IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 27, NO. 2, JANUARY 15, 2015
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Enhanced Blind Equalization for Optical DP-QAM in Finite Precision Hardware Amr M. Ragheb, Mobien Shoaib, Saleh Alshebeili, and Habib Fathallah, Member, IEEE
Abstract— In this letter, we apply the inverse QR decomposition-constant modulus algorithm (IQRD-CMA), blind equalization technique, for impaired 14 GBd DP-16 quadraticamplitude modulation optical modulated signal. The IQRD-CMA is considered here to mitigate the effect of residual chromatic dispersion (CD) and polarization mode dispersion (PMD). We evaluate the performance of the proposed inverse QR decomposition in terms of convergence rate, steady-state/residual mean square error (MSE), and bit error rate (BER), in comparison with standard blind constant modulus algorithm (CMA) and recursive least squares (RLS-CMA). Assuming infinite precision, the IQRD-CMA and RLS-CMA achieve similar performance and induce the same computational complexity; however, both largely outperform standard CMA for wide ranges of CD and PMD. However, for finite precision hardware, compared with RLS-CMA, the proposed blind IQRD-CMA clearly achieves one order of BER magnitude at 8- and 10-b resolutions, and further reduces the steady-state MSE by 3 dB. Moreover, results show that IQRD-CMA maintains the system stability at acceptable number of bits resolutions. In addition, this letter addresses the tradeoff between the complexity and the performance of all these equalizers for several precision settings. Index Terms— Inverse QR decomposition (IQRD), recursive least squares (RLS), coherent optical communications, constant modulus algorithm.
for mitigating their effect. Fractionally spaced equalizer (FSE) with a MIMO structure has been proposed in order to compensate for both: CD and PMD [2]; see Fig. 1. The mostly used types of blind adaptive equalizers, in optical coherent receivers, include the constant modulus algorithm (CMA), the radius directed equalizer (RDE), and the recursive least squares (RLS) algorithm [3]. In this letter, we improve our work in [4] by applying the CMA version of inverse QR decomposition (IQRD) algorithm to mitigate the effect of the residual CD and 1st order PMD, and compare its performance versus complexity with that of standard CMA and RLS-CMA, for 14Gbaud DP-16QAM coherent receiver. We make our investigation from two different, yet important perspectives. The first assumes signal processing with infinite precision, while this has no performance advantage, it is very instructive for the next analysis steps. In our second and most important perspective, we assume finite precision processing, hence more practical and realistic case. In Section II and III, we briefly introduce the channel model and the equalizers respectively. Section IV discusses the measured performance in terms of convergence rate, steady state MSE, and BER. Section V concludes the letter.
I. I NTRODUCTION
F
UTURE coherent optical systems face two main categories of linear impairments. The first is chromatic dispersion (CD), which is usually static in time, but creates large inter-symbol interference (ISI) [1]. The second impairment originates from optical polarization mode dispersion (PMD) and polarization dependent loss (PDL). These impairments respectively broaden the optical pulse, provoke the rotation of the principal state of polarizations (PSPs) in the fiber and lead to a power fluctuation. CD may change from time-to-time because of network reconfiguration and PMD is inherently a time varying phenomenon, in the order of milliseconds. Both require adaptive equalization techniques
Manuscript received September 4, 2014; revised October 16, 2014; accepted October 17, 2014. Date of publication October 24, 2014; date of current version December 24, 2014. A. M. Ragheb, S. Alshebeili, and H. Fathallah are with the Department of Electrical Engineering, King Saud University, Riyadh 11421, Saudi Arabia, and also with the King Abdulaziz City for Science and Technology-Technology Innovation Center in RF and Photonics, King Saud University, Riyadh 11451, Saudi Arabia (e-mail:
[email protected]). M. Shoaib is with the Prince Sultan Advanced Technology Research Institute, King Saud University, Riyadh 11421, Saudi Arabia (e-mail:
[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.2014.2364265
II. S IGNAL M ODEL In Fig. 1, we illustrate the channel block diagram in the left and the equalization structure in the right. The transmitted signal for both polarizations is given by [2] x p (k)g(t − kTs ), p = 1, 2 (1) x p (t) = k
where x p (k) is an independent and identically distributed (iid) sequence of QAM symbols, p = 1, 2 for the two system polarizations X and Y, g(t) is the impulse response of the transmission shaping filter, and Ts is the symbol time. The received signal in vector notation form can be expressed as [2] r(t) = H(t) ∗ x(t) + n(t)
(2)
where H(t), x(t), and n(t) are respectively a 2 × 2 MIMO fiber channel impulse response, the transmitted signal vector, and a circularly symmetric Gaussian noise vector with zero mean and variance N0 [2]. Neglecting the fiber nonlinearities, the frequency response of the fiber channel due to CD and 1st order PMD is expressed as [5] H(ω) = F{H(t)} = HCD (ω)HPMD (ω)
(3)
where F{}is the Fourier transform operator, HCD (ω) and HPMD (ω) are respectively the CD and PMD frequency
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 27, NO. 2, JANUARY 15, 2015
Fig. 1. Model of Coherent optical system (left) and 2 × 2 MIMO equalizer (right) ; y1,n and y2,n are the two polarization input signals, xˆ1,n is the estimated output, R1 is a constant, e1,n is the estimated error, and T is the fractional sampling rate, L is the number of filter taps.
