Hardware-efficient frequency offset and phase noise mitigation in coherent optical quadrature amplitude modulation systems

ELECT A hardware-efficient adaptive algorithm for frequency offset and phase noise mitigation in coherent optical quadrature amplitude modulation systems is presented and analysed. Hardware efficiency for the mitigation of imperfections is achieved by computing directly the complex exponential, thus avoiding a CORDIC processor in the phase recovery loop. The design includes a square-rootand trigonometry-free update. Simulation results substantiate the theoretical findings.

Problem formulation: Consider an equalised and time-synchronised coherent optical quadrature amplitude modulation (CO-QAM) system.This Letter focuses on mitigation of laser phase noise (PN) and frequency offset (FO).Let denote the baud-rate samples of the output of the matched filter at the receiver, where i = −1 √ , k is the discrete index denoting the time kT, T is the symbol period, θ k denotes the PN, x k is an independent and identically distributed (i.i.d.) sequence, Ω = 2πTΔf represents normalised FO (where Δf is the frequency difference between the transmitter laser and the local oscillator (LO)), and w k is a symmetric i.i.d.zeromean Gaussian additive noise with independent real and imaginary parts.The PN, θ k , follows a random walk model where q k is a zero-mean Gaussian distributed i.i.d.sequence with variance s 2 q = 2pT Dv, and Δv is the combined linewidths of the transmitter laser and the LO [1].Let u k be an estimate of θ k available at the receiver, then an estimate of the signal x k is given by z k = y k exp (−i u k ).One of the admissible methods for finding u k is to minimise the fourth-order statistics of the derotated sequence {z k } with respect to u k [2,3].The fourth-power cost is which results in a globally convergent update where μ is the loop gain (LG), and z k,R and z k,I are the in-phase and quadrature components of z k , respectively.Once u k is known, the traditional way is to obtain the complex exponential exp (i u k ) using a CORDIC processor [4], which computes the required trigonometric quantities, i.e. cos ( u k ) and sin ( u k ).Note that a single-axis variant of (3) has been recently used in the vestigial sideband [5] and QAM-based CO-OFDM systems for adaptive PN mitigation [6].
In this Letter, we suggest minimising the cost function J 4 (z k ) with respect to r k := exp (i u k ) to obtain a complex-valued update as follows: where z k = r * k y k .Since r k := exp (i u k ) is readily available, it avoids trigonometric operations.Note that update (4) requires normalisation to unit amplitude which may lead to needing a CORDIC processor to compute the required square-root.To avoid this, in this Letter r k+1 is normalised approximately by projecting it onto a unit polygon: where proj( • ) is performing a low-complexity normalisation, < and ℑ denote real and imaginary operations, respectively, and the choice of ( p 1 , p 2 ) = (15/16, 15/32) ensures a good polygonal fit (see Fig. 1).
Performance analysis: Below, we first analyse a generic synchroniser and then extend the results to the update (5).Let , where e k has similar statistical properties as that of additive noise w k .
Theorem 1: Consider any adaptive synchroniser of the form ( 6) and assume filter operation in steady state with PN but no FO.Assume further that a k = y k exp ( − i u k,opt ), where u k,opt (an optimal estimate of θ k ) varies according to the random-walk model u k+1,opt = u k,opt + q k , and q k (with the same statistics as defined in ( 2)) is independent of {a m } and {y m , u 0,opt } for all m < k.Denoting c k = u k,opt − u k , the following variance relation holds: Proof: See [7] for the energy conservation-based variance relation for an adaptive filter operating in a non-stationary environment.Next, we mention the proposition of Moridi and Sari [8] By combining Theorem 1 and Proposition 1, we have an optimal LG for the mitigation of PN, as summarised in the following theorem.Theorem 2: Considering the variance relation (7) and Proposition 1, we have The optimal LG that tracks the PN (2) with minimum PJV is given by where approximations are based on the assumption that 4ga 2 ≫ b 2 s 2 q .
