and diag(

**x**) is a diagonal matrix with the vector

**x**as the diagonal.

**w**

*= [*

_{i}*w*(0)

_{i}*w*(1) …

_{i}*w*(

_{i}*N*– 1)]

*denotes the additive white Gaussian noise (AWGN) and*

^{T}**T**

*is the linear transform matrix for the*

_{i}*i*th block whose definition is given in Table 1.

**T**

*is a function of the channel matrix ϒ*

_{i}*which is an*

_{i}*N*×

*N*circulant matrix with the first row vector as [

*h*(0) 0 … 0

_{i}*h*(

_{i}*L*– 1)

*h*(

_{i}*L*– 2) …

*h*(1)], where

_{i}*h*(

_{i}*l*) is the

*l*th coefficient of the channel impulse response (CIR) vector

**h**

*= [*

_{i}*h*(0)

_{i}*h*(1) …

_{i}*h*(

_{i}*L*– 1)]

*which has a memory of*

^{T}*L*– 1 for the

*i*th symbol. In Table 1,

**F**= [

**f**

_{0}

**f**

_{1}…

**f**

_{N−1}]

*denotes DFT transformation whose element*

^{H}**f**

*is described as*

_{m}*H*denotes Hermitian transpose and

*m*can also be a negative integer.

#### 2.2. Theory of the proposed methods

The block diagram of the proposed phase noise suppression methods is shown in Fig. 2, the details of which are described in the following subsections.

### 2.2.1. Time-domain channel estimation

To realize effective phase noise suppression, accurate channel state information (CSI) at the receiver becomes indispensable. Since the optical fiber channel varies slowly, CSI can be considered constant within a frame and acquired resorting to the preamble of each frame. Eqn. (1) can be rewritten as

**A**

*is an*

_{i}*N*×

*L*Toeplitz matrix whose definition for CO-OFDM and CO-SCFDE is given in Table 1. The subscript

*i*of CIR vector

**h**is omitted since CSI is regarded invariant across the entire frame. For channel estimation, we assume that phase noise just induces a constant unknown phase rotation, i.e. So the maximum likelihood (ML) estimation of the scaled CIR vector

**h**

*can be computed as [13]*

_{s}**h**

*=*

_{s}**h**

*e*

^{j$\overline{\varphi}$1}.

### 2.2.2. Pilot-aided (PA) phase noise estimation and suppression

Figure 3 presents the signal flow chart for the proposed phase noise suppression with channel compensation which is described as follows. Orthogonal basis expansion with orders larger than one is employed to model the phase noise in the data blocks and the diagonal of Φ* _{i}* in Eqn.(1) denoted by Ψ

*can be expanded as*

_{i}**B**= [

**b**

_{0}

**b**

_{1}…

**b**

_{ζ}_{−1}] and

**b**

_{0}

**b**

_{1}…

**b**

_{ζ}_{−1}are an orthogonal basis of

*ζ*-dimensional space. Γ

*= [*

_{i}*γ*(0)

_{i}*γ*(1) …

_{i}*γ*(

_{i}*ζ*− 1)]

*is the coefficient vector and DFT basis is employed in the proposed methods. Utilizing the property that where the superscript * denotes conjugation, we have*

^{T}*ɛ*is the residual error induced by AWGN as well as phase noise modeling imperfection. Consider the case that pilots are uniformly distributed in the data, let

_{i}*k*denote the index of the pilots and

_{q}*k*=

_{q}*k*

_{0}+

*qD*(

*q*= 0, 1, …,

*M*– 1), where

*D*is the density of the pilots. Then pilot vector ${\mathbf{\text{a}}}_{i}^{p}\hspace{0.17em}=\hspace{0.17em}{[{a}_{i}\hspace{0.17em}({k}_{0})\hspace{0.17em}{a}_{i}({k}_{1})\hspace{0.17em}\cdots \hspace{0.17em}{a}_{i}\hspace{0.17em}({k}_{M-1})]}^{T}$ can be expressed as

**e**

*is an*

_{kq}*N*× 1 vector with the

*k*th entry equal to 1 and the other equal to 0. Based on (9), we derive

_{q}Least squares (LS) principle is applied for phase noise estimation. The circulant matrix ϒ can be diagonalized by DFT matrix **F**, i.e.

