IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 20, OCTOBER 15, 2012
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Polarization-Time Code and 4 × 4 Equalizer-Decoder for Coherent Optical Transmission Mahdi Zamani, Student Member, IEEE, Chuandong Li, and Zhuhong Zhang, Member, IEEE
Abstract— A linear polarization-time code in conjunction with an adaptive multi-tap 4 × 4 joint equalizer-decoder is proposed to average the effects of polarization-dependent impairments on two polarizations. For the first time in the context of polarizationtime coding, there is no assumption of polarization-dependent loss (PDL) of knowledge at the receiver. This method compensates the performance loss penalty due to polarization-dependent impairments in polarization division multiplexed coherent optical single-carrier systems. Using the proposed method, a 1.25-dB optical signal-to-noise ratio gain is achieved in the presence of 6-dB PDL. The proposed 4 × 4 joint equalizer-decoder has twice the complexity of conventional time-domain equalizers. Further changes at the receiver including the demodulator are not required. Since the outputs of the equalizer are noisy quadrature phase shift keying signals, adoption of the method to work with both hard and soft forward error corrections is straightforward. Index Terms— Coherent optical communications, polarizationdependent loss (PDL), polarization-mode dispersion (PMD), polarization-time codes, single-carrier transmission.
I. I NTRODUCTION
I
N HIGH speed long-haul polarization division multiplexed (PDM) coherent optical single-carrier (COSC) systems, channel impairments such as chromatic dispersion (CD) and polarization-mode dispersion (PMD) which are unitary processes, polarization-dependent loss (PDL), laser phase noise, and non-linear effects limit the transmission performance. In PDM-COSC, to mitigate the linear channel impairments, channel equalizations are performed before the decoder, and then, simple decoders such as minimum distance demodulators or quadrature amplitude modulation (QAM) slicers are implemented. Due to PDL and state-of-polarization (SOP) rotation with random angles, two polarizations experience different received signal-to-noise ratios (SNRs). Hence, the polarization with the worse SNR bottlenecks the system performance. When the SOP rotation angle is 45 degrees, the system has its highest performance as the effect of PDL is averaged over Manuscript received July 22, 2012; revised August 22, 2012; accepted August 26, 2012. Date of publication August 31, 2012; date of current version September 21, 2012. The authors are with Huawei Technologies Company, Ltd., Ottawa, ON K2K 3J1, Canada (e-mail:
[email protected]; chuandong.li@ huawei.com;
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2012.2216259
two polarizations. In high speed long-haul transmission, due to the feedback delay, it is seemingly impossible to always keep the system at a specific SOP. On the other hand, since SOP rotation angle varies in time randomly, there is no linear code between the polarizations to always average the PDL effect on two polarizations. Space-time codes have been proposed in wireless communications to achieve diversity gain in multiple-input multipleoutput (MIMO) channels. As PDM-COSC can be considered as a 2×2 MIMO channel [1], the wireless ideas can be carried over and adopted. The Golden code [2] and the Silver code [3] are two full-rate block space-time codes, which have been proposed for 2×2 MIMO channels. All of the previous works that used space-time codes for PDL compensation in optical fiber channels was based on orthogonal frequency division multiplexing (OFDM) systems [4]–[7]. However, OFDM based long-haul optical modems are not commercialized yet due to their stringent sensitivity to fiber non-linearity, phase noise, and frequency offset [8]. In [4]–[7], it is assumed that the receiver accesses the channel state information (CSI) such as CD, PMD, PDL, and SOP rotation angles accurately, or estimates the CSI. In the problem of interest, there is no CSI assumption at the receiver;instead, frequency-domain equalizers (FDEQ) and an adaptive 2×2 time-domain equalizer (TDEQ) are used before the decoder to mitigate the channel impairments. To the best of our knowledge, the only space-time code which has been reported for PDM-COSC systems is [9]; however, there are two major differences between this letter and [9]. First, the proposed scheme uses quadrature phase shift keying (QPSK) slicing for demodulation, while maximumlikelihood (ML) detection is performed in [9]. Second, in our scheme, the receiver is blind and has no information about CSI including PDL and SOP rotation angle. Space-time coded modulations are more complicated than the regular modulators and have many levels in the constellation space. Hence, conventional equalizers cannot mitigate the linear channel impairments, completely. To solve this problem, we propose an adaptive 4×4 equalizer-decoder, and show that it outperforms the conventional separate 2×2 equalizing and space-time decoding. The Golden code is a linear space-time code and is one of the best known codes in 2×2 wireless channels [2]. Since the optical channel has a different impulse response and