complex conjugate transpose, respectively. It is worthy to note that CMA is less efficient for MQAM modulation with M ≥ 8, since the constant modulus property is no longer satisfied [5]. B. Recursive Least Squares (RLS-CMA)
Fig. 2.
The RLS-CMA algorithm converges faster than the SGA-based blind algorithms [3]. It uses the information contained in the input data extending back to the instant of time when the algorithm was initiated. The cost function of the MIMO RLS-CMA algorithm can be written as [7] n 2 H λn−m (7) J p (n) = w p (n)z p (m) − 1 p
Equalizer signal flow graph of (a) CMA; (b) RLS/IQRD-CMA.
responses [5]. The received sampled output can be expressed as in [2] y p=1,2 (nT ) = y p,n =
2 k
m=1
where z p (n) = y(n)yH (n)w p (n − 1)
x j,k c p,q (nT −kTs )+η p (nT ) (4)
q=1
where T is the fractional receiver sampler rate, c p,q (t) = b(t)∗ h p,q (t)∗ g(t), where b(t) is the receiver filter impulse response, hp, q(t) is the response of the pth output polarization to an impulse in the q th input polarization [2], η p (t) is the filtered AWGN noise, and “*” is the convolution operator.
(8)
represents an intermediate data vector, and λ p is called the forgetting factor (0 ≤ λ p ≤ 1) that controls the speed of convergence. The RLS-CMA shows faster convergence than CMA at the expense of higher complexity. However, it is known to suffer from numerically ill-condition when computing the inverse of the input autocorrelation matrix R(n) in finite precision hardware, such as, digital signal processor (DSP) or field programmable gate array (FPGA) [8].
III. B LIND E QUALIZATION In Fig. 2 we illustrate the flow diagram of CMA and RLS-CMA in addition to the proposed IQRD-CMA equalizer. A. Constant Modulus Algorithm (CMA) The CMA is one of the most known blind equalization techniques proposed in 1980’s by Godard [6]. The cost function to be minimized is given by J p (n) = E[(|xˆ p (n)|2 − R p )2 ],
p = 1, 2
(5)
where R p = E(|x p (n)|4 )/E(|x p (n)|2 ) is constant radius which depends on the input data symbols, and xˆ p (n) is the estimated output signal. The equalizer coefficients are adapted as follows [3] w p (n + 1) = w p (n) + μ y(n) e∗p (n),
p = 1, 2
(6)
where μ is the equalizer step size, e p (n) is the error signal, and y(n) = [y1(L)T(n) y2(L)T(n)]T is the received sampled signal vector, the superscript T and H stand for transposition and
C. Proposed Inverse QR Decomposition (IQRD-CMA) In wireless communications, QR decomposition is known as numerically well-conditioned method that is well suited for deterministic algorithms and can solve RLS problem by converting an input data matrix Z(n) of dimension (n + 1) × L into upper or lower triangular matrix, using an (n +1)×(n +1) unitary rotation matrix Q(n). In our case we apply this concept to the input data matrix Z(n) as follows [8] 0 (9) Q(n)Z(n) = (n+1−L)×L U(n) where U(n) is the resulting Cholesky triangular matrix with dimension L×L. It is important to note that this transformation doesn’t affect the cost function in (7) as Q(n) is a unitary matrix [8]. We avoid the ever increasing order in (9) by applying the triangularization using L + 1 × L + 1 Givens rotation matrices Qθ p (n) (for further details see [9]) γ p (n) gTp (n) Qθ p (n) = (10) f p (n) E p (n)
RAGHEB et al.: ENHANCED BLIND EQUALIZATION OF OPTICAL DP-QAM
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TABLE I C OMPUTATIONAL C OMPLEXITY OF D IFFERENT A LGORITHMS
Fig. 3. Convergence analysis (infinite precision) of proposed IQRD versus RLS and CMA in DP-16QAM (θ = π /4); OSNR = 20 dB. (a) MSE vs. iterations. (b) BER vs. iterations. (c) MSE ( res. CD = 250 − 1750 ps/nm). (d) MSE (DGD = 10 − 70 ps).
