Blind polarization demultiplexing by constructing a cost function for coherent optical PDM-OFDM

Zhenming Yu; Minghua Chen; Hongwei Chen; Xingwen Yi; Sigang Yang; Shizhong Xie

doi:10.1364/OE.23.018511

1. Introduction

Coherent optical orthogonal frequency division multiplexing (CO-OFDM) has recently received much attention as a promising modulation format with the increasing interest in digital signal processing (DSP) [1–3]. Together with digital coherent detection, CO-OFDM brings benefits such as simple and efficient channel equalization, the ability to allocate signal power and modulation format on different subcarriers, and the flexible oversampling rate [4,5]. Meanwhile, polarization division multiplexing (PDM) is a very effective method for doubling the spectral efficiency of OFDM system [6,7]. In conventional coherent optical PDM-OFDM system, periodic training symbols (TSs) are inserted at the transmitter to recover the polarization states of the received signals [8]. The TSs are specifically designed, which increase the system redundancy and reduce the spectral efficiency. Moreover, this method tends to fail if the optical channel changes within one period of the TSs, especially for polarization-related effects that can change faster than millisecond. Reduced-guard-interval (RGI) CO-OFDM and zero-guard-interval (ZGI) CO-OFDM have been proposed which have very short cyclic prefix and even no cyclic prefix [5,9]. Some blind phase noise estimations also have been proposed to reduce the pilot subcarrier overhead [10,11]. Then the overhead for polarization demultiplexing becomes dominant. Recently, some blind polarization demultiplexing methods without using TSs have been proposed for coherent optical PDM-OFDM such as polarization demultiplexing in stokes space (SS-PDM) and independent component analysis (ICA) [12,13]. However, SS-PDM has to convert the received signals into Stokes space and the iterative estimation of ICA brings high DSP complexity.

Schmogrow et al. once proposed a novel blind polarization demultplexing algorithm with low computational complexity by constructing a cost function (CCF-PDM) in the time domain to blindly demultiplex optical PDM signals in single carrier system [14]. This algorithm combines the computational simplicity of the constant modulus algorithm (CMA) with the versatility of SS-PDM [15,16].The principle of this method is to construct a cost function by the received PDM signals, take advantage of the maxima of the cost function to find an inverse Jones matrix, and then multiply it with the received PDM signals in the time domain to recover the polarization states. However, the inverse Jones matrix calculated by this method to recover the polarization states is assumed frequency-independent while the polarization effects are generally frequency-dependent and cannot be ignored in wide-bandwidth and long-haul optical system. Therefore, CCF-PDM cannot be applied to PDM signals with frequency-dependent effects. Furthermore, OFDM signals in the time domain is noise-like and could not construct such the cost function to find the inverse Jones matrix. In this paper, we apply CCF-PDM to the received OFDM signals in the frequency domain and for individual subcarriers. The reasons include (i) each subcarrier has a much lower speed and narrower bandwidth, so the polarization effects that it experiences can be treated as flat, (ii) OFDM signals in the frequency domain could construct the effective cost function. Consequently, CCF-PDM can be applied to high-speed, wide-bandwidth OFDM system. This method reduces the system redundancy and has a fast convergence speed. Moreover, since the neighboring subcarriers experience similar polarization effects, we set the matrix parameters calculated by CCF-PDM of one subcarrier as the initial parameters of its neighboring subcarrier for CC-PDM. This approach could help find the maxima of the cost function more quickly and prevent swapping the data of the two orthogonal polarizations. We demonstrate this method in experiment by transmitting 66.6-Gb/s PDM-OFDM signal with 4QAM subcarrier modulation over 5440km SSMF and 133.3-Gb/s PDM-OFDM signal with 16QAM subcarrier modulation over 960km SSMF respectively. The performance of this method is comparable with that of the channel estimation based on TSs. It is also compatible with RGI-CO-OFDM, ZGI-CO-OFDM and the blind phase noise estimations well. Therefore, this method can play an important role in reducing the total overhead of OFDM system together with other methods. We also analyze the speed of convergence of this method.