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 20, OCTOBER 15, 2012
different behavior, the Golden code is no longer the best candidate for such systems. The codes with a higher minimum distance perform better in PDM-COSC channels. As a result, the Silver code outperforms the Golden code in coherent optical systems [4], [5], [7]. In contrast with the Golden code, the Silver code is not a linear space-time code and its linear joint equalization-decoding is more complicated in PDM-COSC receivers. Thus, inspired by the Silver code, we propose a new linear polarization-time block code (PTC), namely Semi-Silver code, whose performance is the same as that of the Silver code in PDM-COSC. Simulation results show that the proposed Semi-Silver code averages the polarizationdependent impairments effects on two polarizations. Note that space-time code and polarization-time code expressions are used interchangeably throughout this letter. II. S YSTEM M ODEL A relationship between the transmitted and received signal in frequency domain can be considered as follows, n˜ X ( f ) H X X ( f ) HY X ( f ) S˜ X ( f ) r˜X ( f ) + = (1) r˜Y ( f ) H X Y ( f ) HY Y ( f ) S˜Y ( f ) n˜ Y ( f ) where s and r are the complex-valued transmitted and received signals, respectively. n ∼Cℵ(0, σn2 ) is the zero mean circularly symmetric complex additive white Gaussian noise (AWGN), which is commonly used for modeling the amplified spontaneous emission (ASE) at each amplifier. The linear channel matrix, H( f ), is composed of a transfer function T ( f ), concatenated elements of PMD denoted by Ui ( f ), and PDL denoted by [10]. Mathematically, N Ui ( f ). H( f ) = T ( f )i=1
(2)
The linear function T ( f ) is composed of the all-pass transfer function model of chromatic dispersion, shaping filters at the transceiver, and all band-pass filters during transmission. Ui ( f ) is a unitary matrix in the form of jπ f τ ( f ) i 0 e R −1 (θi , ϕi ) (3) Ui ( f ) = R(θi , ϕi ) 0 e− j π f τi ( f ) where τi follows a Maxwellian distribution, and cos θ sin θ e j ϕ R(θ, ϕ) = −sin θ e− j ϕ cos θ
(4)
describes the local orientation mismatch between the principal state-of-polarization (PSP) of the fiber and the signal polarizations. The PDL matrix can be written as √ 1+λ √ 0 (5) = R(θ , ϕ ) R(θ , ϕ ) 1−λ 0 and PDL value in dB refers to PDL = 10log10 (1 + λ/1 − λ). As we shall show later, the optical signal-to-noise ratio (OSNR) penalty due to PDL changes with the SOP rotation angle θ .
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III. S EMI -S ILVER C ODE In polarization-time codes, the modulated symbols are coded in both polarization and time domains. For example, a full rate 2×2 polarization-time code transmits four independent symbols from two polarizations in two consecutive time slots, whose spectral efficiency is one symbol per polarization per time slot. Therefore, the rate and spectral efficiency is not reduced. This code leads to experience all channel effects on all symbols equally, which means the polarization-dependent impairments are averaged on all symbols. A 2×2 polarizationtime code can be specified by four functions of four modulated symbols as f1 (S1 , S2 , S3 , S4 ) f 2 (S1 , S2 , S3 , S4 ) . (6) f3 (S1 , S2 , S3 , S4 ) f 4 (S1 , S2 , S3 , S4 ) In order to decode such codes, CSI is required at the receiver. Here, instead, we jointly equalize and decode the received coded signals using an adaptive equalizer. To this goal, we restrict our attention on linear PTCs, in which f 1 , f 2 , f 3 , and f 4 are linear functions of symbols (no conjugation), to be able to decode them using an adaptive equalizer. We propose the Semi-Silver code as follows. To transmit four modulated symbols [S1 S2 S3 S4 ], let us first define 1 1 + i −1 + 2i S3 Z3 √ . (7) Z4 S4 7 1 + 2i 1 − i At the first time slot, X (resp. Y ) polarization transmits S 1X = S1 + Z 3 (resp. SY1 = S2 − Z 4 ). At the second time slot X (resp. Y ) polarization transmits S 2X = −S2 − Z 4 (resp. SY2 = S1 − Z 3 ). Fig. 1 depicts the performance of the Semi-Silver code compared to the Golden code and the uncoded system in an ideal assumption of CSI at the receiver, where SOP rotation angle is 0. Indeed, this is not a practical assumption in PDMCOSC; however, Fig. 1 shows the potential ability of the SemiSilver code in PDL compensation. IV. A DAPTIVE 4 × 4 J OINT E QUALIZER -D ECODER Using an M level modulation, the Semi-Silver coded signal has M 3 levels. For example, for QPSK modulation, the SemiSilver code has 64 levels. As these coded levels are very close to each other, an adaptive 2×2 equalizer cannot mitigate the linear channel impairments, accurately.