Proof is simple and thus skipped.Note from Theorem 2 that both PJV and LG are proportional to the standard deviation of the PN.The larger the deviation, the larger the LG to track.Next, we state a generic theorem for the evaluation of optimal LG for the system exhibiting a fixed FO.Theorem 3: Consider any adaptive synchroniser of the form (6) with residual c k = u + kV − u k (where u denotes a possible fixed phase offset), and assume a filter operation in steady state.The PJV is Eψ ∞ is the tracking error and the optimal LG that minimises (11) is where ELECTRONICS LETTERS 21st January 2016 Vol.52 No. 2 pp.131-133 To mitigate both PN and FO, we have the following theorem.Theorem 4: Consider any adaptive synchroniser of the form (6) with residual c k = u k + kV − u k , and assume a filter operation in steady state.The PJV is minimisation of which yields an optimal LG where Finally, for the update (5), we have the following result.Proposition 2: Considering the proposed synchroniser ( 5), and assuming Taking conjugate and multiplying with exp (iθ k ), we get exp Assuming exp(iψ k ) ≈ 1 + iψ k , and comparing the coefficients, we get a real-valued update , an exact expression for E[E 2 k |c k ] is evaluated but the result is too long to be included here.Exploiting Taylor's series-based simplifications sin (ψ) ≈ ψ and cos (ψ) ≈ 1 − 0.5ψ 2 , and statistics Ee 2 R = s 2 e , Ee 4 R = 3s 4 e , and Ee 6 R = 15s 6 e for the quadrature components of additive Gaussian noise e k , however, we obtain Owing to our analytical findings, we can evaluate the steady-state performance of the proposed update (5) in the presence of PN or FO or both for any given LG, QAM size, and SNR values.Since the analysis assumes an open-eye condition, therefore, we are not considering other optical interferences like polarisation mode and chromatic dispersions.
Simulation results: We consider a 2.5 Gbit/s, Gray-encoded, 16-QAM signalling (i.e. two uncorrelated 625 Mbaud quaternary data streams) under four different simulation scenarios as depicted in the legend of Fig. 2, compute the simulated PJV (for one of the polarisations), and compare it with analytical PJV.We also consider differential encoding to resolve the four-fold phase ambiguity.As the proposed update attempts to calculate directly the complex exponential, the steady-state value of PJV (for the given LG, μ) is estimated as where c = 2 p p/4 −p/4 cos (t) 2 dt = (2 + p)/(2p) = 0.8183 is obtained by exploiting the mean-value theorem and Lipschitz continuity of sin(t), i.e. by realising the fact that E(t 1 − t 2 ) 2 ≥ (1/c)E(sin (t 1 ) − sin (t 2 )) 2 , for t 1 , t 2 ∈ [−π/4, π/4), and c > 0.Moreover, θ k,j and r k,j , respectively, denote the realisation of PN and the estimated complex-exponential in the kth iteration of the jth run, where N runs = 500 and N iter = 5000.Each simulation point is thus obtained as an average of N runs runs each having N iter iterations with independent and random realisation of PN, additive noise, and signal points.Our analytical findings are in close agreement with the simulation results, as seen in Fig. 2.

a 6 PJV, E 2 •Fig. 2
Fig.2Plots of PJV against LG for 16-QAM signalling Conclusion: A hardware-efficient adaptive synchroniser for CO-QAM systems is proposed for the first time, is analysed and demonstrated to mitigate laser PN and FO.Compared with the traditional synchroniser, the proposed update computes the complex exponential directly without requiring any trigonometric and square-root operations.Results for the PJV are obtained for a wide range of LG values under different simulation scenarios, where the analytically obtained optimal LGs are found to be in close agreement with the simulated ones.Our proposed technique is proved as a substitute for the traditionally used ones.
in relation to the statistics involved in Theorem 1.□ [9]ig.1Polygon fit to constant modulus[9].On xy-plain, the unit-radius polygon is expressed as (15/16) max(|x|, |y|) + (15/32) min(|x|, |y|) = 1 Proposition 1: Consider any adaptive synchroniser of the form (6) and assume a filter operation in steady state, it is true that Ec k E k = aEc 2 k , and EE 2 k = bEc 2 k + g, with appropriate α, β, and γ.The steady-state phase jitter variance (PJV) is Ec 2 1 = mg/(2a − mb), and the convergence speed is characterised by the time constant On the basis of the assumption |r k | ≈ 1, we write (below