2. Principle of CCF-PDM for CO-OFDM

As the optical waves propagate over the optical fiber, their polarization states evolve due to birefringence but remain nearly orthogonal. The differential group delay (DGD) in fiber depends on the optical frequency and the bandwidth of the principal states $Δ λ_{p s p} \approx 1 n m / 〈 Δ τ 〉 [p s]$ is a wavelength range over which the DGD is reasonably constant, where $Δ τ$ is the differential group delay [17]. In the long-haul optical system, the DGD can reach 10~100ps easily and the bandwidth of the principle states becomes very narrow. Consequently, the polarization rotation has to be considered frequency-dependent while CCF-PDM is frequency-independent. Therefore CCF-PDM cannot be applied to single carrier signals with wide-bandwidth and long-haul transmission. Fortunately, in OFDM, a large number of subcarriers are usually used so that the frequency-domain transfer function of a given transmission channel for each subcarrier can be regarded as constant or flat. The data modulation is also performed in the frequency domain and converted in the time domain by an inverse FFT. The individual OFDM subcarriers could construct such the cost function to find the inverse Jones matrix. These motivate us to apply CCF-PDM to the received OFDM signals in the frequency domain and for individual subcarriers. Within one OFDM frame in the frequency domain, we use i and k to represent the indices of the OFDM symbol and subcarrier. Jones vector ${\overset{⇀}{s}}^{t} = {[s_{x} s_{y}]}^{T}$ represents the PDM signal with two polarizations, i.e., $s_{x}$ and $s_{y}$ . After the FFT window synchronization, the channel model for individual subcarriers is [18],

{\vec{s}}_{_{k i}}^{r} = \exp (j φ_{i}) \cdot \exp (j Φ_{k}) \cdot M_{k} \cdot {\vec{s}}_{_{k i}}^{t} + n_{k i} .

where

Φ

is the phase dispersion owing to the fiber chromatic dispersion,

φ

is the laser phase noise, M is a

2 \times 2

matrix representing polarization rotation and PDL, n is the random noise. From this channel model, the polarization rotation matrix is frequency-dependent on the subcarrier index. The thrust is to find the inverse matrix

M^{- 1}

and apply

{\vec{s}}^{t'}_{k i} = M_{k}^{- 1} \cdot {\vec{s}}^{r}_{k i}

for polarization alignment.

{\vec{s}}^{r}_{k i}

and

{\vec{s}}^{t'}_{k i}

are the PDM signals of the k^th subcarrier before and after polarization demultiplexing.

If we neglect polarization dependent loss (PDL), M is a unitary matrix and M⁻¹ of the k^th subcarrier can be defined as $M {(k)}^{- 1} = [a^{*} (k) - b (k); b^{*} (k) a (k)]$ . The PDM signals of the k^th subcarrier after polarization alignment are represented as:

(\begin{array}{l} s_{k i, x}^{t'} \\ s_{k i, y}^{t'} \end{array}) = (\begin{matrix} a^{*} (k) & - b (k) \\ b^{*} (k) & a (k) \end{matrix}) (\begin{array}{l} s_{k i, x}^{r} \\ s_{k i, y}^{r} \end{array}) = (\begin{matrix} \cos α & - e^{- j ϕ} \sin α \\ e^{j ϕ} \sin α & \cos α \end{matrix}) (\begin{array}{l} s_{k i, x}^{r} \\ s_{k i, y}^{r} \end{array}) .

where

α

is the rotation angle and

ϕ

is the phase-retardation angle. If the phase-retardation angle

α

(

0 \leq α < π / 2

) and phase-retardation

ϕ

(

0 \leq ϕ < 2 π

) are chosen correctly to make

M^{- 1} M = I

, the inter-polarization crosstalk between the two orthogonal polarization states would disappear. Then we construct a cost function to find the correct

α

and

ϕ

[14]:

J (α, ϕ) = 〈 | s_{k, x}^{t'} | 〉 + 〈 | s_{k, y}^{t'} | 〉 = \frac{1}{I} \sum_{i = 1}^{I} [| s_{k i, x}^{t'} | + | s_{k i, y}^{t'} |] .