ZAMANI et al.: PTC AND 4 × 4 EQUALIZER-DECODER FOR COHERENT OPTICAL TRANSMISSION
EO Convertor
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Fig. 3. BER in X and Y polarizations versus OSNR (dB) without and with the Semi-silver code (PDL = 6 dB).
At the receiver, instead of equalizing and PTC decoding the signals separately, we jointly equalize and decode them using a multi-tap 4×4 adaptive equalizer. The received signals at X and Y polarizations are denoted by r X and rY , respectively. Each of the received signals is divided into two signal sequences based on the associated time order (1st or 2nd ) in polarization-time encoder. For example, in a T /2-spaced case, if the remainder of t divided by 4 is zero or one, r X (t) goes to r X1 , otherwise it goes to r X2 . Similarly, rY (t) is divided into rY1 and rY2 . Mathematically, r X1 = r X (t)|(t mod 4) {0, 1} r X2 = r X (t)|(t mod 4) {2, 3} rY1 = rY (t)|(t mod 4) {0, 1} (8) rY2 = rY (t)|(t mod 4) {2, 3} . This process can be adopted accordingly for the general case of T /K -spaced case, directly. These four signal sequences are equalized using an adaptive 4×4 MIMO equalizer to not only equalize the channel impairments, but also decode the Semi-Silver coded symbols. The 16 tap-vectors are updated by the error computed by comparing the equalized signals and decision signals using a QPSK slicer [11]. Finally, the output of the equalizer is a noisy version of the original QPSK constellation points, corrupted by additive Gaussian noise. Fig. 2 shows a simple system block diagram including the proposed blocks. The proposed joint equalizer-decoder has
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Fig. 4. BER versus OSNR (dB) without and with the semi-silver code (PDL = 6 dB), and QPSK theoretical BER.
double complexity compared to a conventional 2×2 equalizer. The complexity of the other parts including FDEQ which compensates CD and filtering effects in not changed. Inputs of TDEQ, H and V branches, are the dispersion free distorted coded signals, which are equalized and decoded in 4×4 MIMO FIR. After phase recovery, the decision is made using a simple QPSK slicer. As Fig. 5 shows, the proposed equalizer-decoder with simple QPSK slicer (square marked curve) outperforms separate equalizing and optimum PTC decoding of the Semi-Silver code (dashed-dotted curve). Note that the former has also a lower complexity. V. S IMULATION R ESULTS The proposed Semi-Silver code and adaptive 4×4 equalizerdecoder is simulated in PDM-COSC with 32 GBaud QPSK modulation. The transmitter consists of digital-to-analog convertor (DAC) with zero-order-hold (ZOH) filter model, T /8 up-sampler, 5th order Bessel filter for electronic driver, laser carrier, and Mach-Zehnder modulator (MZM). The receiver has a 4th order Gaussian optical filter with 40 GHz bandwidth, an optical hybrid, an electrical 5th order Bessel filter, downsampler, analog-to-digital convertor (ADC), and digital signal processing (DSP) unit. Pre-compensator and post-compensator including squared raised-cosine filter, inverse Bessel filter, and CD compensator are implemented at the transceivers. The laser phase noise is modeled as a random walk process with Gaussian distribution and 200 KHz laser line width. 216 symbols with random data are transmitted to achieve a
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 20, OCTOBER 15, 2012
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Fig. 5. Required OSNR (dB) to achieve BER = 3 × 10−3 versus the SOP rotation angle (PDL = 6 dB).
