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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 1981–2003
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Carrier phase estimation for optically coherent QPSK based on Wiener-optimal and adaptive Multi-Symbol Delay Detection (MSDD)

Netta Sigron, Igor Tselniker, and Moshe Nazarathy  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 1981-2003 (2012)
http://dx.doi.org/10.1364/OE.20.001981


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Abstract

The MSDD carrier phase estimation technique is derived here for optically coherent QPSK transmission, introducing the principle of operation while providing intuitive insight in terms of a multi-symbol extension of naïve delay-detection. We derive here for the first time Wiener-optimized and LMS-adapted versions of MSDD, introduce simplified hardware realizations, and evaluate complexity and numerical performance tradeoffs of this highly robust and low-complexity carrier phase recovery method. A multiplier-free carrier phase recovery version of the MSDD provides nearly optimal performance for linewidths up to ~0.5 MHz, whereas for wider linewidths, the Wiener or LMS versions provide optimal performance at about 9 taps, using 1 or 2 complex multipliers per tap.

© 2012 OSA

1. Introduction

Our objective is a novel CR technique for QPSK optically coherent links, based on Multi-Symbol-Delay Detection (MSDD), called Multi-Symbol-Differential Detection (also with MSDD acronym) in the wireless literature, alternatively referred to by the synonym term Multi-Symbol-Phase Estimation (MSPE) which was also used in photonic applications.

It turns out that, beyond QPSK, our MSDD methodology is also applicable to QAM coherent detection, as well as to carrier frequency offset (CFO) estimation in addition to phase estimation. Nevertheless, for ease of exposition of the initial concept, in this work we focus exclusively on thoroughly deriving and explaining MSDD carrier phase estimation (CPE) principle for QPSK coherent detection, relegating to a future publication the additional MSDD extensions to QAM and to CFO tracking and correction. The MSDD CPE method is theoretically derived and simulated here in the QPSK transmission context, however we emphasize that our method is actually “QAM-ready”—the block diagrams developed here will function for QAM as well, however QAM extensions our outside the QPSK-oriented paper scope.

We aim to establish MSDD as a preferred alternative for accurate yet simple QPSK carrier phase estimation and correction. Unlike prior methods, our MSDD method is optimal in the Minimum-Mean-Square-Error (MMSE) sense, in the wake of channel statistics consisting of a combination of ASE-induced phase noise (PN) and laser phase noise (LPN), i.e. the MSDD CR will exhibit the best possible OSNR performance and tolerance to laser linewidth (LW). The adaptive LMS version, as derived here in detail for the first time, requires no prior knowledge of channel statistics – it learns the channel whatever the relative strengths of ASE and LPN (OSNR vs. LW) are, automatically adjusting the taps for optimal performance.

Notice that we inevitably require multiple, L, taps in order to suppress the phase noise by an effective averaging effect. The computational complexity of our optimized algorithm is about one complex multiplier (CM) per tap for the Wiener-optimal version with fixed coefficients and about 2 CMs per tap for the LMS-adaptive version. However, at the expense of slight (or in some cases negligible) reduction in performance, if we give up optimized coefficients but rather make all tap coefficients equal to unity, we obtain an MSDD variant of ultimate simplicity: The CPE becomes multiplier-free. This version has negligible performance penalty relative to a fully optimized MSDD, in the prevalent scenario that for coherent-grade lasers with 100 KHz linewidth are used in the transmitter and for LO, and even up to 0.5 MHz linewidth for a parallelization factor of 16. In addition to performance and complexity metrics, we should also mention that the MSDD CR method is robust, providing uninterrupted operation, as MSDD processing is essentially linear time-varying, rather than non-linear, thus cycle slips and other non-linear phase-wrapping artifacts of the competing leading M-power (Viterbi&Viterbi) method for QPSK CR, are completely eliminated.