Figures 1(a) and 1(b) visualize the cost function

J (α, ϕ)

by analyzing the experimental data of one subcarrier with 4QAM modulation. This

J (α, ϕ)

is independent of noise. The number of the symbols we use is I = 100. In fact, The cost function is periodic in

α

and

ϕ

. Inside the period of

0 \leq α < π / 2

and

0 \leq ϕ < 2 π

, there are only two maxima which both correspond to the correct parameters

α

and

ϕ

for polarization alignment [14]. Note that one of the two maxima leads to the situation that the two orthogonal polarizations of received OFDM signals are exchanged. This can be readily corrected by a logical comparison. In order to find the extremum in Fig. 1(a) and 1(b), the gradient algorithm is employed. The gradient of J is defined as:

\nabla J = \frac{\partial J}{\partial α} e_{α} + \frac{\partial J}{\partial ϕ} e_{ϕ} .

where

e_{α}

and

e_{ϕ}

are the two orthogonal unit vectors [15]. The maxima and the corresponding

α

and

ϕ

could be found quickly along the opposite direction of the gradient. Since the neighboring subcarriers have similar impacts by the polarization effects which means they have similar parameters

α

and

ϕ

, we set the obtained parameters

α

and

ϕ

of one (k^th) subcarrier as the initial parameters of its neighboring ((k + 1)^th) subcarrier for the gradient algorithm. This measure could accelerate the speed of finding the maxima significantly. Moreover, only making the logical comparison for the first subcarrier can help prevent the situation of swapping the data of the two orthogonal polarizations of the following subcarriers. After we obtain the

M^{- 1}

for each subcarrier, the following DSP processes, i.e., PDL compensation, channel compensation and laser phase noise compensation are conducted to recover the transmitted data [18–20]. Note that channel compensation only includes simple constellation rotation. Figure 1(c) and 1(d) are the data of received PDM-OFDM signals with 4QAM before and after CCF-PDM by analyzing the experimental data, respectively. Figure 1(e) is after PDL compensation, phase noise compensation and channel compensation. By applying the CCF-PDM on individual subcarrier, CCF-PDM can deal with the frequency-dependent polarization effects and has a very large tolerance to DGD (>>100 ps) in OFDM system but it is difficult to be applied in single carrier system.

Fig. 1 Cost function $J (α, ϕ)$ for one OFDM subcarrier with 4QAM modulation (a) three-dimensional view and (b) two-dimensional view; Constellations of received PDM-OFDM signals (c) before and (d) after CCF-PDM, and (e) after PDL compensation, phase noise compensation and channel compensation, respectively.

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3. Experimental setup and results

The experimental step of coherent optical PDM-OFDM is shown in Fig. 2. At the transmitter, one laser source is fed in to the optical IQ modulator to carry the OFDM signal. We use MATLAB program to generate transmitted signals off-line with a data sequence of 2¹⁵-1 pseudo-random binary sequence (PRBS) and map them onto 4QAM or 16QAM constellation. An arbitrary waveform generator (AWG) is used to produce I/Q RF signals at 10 GS/s. An optical intensity modulator is used to further duplicate the signal to three copies. The frequency of the driving sine wave signal is at 6.71875 GHz, which is intentionally set to satisfy the condition of orthogonal band multiplexing. All the clock resources are phase-locked to the AWG using a 10 MHz reference clock. Each polarization components was mapped onto 86 OFDM frequency subcarriers and six out of them are used to compensate for the laser phase noise. We use 128-point IFFT to convert the frequency domain signal to the time domain. We also use 1/8 of OFDM symbol period for cyclic prefix to avoid the fiber dispersion. Then the optical OFDM signal is polarization multiplexed by a pair of polarization beam splitter/combiner with one branch delayed by one OFDM symbol. The transmission link is a fiber recirculation loop, which contains four spans of 80km SSMF whose loss is compensated by Raman amplifiers. At the receiver side, a local oscillator is coupled into polarization diversity optical hybrid to mix with the signal. The signal is detected by typical coherent receivers. The four RF signals for the two IQ components are then fed into a real-time oscillator scope and are acquired at 50 GS/s and processed off-line with the MATLAB programs.

Fig. 2 Experimental setup. PBS/C: Polarization beam splitter/combiner; ECL: External cavity laser; SW: switch; OTF: optical tunable filter.