*N*points-DFT to CIR vector

**h**. In principle, the LS estimation of Γ

*can be obtained by minimizing $\Vert {\varepsilon}_{i}^{p}\Vert $. However, the true channel matrix*

_{i}**H**can not be directly acquired considering the impact of phase noise. Define diagonal matrix

*N*points-DFT to

**ĥ**

*to derive the diagonal of Λ and we obtain*

_{s}**R**

*relates to Λ and is shown in Table 1 for two different transmission schemes. $\mathrm{\Delta}{\mathrm{\Gamma}}_{i}\hspace{0.17em}=\hspace{0.17em}{\mathrm{\Gamma}}_{i}^{*}{e}^{j{\overline{\varphi}}_{1}}$ is scaled ${\mathrm{\Gamma}}_{i}^{*}$ due to the difference between Λ and*

_{i}**H**. So the LS estimation of ΔΓ

*is shown as [13]*

_{i}**a**

*is derived as where*

_{i}*Q*(·) is a quantization operation performed in the threshold detector.

### 2.2.3. Data-enhanced (DE) phase noise suppression

To make observation equation (16) overdetermined, the dimension of Γ* _{i}* is required to be less than the number of pilots. This leads to significant modeling error for PA phase noise suppression method when relatively serious phase noise presents. For this case, utilizing both data and pilots to augment the observation could be a good solution to solve the problem. After PA phase noise suppression, the derived

**â**

*could be fed back to replace the pilots to update the LS estimation of ΔΓ*

_{i}*as*

_{i}**B**

*is a matrix composed by DFT basis as [*

_{d}**b**

_{0}

**b**

_{1}…

**b**

_{$\tilde{\zeta}$}_{−1}]. Here

*$\tilde{\zeta}$*satisfies

*$\tilde{\zeta}$*>

*ζ*since more data can be used to compute the LS estimation. And the final estimate of

**a**

*is also derived by Eqn. (19) (*

_{i}**B**

*is employed at this time) and the threshold detection.*

_{d}Based on the theory above, three methods are available for phase noise suppression according to the property of laser phase noise. When phase noise is mild, just constant phase shift (CPS) suppression needs to be taken into account. For this case, *ζ* = 1 and then CPS suppression algorithm is suitable. When phase noise is more serious, PA phase noise suppression algorithm with *ζ* > 1 should be employed. For the case that laser linewidth is very large and phase noise is very serious, extra DE phase noise suppression is performed to assist PA phase noise suppression algorithm (PA+DE phase noise suppression algorithm) corresponding to *ζ* > 1 and *$\tilde{\zeta}$* > *ζ*.

### 2.2.4. Complexity analysis

For the proposed phase noise suppression algorithms, the computational complexity is mainly induced by the calculation of **C**
* _{i}*/

**C̃**

*(Eqn. (18)/Eqn. (22)), Δ$\widehat{\Gamma}$*

_{i}*(Eqn. (17)/Eqn. (21)) and*

_{i}**z**

*(Eqn.(19)) which include both matrix multiplication and inversion. To be specific, in the PA phase noise suppression method, the addition/multiplication needed by computing*

_{i}**C**

*, Δ$\widehat{\Gamma}$*

_{i}*and*

_{i}**z**

*are respectively*

_{i}*O*(

*ζN*log(

*N*)),

*O*(

*Nζ*

^{2}) +

*O*(

*ζ*

^{3}) and

*O*(

*N*log(

*N*)). While in the extra DE phase noise suppression employed when phase noise is very serious, the addition/multiplication needed to calculate

**C̃**

*, Δ$\widehat{\Gamma}$*

_{i}*and*

_{i}**z**

*are respectively*

_{i}*O*(

*$\tilde{\zeta}$ N*log(

*N*)),

*O*(

*N$\tilde{\zeta}$*

_{2}) +

*O*(

*$\tilde{\zeta}$*

_{3}) and

*O*(

*Nlog*(

*N*)). For the above complexity evaluation, we assume that fast Fourier transform (FFT) is employed for the multiplication with

**F**/

**F**

*. Since phase noise is always narrowband,*

^{H}*ζ*and

*$\tilde{\zeta}$*are both sufficiently small and computational efforts induced here are very limited.