where γ p (n) is a scalar, and g p , f p , and E p depend on the type of triangularization process (i.e., upper or lower). This methodology allows the QR algorithm to attain the same convergence speed as that of the RLS; however it provides a numerically well-conditioned solution unlike the RLS algorithm, especially in finite precision environment. The IQRD-CMA algorithm avoids an extra back propagation step used in QR-decomposition by directly updating the inverse of the Cholesky factor, U(n) [9]. The weights vector update equation is given by [8] w p (n) = w p (n − 1) − γ p (n)e∗p (n)u p (n)
(11)
where γ p (n)u p (n) represents the Kalman gain, and u p (n) = 1/2 -H −γ p (n)λ p U−1 p (n−1)a p (n), where a p (n) = U p (n−1)z p (n). IV. S IMULATION R ESULTS AND D ISCUSSIONS In order to make an appropriate performance comparison, we simulate all the three blind equalizers to compensate for the residual CD and PMD, in 14Gbaud DP-16QAM coherent optical system operating at 112Gbps as in [5]. We use standard simulation parameters including a square root raised cosine filter with roll-off factor α = 1, a step size of μ = 10−3 for the CMA, a forgetting factor λ = 0.999 for both RLS and IQRD, and seven taps MIMO structure. We first analyze the convergence with infinite precision hypothesis then using finite precision case. A. Convergence Analysis in Infinite Precision In Fig. 3(a), we illustrate the MSE evolution versus the number of iterations of the three equalizers for two OSNR values: 18 and 20 dBs. The fiber is assumed to have residual (res.) CD, and differential group delay (DGD), of 1500 ps/nm and 50ps, respectively as in [5]. The angle between the input signal polarization and the fiber’s PSPs θ is π/2 (worst case). Simulation results show that the proposed IQRD achieves the same convergence rate as RLS, i.e., about 5000 iterations. The CMA, however, takes almost 10 times the number of
Fig. 4. Convergence analysis (finite precision-8bit) of proposed IQRD versus RLS and CMA in DP-16QAM (θ = π /4); OSNR = 20 dB. (a) MSE vs. iterations. (b) BER vs iterations. (c) MSE (res. CD = 0 − 1750 ps/nm). (d) MSE (DGD = 0 − 70 ps).
iterations to converge to its steady state. Moreover, it is also important to note that IQRD and RLS converge to practically the same value of residual MSE at their steady state regime that is about −8.5dB. In Fig. 3(b), we observe the evolution of BER during the convergence time of the equalizers and at their steady state. Observation of the BER confirms the obtained results from Fig. 3(a) in terms of convergence rate. The solid lines in Fig. 3(b) show the theoretical BER level of 16QAM AWGN system. It is important to remind that RLS and IQRD have the same computational complexity of O(L 2 ) in contrast to O(L) for the CMA. Table I summarizes the computational complexity for each algorithm in terms of complex addition and multiplication operations in addition to the number of iterations required for convergence. In Fig. 3(c) and (d), we extend our analysis of IQRD and RLS convergence to a wide range of CD and PMD, values respectively. In the first, we observe that the convergence time of IQRD and RLS is varying from 2000 to 7000 iterations when CD varies from 500 to 1750 ps/nm, respectively, and PMD close to zero. In Fig. 3(d), we fix the CD to negligible value and vary the PMD value from 30 up to 70ps and we measure a convergence rate varying from 2000 to 4000 iterations, respectively. At very low values of CD and PMD, the RLS and IQRD equalizers are in idle state. B. Convergence Analysis in Finite Precision Setting In Fig. 4, we analyze the convergence of CMA, RLS and IQRD in 8 bits finite precision conditions. In (a) we start to see a dramatic difference in performance between our
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 27, NO. 2, JANUARY 15, 2015
Fig. 5. MSE and BER (infinite precision) for 8, 10, and infinite bit resolution in DP-16QAM (θ = π /4); OSNR = 20 dB. (a) MSE vs. iterations. (b) BER vs. iterations.
Fig. 7. Tracking behavior of different equalizers for res. CD = 0; OSNR 20dB. (a) Infinite precision. (b) Finite precision-8 bits.
we show the tracking capability for different equalizers for infinite and finite precision, respectively for res. CD = 0 and 20 dB OSNR. The algorithms were able to track angular frequencies (ω) of 1Mrad/s and 0.5Mrad/s for infinite and finite precision, respectively, which is enough to track maximum variation of state of polarization (SOP) of ~10’s of radians in installed systems [10]. V. C ONCLUSION Fig. 6. BER analysis of IQRD versus RLS for dual polarization (θ = π /4), OSNR = 20 dB, at infinite and 10 bit resolution. (a) BER (res CD = 0 − 1500 ps/nm). (b) BER (DGD = 0 − 120 ps).