The paper is structured as follows: Section 2 reviews generic CR concepts and discusses the naïve Delay Detector (DD), which is extended in section 3 to the more advanced MSDD concept, explaining the MSDD principle of operation. Section 4 develops a Wiener filtering solution, optimizing the MSDD coefficients for a channel affected by both ASE-induced and laser source phase-noises. In Section 5 we derive an LMS adaptive algorithm for the MSDD coefficients. Section 6 introduces efficient implementations and evaluates computational complexity of the MSDD. Two hardware structures are derived: a very low complexity multiplier-free CPE which is non-adaptive and non-optimized (but displays nearly optimal performance for low linewidths) and a more complex optimally performing Wiener or LMS-adaptive version. Section 7 develops the polyphase hardware parallelization of the MSDD. Section 8 presents numeric simulation performance results and Section 9 concludes the paper.

Appendix A reviews some differential precoding mathematical properties, Appendix B details the derivation of the Wiener optimal solution and Appendix C collects the relevant abbreviations used in this paper.

2. Carrier recovery (CR) concepts – naïve delay detector (DD)

2.1 Differential precoding

Differential precoding is used in Direct Detection Differential Phase Shift Keying (DPSK) systems, yet here we are interested in CR for coherent rather than direct detection. Our motivation for reviewing and expanding the DP concept is that MSDD carrier recovery may be viewed as a generalization of DPSK, retaining some of the DPSK advantages while overcoming the sensitivity disadvantage of DPSK. A coherent QPSK transmitter (Tx) intended to operate with an MSDD based receiver (Rx), should include a Differential Precoder (DP) (Fig. 1
Fig. 1 QPSK/QAM transmitter scheme with (modulus-preserving) differential precoder, compatible with a MSDD-based receiver.
). Each information symbol, as selected out of the QPSK complex alphabet, s˜k{A,jA,A,jA} is mapped into a line symbol from the same alphabet, A˜k{A,jA,A,jA}, according to following recursion which defines the DP mapping:
|A˜k|=|A˜k1|=A;A˜k=s˜k+A˜k1s˜k=A˜kA˜k1
(1)
The line symbols A˜kgenerated at the DP output are pulse-shaped and optically transmitted.

The DP recursion [Eq. (1)] amounts to an additive accumulator in the phase domain: The QPSK information phase s˜ksets the difference A˜kA˜k1 between two successive phases of the line symbols, i.e. information is encoded in the phase differences transmitted on the line.

A more mathematically abstract formulation of the DP, amenable to generalizing the current QPSK MSDD to a higher-order QAM constellation, is obtained in terms of the following unimodular normalization operation, referred to as “Uop”,
z˜U{z˜}z˜/|z˜|=ejz˜=z˜/z˜*
(2)
which normalizes a given phasor (complex-number z˜) into a unimodular output phasor (unimodular means unity modulus, |z˜|1), retaining the same angle (argument): z˜=z˜.

A modulus-preserving differential precoder (MP-DP) applicable to both QPSK and QAM was proposed by N. Kikuchi [26

26. N. Kikuchi and S. Sasaki, “Highly sensitive optical multilevel transmission of arbitrary quadrature-amplitude modulation (QAM) signals with direct detection,” J. Lightwave Technol. 28(1), 123–130 (2010). [CrossRef]

]. In the polar domain this MP-DP is described as accumulating the phase, A˜k=m=0ks˜mwhile preserving the modulus, |A˜k|=|s˜k|. In the Uop based complex notation, the MP-DP is compactly represented as A˜k=s˜kA˜k1:
A˜k=s˜kA˜k1|A˜k|=|s˜k|andA˜k=s˜k+A˜k1.
(3)
Here we prefer to express Kikuchi’s MP-DP description, originally expressed in the polar (magnitude and phase) domain, in the more abstract equivalent form A˜k=s˜kA˜k1, formulated terms of the Uop. The mathematics of the Uop and MP-DP modules are developed in Appendix A.

In this paper our exclusive focus is on QPSK coherent transmission. It is readily verified that the MP-DP transformation, Eq. (3), generally applicable to QAM, reduces to Eq. (1) in the special case of a QPSK constellation, wherein |s˜k|=A. Henceforth, for brevity, we use the term DP in the sense of MP-DP (such as in Fig. 1). Resorting to the complex description A˜k=s˜kA˜k1 for the DP will facilitate of mean-square optimal MSDD coefficients derivation in section 4.