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To verify the effectiveness of polarization demultiplexing by constructing a cost function (CCF-PDM) for PDM-OFDM, we also conduct the received experimental PDM-OFDM signals with channel estimation based on TSs which uses 20 training symbols in every 200 symbols with time-domain averaging for comparison [4]. We use 100 symbols to construct the cost function for CCF-PDM. Figure 3(a) shows the back-to-back (b2b) performance and transmission over 5440km SSMF performance of PDM-OFDM with TSs and CCF-PDM when the subcarriers are modulated by 4QAM. Figure 3(d) shows the back-to-back (b2b) performance and transmission over 960km SSMF performance of PDM-OFDM with TSs and CCF-PDM when the subcarriers are modulated by 16QAM. Figures 3(b), 3(c), 3(e) and 3(f) are the constellations for the two polarizations of the recovered subcarrier 4QAM modulated and 16QAM modulated signals by CCF-PDM, respectively. The performance of both b2b and after fiber transmission by CCF-PDM is comparable with that by TSs.

Fig. 3 (a) BER results of 66.7-Gb/s PDM-OFDM with 4QAMsubcarrier modulation using CCF-PDM and TSs respectively; (b) and (c) Constellations after transmission over 5440km with 4QAM subcarrier modulation using CCF-PDM vs. OSNR 18.42dB; (d) BER results of 133.3-Gb/s PDM-OFDM with 16QAM subcarrier modulation using CCF-PDM and TSs respectively; (e) and (f) Constellations after transmission over 960km with 16QAM subcarrier modulation using CCF-PDM vs. OSNR 26.75dB.

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Another advantage of CCF-PDM is that it has a very fast convergence speed. After the FFT window synchronization, the data on each subcarrier exclude the transition between symbols, unlike the single-carrier systems with oversampling. Since the CCF-PDM is implemented after the FFT window synchronization and FFT, we can construct the cost function to find $M^{- 1}$ without using many date points. Figure 4(a) shows the absolute values of the first row elements of the matrix $M^{- 1}$ , i.e., $| a |$ and $| b |$ vs. the number of OFDM symbols to analyze the speed of convergence with 4QAM subcarrier modulation. It needs only tens of symbols, on the order of microsecond in our experimental system, which is enough to track the dynamic change of polarization in field fibers. Figure 4(b) shows the number of iteration for the gradient algorithm to find the correct matrix parameters of one subcarrier. In the termination error of 0.0005 and step size of two, the correct matrix parameters are obtained after 31 iterations when the initial $α$ and $ϕ$ are both set to one while it only needs 16 iterations when the initial $α$ and $ϕ$ are set to the obtained matrix parameters of the neighboring subcarrier.

Fig. 4 (a) Speed of convergence of CCF-PDM of 4QAM subcarrier modulation. (b) Number of iteration for the gradient algorithm to find the correct matrix parameters of one subcarrier.

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4. Conclusion

In this paper we have proposed a TSs-free polarization demultiplexing method by constructing a cost function for coherent optical PDM-OFDM. Since single subcarrier has a much lower speed and narrower bandwidth, the polarization effects that it experiences can be treated as flat, and OFDM signals in frequency domain could construct the effective cost function. We apply polarization demultiplxing by constructing a cost function to the received OFDM signals in the frequency domain and for individual subcarriers. This method is applicable for high-speed, wide-bandwidth OFDM signals, all common subcarrier modulation formats and long-haul transmission and reduces the system redundancy for it requires no training symbols. We verify this method in the experiments of 66.6-Gb/s coherent optical PDM-OFDM with 4QAM subcarrier modulation and 133.3-Gb/s coherent optical PDM-OFDM with 16QAM subcarrier modulation. The experiments results show that the performance of CCF-PDM is comparable with that of channel estimation based on TSs. We analyze the speed of convergence of this method, which is tens of symbols, only on the order of microsecond. Since the neighboring subcarriers experience similar polarization effects, the approach of setting the initial matrix parameters by the neighboring subcarrier for the gradient algorithm also could reduce the number of iteration significantly and prevent swapping the data of the two orthogonal polarizations. Note that it still needs some training symbols to compensate for some other channel impairments in a practical OFDM system at present while the number can be much fewer and the designing can be much simpler. Our future work is to estimate the other channel impairments blindly and minimize the number of training symbols.

Acknowledgement

This work was supported by the NSFC under Contract (61132004, 61090391, 61335002, 61322113), by the young top-notch talent program sponsored by Ministry of Organization, China, and by Tsinghua University Initiative Scientific Research Program.

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Blind polarization demultiplexing by constructing a cost function for coherent optical PDM-OFDM

Abstract

1. Introduction

2. Principle of CCF-PDM for CO-OFDM

3. Experimental setup and results

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (4)

Equations (4)

Optics Express