proposed IQRD and standard RLS in terms of convergence rate and MSE steady state value. They achieve convergence after 4000 and 2000 iterations, respectively. However, the steady state MSE value of IQRD is about 3dB lower than RLS at 8 bit resolution. This will have a larger impact on the BER performance after conversion. The CMA takes almost 10 times the number of iterations to converge to its steady state. Fig. 4(b) shows one order of BER magnitude of IQRD lower than RLS. This confirms the advantage of IQRD compared to RLS in terms of convergence rate and steady state values. In Fig. 4(c), we examine the steady state MSE of the three equalizers for negligible and wide range of residual CD values ranging from 750 to 1750 ps/nm. Specifically, we have −4 to 2dB for RLS and -6 to 0dB for IQRD. For all these simulated CD values, IQRD constantly improved the MSE steady state two times compared to RLS. In Fig4 (d), we repeat the experiment of Fig. 4(c) but by fixing CD to negligible value and vary the PMD from 30 to 70ps. In Fig. 5(a) and (b), we observe the MSE and BER convergence, respectively, of RLS and IQRD equalizers for 8 and 10 bit resolution setting as well as the infinite resolution for better understanding and interpretation. For 10 bit resolution, the IQRD achieves steady state MSE value with 0.5dB penalty with respect to infinite precision, however, RLS has 2dB penalty at same bit resolution. In Fig 6 (a) and (b), we summarize the key results associated to BER performance after convergence of the IQRD and RLS algorithms for infinite and finite precision, respectively. For an infinite precision, both algorithms can tolerate up to 1500 ps/nm of residual CD and 80 ps DGD distortions with BER less than 10−4 at 20 dB OSNR. However, at 10 bit finite precision, we clearly observe that IQRD BER overcomes that of RLS. IQRD can tolerate up to 1250 ps/nm of residual CD and 70 ps of DGD distortion with BER less than 3 × 10−4 at 20dB OSNR. However, for the same bit resolution, RLS fails to achieve similar BER performance. In Figure 7 (a) and (b),
In this letter, we have proposed and investigated the performance of IQRD-CMA algorithm for blind equalization of CD and PMD distortions in 14Gbaud DP-16QAM optical coherent system. It has been shown through simulation that for infinite precision, the IQRD-CMA and RLS-CMA algorithms achieve similar performance in terms of convergence rate, residual MSE, and BER. However for finite precision, our proposed IQRD outperforms the RLS for approximately 3 dB in steady state MSE and one order of magnitude in BER. Furthermore, with respect to RLS-CMA, the IQRD-CMA can maintain the system stability at acceptable number of bits resolutions, which leads to lower power consumption in finite precision hardware such as DSPs and FPGAs. R EFERENCES [1] S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Topics Quantum Electron., vol. 16, no. 5, pp. 1164–1179, Sep./Oct. 2010. [2] E. M. Ip and J. M. Kahn, “Fiber impairment compensation using coherent detection and digital signal processing,” J. Lightw. Technol., vol. 28, no. 4, pp. 502–519, Feb. 15, 2010. [3] I. Fatadin, D. Ives, and S. J. Savory, “Blind equalization and carrier phase recovery in a 16-QAM optical coherent system,” J. Lightw. Technol., vol. 27, no. 15, pp. 3042–3049, Aug. 1, 2009. [4] A. Ragheb, M. Shoaib, S. Alshebeili, and H. Fathallah, “Inverse QR decomposition (IQRD) blind equalizer for QAM coherent optical systems,” in Proc. 9th Int. Conf. High Capacity Opt. Netw. Enabling Technol. (HONET), Dec. 2012, pp. 221–225. [5] M. Selmi, C. Gosset, M. Noelle, P. Ciblat, and Y. Jaouen, “Block-wise digital signal processing for PolMux QAM/PSK optical coherent systems,” J. Lightw. Technol., vol. 29, no. 20, pp. 3070–3082, Oct. 15, 2011. [6] D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun., vol. 28, no. 11, pp. 1867–1875, Nov. 1980. [7] Y. Chen, T. Le-Ngoc, B. Champagne, and C. Xu, “Recursive least squares constant modulus algorithm for blind adaptive array,” IEEE Trans. Signal Process., vol. 52, no. 5, pp. 1452–1456, May 2004. [8] J. A. Apolinário, Jr., QRD-RLS Adaptive Filtering, 1st ed. New York, NY, USA: Springer-Verlag, 2009. [9] M. Lucena, J. A. Apolinário, Jr., and M. Shoaib, “The inverse QRD setmembership RLS algorithm,” in Proc. EUSIPCO, vol. 21. Sep. 2013, pp. 1–5. [10] S. Salaun, F. Neddam, J. Poirrier, B. Raguenes, and M. Moignard, “Fast SOP variation measurement on WDM systems are the OPMDC fast enough?” in Proc. ECOC, Sep. 2009, pp. 1–2.