2.2 Link model including the CR

p˜k(1+η˜k)ejϕkLPN=|1+η˜k|ejϕkASEejϕkLPNejϕk;ϕk=ϕkLPN+ϕkASE;ϕkASE=(1+η˜k)
(7)

Figure 3
Fig. 3 (a): Carrier Recovery (CR) system operation. (b): Demodulator as complex conjugate multiplier.
reviews the top-level internal structure of a generic CR module, essentially comprising a Carrier Phase Estimation (CPE) module followed by a demodulator.

2.3 Naïve delay detector

A simple CPE strategy is to use delay-detection (DD). The simple CR s˜k=r˜kr˜k1*is variously referred to as delay detector, differential detector or delay demodulator (all abbreviated as DD). The received signal r˜k=|r˜k|ej(A˜k+ϕk)is then conjugate-multiplied with a unit-time delayed version of itself, r˜k1=|r˜k1|ej(A˜k1+ϕk1), yielding
s˜k=r˜kr˜k1*=|r˜k||r˜k1|ej(A˜kA˜k1+ϕkϕk1)=|r˜k||r˜k1|ej(s˜k+ϕkϕk1)
(8)
where the differential precoding relation Eq. (1) was used in the last equality. The rounded hat on the CR output s˜k indicates that this is an “analog” estimate of the transmission symbol s˜k, attempting to approximate at least the phase s˜k=A˜kA˜k1of s˜k as faithfully as possible. This noisy estimate, s˜kis then sliced (its phase is quantized) in order to extract the decision s˜^k (pointed hat denotes decision, rounded hat denotes the CR output – noisy estimate of s˜k to be input into the slicer). As is well known, the naïve DD is too noisy (approximately doubles the input ASE noise power) thus fails to provide a useful CR for coherent detection. Nevertheless the delay-detection concept is the starting point leading to the high-performance MSDD CR realization, interpreted as a generalization of the naïve DD.

3. From the naïve DD to MSDD carrier recovery

3.1 MSDD principle: Generation of an improved reference from prior received samples

In a naive DD, the last sample,r˜k1, is just too noisy a phase reference. Let us then also process the earlier samples, i.e. generate our CPE by acting on moving window of L past samples, r˜k1,r˜k2,r˜k3,....,r˜kL, in order to form an improved reference, R˜k1, and demodulate the received samples with it, forming an improved decision variable s˜k=r˜kR˜k1* to be presented to the slicer (Fig. 4(a)
Fig. 4 (a): MSDD CR system operating on a window of L past samples in order to generate an improved reference for demodulation. (b): The MSDD CPE may be alternatively realized by acting on each past sample at a time, rotating the respective past sample in order to bring it in approximate alignment with the last sample (the conventional DD reference). Either of these rotated-into-alignment past samples may be selected to serve as a “partial” reference. The improved reference will be obtained by taking a linear combination of all of the partial reference (Fig. 5). The multipliers vertically arrayed on the left indicate that, for each past noisy sample, rotating the sample before it by a corresponding transmission symbol yields a result approximately aligned with the respective past sample.
). Hopefully, the improved reference, R˜k1, will be quieter than the original reference, r˜k1. The problem we are facing in processing the earlier samples in order to form R˜k1, is that the prior samples r˜kiare not aligned with r˜k1due to the data modulation, thus if we attempt to use the r˜kiterms instead of r˜k1in the delay demodulation, then the demodulation will not function properly, unless we first rotate each of the r˜ki terms in order to make them aligned with r˜k1. In Fig. 4(b) we show how this alignment process is applied one prior sample at a time. We already know we can accomplish proper (albeit noisy) delay detection with r˜k1as phase reference. Next, let us consider r˜k2as potential phase reference. The key is to revisit the DP recursion A˜k=s˜kA˜k1 Eq. (1) for the transmitted line symbols. A similar recursion approximately holds between the received symbols, r˜ks˜kr˜k1, as according to Eq. (5), the r˜k-s are just noisy versions of A˜k. Shifting the approximate recursion r˜ks˜kr˜k1back one time-unit (substituting kk1) yields r˜k1s˜k1r˜k2, indicating that if r˜k2is rotated by the complex information symbol s˜k, we then obtain a rotated symbol, R˜k1(2)s˜k1r˜k2, which is roughly aligned with the DD reference, r˜k1. Similarly, (taking r˜k1s˜k1r˜k2 yet another time-unit back) r˜k2s˜k2r˜k3, thus compounding the last two equations yields the relation r˜k1s˜k1r˜k2=s˜k1s˜k2r˜k3, which indicates that R˜k1(3)s˜k1s˜k2r˜k3may also be alternatively used as phase reference, as it is also nearly aligned with r˜k1. More generally,
R˜k1(i)s˜k1s˜k2...s˜ki+1r˜kii=1,2,...,
(9)
may be used as alternative phase references (for i = 1 we simply retrieve the original reference R˜k1(1)r˜k1). Each of these references are nearly aligned with r˜k1and may be used instead of the originally considered R˜k1(1)r˜k1reference in order to delay-demodulate the received symbol r˜k. Demodulation with R˜k1(i) yields the “partial estimates” s˜k(i)r˜kR˜k1(i)*each of which may be sliced in order to extract the k-th decision, s˜^k.