reliable bit error rate (BER) above 10−3 . In simulations, the total CD is 33 773 ps/nm, mean DGD is 60 ps, PDL = 6 dB. Fig. 3 shows BER versus OSNRin X and Y polarizations. Square and circle marked curves represent the cases with and without the Semi-Silver code, respectively. As it is shown, BER in X and Y polarizations are different in the system without the Semi-Silver code, and the performance is limited by the worse polarization; however, in the system with the Semi-Silver code, BER in two polarizations are the same. In the other words, the effect of PDL is averaged over two polarizations. Fig. 4 shows BER versus OSNR for no SOP rotation. The diamond marked curve is the theoretical QPSK performance, and the circle marked curve represents the cases without the Semi-Silver code. The square marked curveis the performance of the Semi-Silver code in conjunction with the proposed joint equalizer-decoder and QPSK slicer. At BER = 3 × 10−3 , which is a typical high performance hard decision forward error correction (FEC) threshold, 1.25 dB OSNR gain is achieved. As pointed out earlier, the results of [9] are based on the knowledge of PDL matrix at the receiver and ML detection. With such assumptions, the performance of the proposed Semi-Silver code is depicted by the star marked curve in Fig. 4. The reason behind OSNR penalty due to no knowledge of PDL at the receiver is noise coloring in both space and time in the equalization process. Fig. 5 shows the required OSNR to achieve BER = 3×10−3 versus the SOP rotation angle. The square and circle marked curves represent the cases with and without the Semi-Silver code, respectively. The dashed-dotted curve is the SemiSilver code performance with separate equalization and PTC detection. The star marked curve is the performance of the Semi-Silver code with the assumption of PDL matrix knowledge at the receiver and ML detection (for details see [9]). The diamond marked curve shows the required OSNR where PDL = 0 dB. As it can be seen from Fig. 5, the performance of the system without Semi-Silver code depends on the SOP rotation angle. As the SOP rotation angle varies with time slowly compared
to the symbol rate and is not available at the transmitter, no ergodic capacity in the sense of Shannon is defined for such channels. Instead, the performance is evaluated by the outage capacity, which is the maximum achievable rate which guaranties that the outage probability is less than a predetermined small value ε. Therefore, the performance is limited by the worst case scenario which is at 0 degrees and ±90 degrees SOP rotation angles. Fig. 5 shows that the performance of the proposed SemiSilver code in conjunction with the obtained 4×4 equalizerdecoder is superior, and also is not changed with SOP rotations. It also shows that 1.25 dB OSNR gain is achieved at BER = 3×10−3 for the proposed method. Moreover, the SemiSilver code provides 2.75 dB gain if the receiver accesses the PDL matrix and performs ML detection.If one is interested in the performance for average of SOP rotation angles, averaging over the horizontal axis of Fig. 5 is taken. VI. C ONCLUSION In PDM-COSC systems, a linear polarization-time code is proposed to compensate the OSNR loss penalty due to random polarization-dependent impairments. For further improvement, a corresponding adaptive multi-tap 4×4 joint equalizerdecoder is obtained. At BER = 3 × 10−3 , 1.25 dB OSNR gain is achieved in the presence of 6 dB PDL. For the proposed scheme, for the first time in the context of polarization-time codes, no CSI including PDL and/or SOP rotation angle values are required to be available at the transmitter and/or receiver. R EFERENCES [1] Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express, vol. 13, no. 19, pp. 7527– 7534, Sep. 2005. [2] J. C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: A 2×2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1432–1436, Apr. 2005. [3] O. Tirkkonen and A. Hottinen, “Improved MIMO performance with nonorthogonal space-time block codes,” in Proc. IEEE GLOBECOM Conf., San Antonio, TX, Nov. 2001, pp. 1122–1126. [4] S. Mumtaz, G. R. B. Othman, and Y. Jaouen, “Space-time codes for optical fiber communication with polarization multiplexing,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5. [5] E. Meron, A. Andrusier, M. Feder, and M. Shtaif, “Use of spacetime coding in coherent polarization-multiplexed systems suffering from polarization-dependent loss,” Opt. Lett., vol. 35, no. 21, pp. 3547–3549, Nov. 2010. [6] I. B. Djordjevic, L. Xu, and T. Wang, “Alamouti-type polarizationtime coding in coded-modulation schemes with coherent detection,” Opt. Express, vol. 16, no. 18, pp. 14163–14172, Sep. 2008. [7] S. Mumtaz, G. R. B. Othman, and Y. Jaouen, “PDL mitigation in PolMux OFDM systems using golden and silver polarization-time codes,” in Proc. Nat. Fiber Opt. Eng. Conf., San Diego, CA, Mar. 2010, pp. 1–3. [8] S. L. Jansen, I. Morita, K. Forozesh, S. Randel, D. Van Den Borne, and H. Tanaka, “Optical OFDM, a hype or is it for real?” in Proc. 34th ECOC, Brussels, Belgium, Sep. 2008, pp. 1–4. [9] S. B. Rayana, H. Besbes, G. R. B. Othman, and Y. Jaouen, “Joint equalization and polarization-time coding detection to mitigate PMD and PDL impairments,” in Proc. OSA Signal Process. Photon. Commun., Colorado Springs, CO, Jun. 2012, pp. 1–3. [10] F. N. Hauske, M. Kuschnerov, B. Spinnler, and B. Lankl, “Optical performance monitoring in digital coherent receivers,” J. Lightw. Technol., vol. 27, no. 16, pp. 3623–3631, Aug. 15, 2009. [11] N. Al-Dhahir, “FIR channel-shortening equalizers for MIMO ISI channels,” IEEE Trans. Commun., vol. 49, no. 2, pp. 213–218, Feb. 2001.