## 3. Performance analysis

Standard Monte Carlo simulation is employed to evaluate the performance of the proposed phase noise suppression methods. For the transmission systems under consideration, following the setting in [9], the sampling rate is 10 GS/s (giga samples per second), modulation format of QPSK is considered and the size of data block is 256. In every data block, 25 pilots are equally inserted. ISI in the optical fiber channel is induced by the residual CD of 2000 ps/nm and the corresponding length of CP is 32 in each data block.

In the following performance analysis, we consider CPS suppression algorithm corresponding to *ζ* = 1, PA phase noise suppression algorithm corresponding to *ζ* = 3 and PA+DE phase noise suppression algorithm corresponding to *ζ* = 3, *$\tilde{\zeta}$* = 9. Figure 4 presents the OSNR penalty curves for the CO-SCFDE system with BER=10^{−4}. The OSNR penalty of the system utilizing the ML channel estimation based equalization method (ML channel equ.) is also presented as a comparison, which is described as

*N*-points DFT on

**ĥ**

*. It is shown that significant OSNR gain is brought by the proposed family of phase noise suppression algorithms. For the case that laser linewidth (LW) is 200 KHz, the three methods achieve similar performance. So when phase noise is mild, performing CPS compensation is enough. When LW is lager than 200 KHz, the OSNR gain brought by PA and PA+DE methods becomes more and more obvious along with the increase of LW. When LW is about 950 KHz, 12.4 dB penalty is observed after the proposed CPS compensation and if the phase noise is more serious, BER floor appears for this case. At the same time, PA and PA+DE algorithms keep the OSNR penalty below 3.9 dB and 1.4 dB respectively when LW is less than 1 MHz. If the phase noise is very serious, i.e. LW ≥ 1 MHz, a second round data feedback is necessary at the DE phase noise suppression step (*

_{s}*$\tilde{\zeta}$*is set to be 11 for the second round feedback). At this time, PA+DE method can bring 8.6 dB OSNR gain for LW=1.2 MHz compared with utilizing PA method only. When LW is large than 1.2 MHz, just performing PA method suffers from BER floor which can be reduced successfully by the extra DE processing.

OSNR penalty curves for DFT based CO-OFDM system with BER=10^{−4} are presented in Fig. 5. It can also be found that the proposed algorithms are very effective and among them, suitable method should be chosen corresponding to different LW values. Compared with the case of the CO-SCFDE system, the proposed family of phase noise suppression algorithms is much more effective in the CO-OFDM system. Even when LW achieves 2 MHz, PA algorithm keeps the OSNR penalty below 3.1 dB and PA+DE algorithm keeps the OSNR penalty below 0.8 dB while more than one round data feedback at the DE phase noise suppression step are not needed at all. Compared with the proposed orthogonal basis expansion-based (OBE) phase noise suppression algorithm for CO-OFDM Systems in [9], PA and PA+DE algorithms both have better performances as shown by the OSNR penalty curves. OBE method keeps the OSNR penalty at BER=10^{−4} below 1.2 dB when LW is no larger than 1 MHz [9] while PA+DE algorithm keeps the OSNR penalty below 0.5 dB.

## 4. Conclusion and discussion

In this paper, a unified phase noise suppression framework is proposed for coherent optical block transmission systems. Three new phase noise suppression algorithms, i.e. CPS, PA and PA+DE algorithms are derived under this framework. Compared with OBE method in [9], PA and PA+DE algorithms both bring better performances. Numerical results prove that the proposed algorithms are effective in mitigating laser phase noise obviously in both CO-OFDM and CO-SCFDE systems.

An interesting extension of the framework could be the phase noise suppression of polarization-division multiplexed (PDM) systems. For PDM CO-OFDM and CO-SCFDE systems, the *i*th received block **r**
* _{i}* = [

**r**

_{x,i}

**r**

_{y,i}]

*can be described as, i.e. Eqn. (1) should be revised as*

^{T}*x*and

*y*respectively denote the two orthogonal polarization components. And based on this system model, the phase noise suppression for PDM coherent optical block transmission systems could also follow the unified framework presented in this paper. However, to obtain the closed-form suppression algorithm involves inversion of complex matrices, e.g. the channel transfer matrix in Eqn. (24). New results which address this issue will be included in another submission.

## Acknowledgments

This work was supported by the National Basic Research Program of China under Contracts 2010CB328201 and by the National Natural Science Foundation of China under Grant 60907029.

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