Which of the alternative phase references R˜k1(i),i=2,3,...should be used for demodulation? It turns out that no particular one is preferred; however, the question arises whether we can take advantage of them all, combining these partial references into an improved reference generating a higher quality decision. In the case ASE-induced PN is a significant component of the overall PN (which usually holds when coherent-grade lasers are used), as white ASE noise is dominant, the partial references are essentially mutually independent. In this case it is advantageous to form a linear combination of these partial references (in the simplest case take their sum), generating an improved reference, as follows:
R˜k1=c1r˜k1R˜k1(1)+c2s˜^k1r˜k2R˜k1(2)+c3s˜^k1s˜^k2r˜k3R˜k1(3)+c4s˜^k1s˜^k2s˜^k3r˜k4R˜k1(4)+...
(11)
This improved reference is used demodulate the received samples, generating an improved decision variable s˜kr˜kR˜k1*which is then input into the slicer. The resulting CR system, as illustrated in Fig. 5(a)
Fig. 5 (a): MSDD CR system generating an improved demodulation reference by a linear combination of past samples rotated into approximate alignment with the last sample. (b): Phasor diagram description of the process of rotation-into-alignment of past samples. The improved reference is obtained here as a sum (special case of general linear combination) of the aligned past samples. The summing process yields an averaging effect, improving the signal to noise ratio of the resulting reference, relative to just using the last sample as delay-demodulation reference.
, is the MSDD.

The improved reference R˜k1 Eq. (11) is seen to be formed as a linear combination of L partial references, namely prior samples, phase rotated into alignment. A phasor diagram presenting the rotation (alignment) process of the various past samples is shown in Fig. 5(b). In this figure for simplicity, all linear combination coefficients are taken equal, ci=1, such that the linear combination forming the improved reference reduces to a sum of prior rotated samples. If there were no noise, the prior samples r˜k2,r˜k3,....,r˜kLwould become perfectly aligned with r˜k1. If the PN is entirely white (ASE-induced, i.e. no LPN, i.e., Eq. (5) reduces to r˜k=A˜k+n˜k), then these noisy phasors are no longer perfectly aligned, yet are nearly collinear with r˜k1(which is itself perturbed by additive noise). Nevertheless, the additive white noise perturbations, n˜k, added to the noiseless symbols Ak1,Ak2,Ak3,....,AkL, to form the prior samples r˜k1,r˜k2,r˜k3,....,r˜kL, are mutually uncorrelated, thus add up incoherently; in amplitude, the noises add up on an RMS basis, such that the total RMS noise grows as Las the window size of past samples, L, is increased. In contrast, the noiseless components Ak1,Ak2,Ak3,....,AkLof the received samples, are all collinear and have equal lengths, thus the total signal component of the improved reference grows up linearly in L, therefore the SNR of improved reference grows up as (L/L)2=L.

It appears advantageous to accrue the noise averaging effect over arbitrarily long windows (though in practice, we would get diminishing returns beyond a certain window size, and the computational complexity must also be taken into account). However, when LPN is present, an opposite effect is at work, namely the longer the record of past samples used in forming the improved reference, the worse the LPN induced degradation. Thus, a “block length” effect emerges – it does not pay to increase the block length L indefinitely, but there is an optimal block length, L, as determined by the balance of the ASE and laser phase noises. In this simplified analysis we assumed equal coefficients, ci(taken as unity without loss of generality), but more generally the linear combination coefficients may be arbitrarily selected, in the combined presence of ASE and LPN phase noise sources. In section 4 we apply Wiener filtering theory in order to determine unequal optimal coefficientsc^iwhich yield the best performance for any given block length L, striking the best balance between the opposing effects of ASE and LPN.

3.2 MSDD alternative formulation in terms of partial DD estimators

To derive an alternative point of view of the MSDD demodulation process, let us substitute the improved reference Eq. (11) into the demodulation relation, s˜k=r˜kR˜k1*, yielding
s˜k=r˜k[i=1LciR˜k1(i)]*=i=1Lcir˜kR˜k1(i)*=i=1Lcis˜k(i)
(12)
where we used Eq. (9) for the i-th partial reference and introduced the i-th partial estimator
s˜k(i)=r˜kR˜k1(i)*=[s˜k1s˜k2...s˜ki+1]*r˜kr˜ki*
(13)
It is apparent in Eq. (12) that the MSDD estimate s˜k of the transmitted symbol may be expressed as a linear combination of partial estimators s˜k(i), each obtained by demodulation with a partial reference, each of which could by itself provide a valid, albeit noisier, estimate for the information symbol, as described in the block diagram of Fig. 5(a). It turns out that this alternative equivalent realization of the MSDD would be less desirable for efficient computation than the original MSDD block diagram of Fig. 5(a), which generates first the improved reference, Eq. (11), as a linear combination of partial references, then demodulates with it. Nevertheless the alternative partial-estimators formulation Eq. (12) is more amenable to Wiener optimization, as pursued next.

4. Optimal Wiener-filtering based Minimum Mean Square Error (MMSE) solution

In this section we derive the MMSE optimal solution, which aims at minimizing the Mean Square Error (MSE) between the QPSK or QAM symbols, s˜k, and their estimates s˜k, as generated at the MSDD output (slicer input). Introducing the estimation error ε˜k=s˜ks˜k , we seek the optimal MSDD coefficients minimizing the MSE, |ε˜k|2=|s˜ks˜k|2.

Note that, for the purpose of QPSK detection, we have heretofore ignored the magnitude (modulus) of the improved estimate s˜k=r˜kR˜k1*, which is generated by mixing the received symbol r˜kwith the improved reference R˜k1=i=1LciR˜k1(i), in the process of the generation of which the magnitudes were not normalized. As the QPSK slicer essentially acts on the angle of s˜k, ignoring the magnitude |s˜k|does not pose a problem. A different length of the reference phasor will just scale the modulus of estimate s˜k without affecting its phase. However, once QPSK transmission is extended to QAM, the references magnitudes do become important. Even in the current QPSK context, proper processing of references magnitudes does become essential in the MMSE formulation and derivation. Indeed, although the phase of our slicer input s˜k generated by the MSDD tends to be close to that of the actual transmission symbol, s˜k, nevertheless if the magnitudes of s˜kand s˜kare disparate, then a large MSE deviation may still be generated, defeating the minimization process. Thus, in order to properly optimize the MSDD coefficients, it is imperative to properly scale magnitudes, such that the estimate s˜kbe made to approach s˜knot only in phase but also in modulus, and a small residual estimation error may be generated. Here we use the Uop normalization Eq. (2) as a key step enabling to devise a modified MSDD structure for QPSK (also applicable to QAM), suitable for attaining the MMSE condition. To this end, we propose to apply the Uop to the partial references, R˜k1(i), now to be replaced by Uop-normalized versionsR˜k1(i)(which preserve the original angles of R˜k1(i), i.e., are still nearly aligned with r˜k1, hence are also suitable to form an improved reference):

R˜k1(i)=U{r˜kis˜ki+1s˜ki+2...s˜k1}=r˜kis˜ki+1s˜ki+2...s˜k1
(14)

The resulting MSDD improved reference is then formed by the linear combination
R˜k1=i=1LciR˜k1(i)
(15)
Here, the inverted under-hat at the bottom of R˜k1, though resembling the inverted over-hat used to denote the Uop, does not actually indicate, that R˜k1is a normalized unimodular quantity, but it rather signifies here that it is formed as a linear combination of quantities which are themselves unimodular. Notice that a linear combination of Uop-normalized quantifies is generally not unimodularly normalized itself (unimodularity is not preserved under a linear combination). In fact, whether or not R˜k1 is unimodular depends on the c-coefficients selection, which for the MMSE solution pursued below, will assume optimal values which make R˜k1nearly unimodular (notice that |R˜k1|1ifi=1Lci1). This modified version of the MSDD will be referred to as “U-notU”, as the partial references are Uop normalized, whereas the improved reference R˜k1is not necessarily Uop normalized.

Using the U-notU magnitude normalizations proposed here, the modified “U-notU” MSDD is analyzed in Appendix B in terms of the phase-noisy memoriless channel model.

The overall improved estimate Eq. (12) is a linear combination of the partial estimatess˜k(i). It is then useful to explicitly express dependency of the partial estimates on the phase noise Eq. (7):
s˜k(i)r˜kR˜k1(i)*=r˜kU{R˜k1(i)*}=r˜kU{r˜kis˜ki+1s˜ki+2...s˜k1}*=r˜kr˜ki*s˜ki+1*s˜ki+2*...s˜k1*=(A˜kp˜k)(A˜ki*p˜ki*)s˜ki+1*s˜ki+2*...s˜k1*=A˜kA˜ki*s˜ki+1*s˜ki+2*...s˜k1*p˜kp˜ki*=s˜kp˜kp˜ki*
(16)
where in the last expression we used the generalized delay-detection relation Eq. (34) derived in Appendix A, namely s˜k=A˜kA˜ki*s˜ki+1*s˜ki+2*...s˜k1*. The resulting Eq. (16) indicates that s˜k(i),i=1,2,...,L indeed qualify as partial estimators for s˜k, as they essentially coincide with the transmitted symbols s˜k, apart from multiplicative phase noise perturbations p˜kp˜ki*.

5. LMS algorithm for the MSDD coefficients

In practice, the channel phase-noise statistics (balance of laser phase noise, ASE, and also nonlinear phase noise contributions) is unknown and may even be time-varying. Therefore, it is advantageous to devise an adaptive method to approach the optimal MSDD coefficients automatically. Here we derive an LMS algorithm for the “U-notU” MSDD coefficients.

6. Efficient hardware implementations

In this section we derive an efficient hardware implementation for the MSDD sub-system, as illustrated in Fig. 6 below. In this block diagram, the number of complex multiplications is reduced below the one implied by Eq. (11), which indicates that the MSDD must calculate, in every clock cycle, the linear combination Eq. (15) of partial references. The direct evaluation of the i-th partial reference, Eq. (9),R˜k1(i)s˜^k1s˜^k2...s˜^ki+1r˜ki, seems to require i-1 multiplications by s-symbols to be applied to r˜ki, per clock cycle. The diagram of Fig. 6 presents a more efficient realization, first disclosed in [30

30. N. Sigron, I. Tselniker, M. Nazarathy, A. Gorshtein, D. Sadot, and I. Zelniker, “Ultimate single-carrier recovery for coherent detection,” in OFC’11 Conference on Optical Fiber Communication, OMJ2 (2011).

], wherein just a single complex multiplication of r˜kiwith an s-symbol is performed per clock cycle, rather than i-1 complex multiplications. This complexity reduction is achieved by a skillful arrangement of multipliers interspersed with delay line at the top of the figure, used to generate the partial references.

In addition, we modify the block diagram of Fig. 6 to incorporate the Uop required in the modified MSDD structure introduced in the last section, enabling either the MMSE optimal solution derived in section 4 or the LMS based adaptive solution, as per section 5. Uops are also required for extending QPSK to QAM transmission, to be pursued in future publication, thus our QPSK MSDD structure is “QAM-ready”.

The block diagram further features a coefficients control module tasked with generating the optimal coefficients, ci, whether by an offline MMSE calculation (solution of the W-H equation as derived in the last section, or alternatively (preferably) by means of the adaptive LMS algorithm Eq. (28). In addition, in order to implement the U-notU MSDD modification, a Uop acting on the received samples, r˜k, is inserted ahead of the partial references delay line at the top of the figure.

6.1 MSDD hardware realization complexity (excluding the adaptive coefficients control)

Inspecting Fig. 6, let us initially ignore the complexity of the coefficients control module. We then count, in the core MSDD system, L complex multipliers (CM) of the partial estimates with the cicoefficients, as well as L multipliers performing rotations by the QPSK decision symbols. As multiplications by the QPSK constellation points {±1,±j}are trivial, those do not contribute complexity. In addition, we have an extra full-fledged CM for the demodulation, s˜kr˜kR˜k1*. We should also account for the Uop complexity, which was estimated in Appendix A to consist of four real-multipliers, amounting to 113 CMs, expressing complexity in equivalent CM terms. Thus, overall there are L+213 complex multiplications to be performed per clock-cycle. We further note that the complexity of the L multiplications with the cicoefficients may be reduced by quantizing the coefficients to various degrees, setting a tradeoff between complexity and precision (i.e., CR performance).

A simplified system is obtained for ci=1/L, replacing the L coefficients multiplications by a single scaling multiplication performed prior to the demodulation, as indicated in Fig. 7. In this reduced-complexity MSDD version, as the coefficients are no longer optimized, the Uop may be discarded. Thus, this highly efficient non-optimal QPSK MSDD version, with uniform coefficients, requires just a single heavy complex multiplication (the one used for the demodulation). Evidently, as non-optimal coefficients are used, there is some performance degradation. Nevertheless, if the laser linewidth does not exceed 0.5 MHz, the resulting performance penalty will be quite small or even negligible (see section 8). This reduced complexity MSDD implementation is preferred for 100G QPSK low-cost applications.

6.2 MSDD with adaptive coefficients control and its total complexity

At the high-end extreme, consider a high performance system with its coefficients LMS-optimized, as described in Fig. 8
Fig. 8 QPSK/QAM receiver with “U-notU” adaptive MSDD, including full detail on the coefficients adaptation mechanism. This scheme, intended here for QPSK, is “QAM-ready”. For QPSK, the Uop-2 module may be discarded (and Uop-3 is always a trivial lookup-table) thus just Uop-1 is required to implement the QPSK U-notU version.
, which details the inner workings of an adaptive realization of the “coefficients control” module of Fig. 6. This diagram precisely implements the functionality of the adaptive algorithm described in section 5.

Accounting now for the contribution of the coefficients adaptation to complexity, we must consider additional CMs: Another full-fledged CM, ε˜k*r˜k, an easy CM generating fixed scaling by μ(which may be quantized to a convenient value, with few one-bits, which is simple to multiply by, so it will not be counted), then another full-fledged CM r˜kR˜k1*required for generating the error, ε˜k=s˜kr˜kR˜k1* Eq. (29), plus L full-fledged CMs generating the coefficient updates, Ui[k]=με˜k*r˜kR˜k1(i)*. Thus L + 2 extra full-fledged multiplications are required for the adaptive part, which when added to the L+213non-adaptive multiplications, yields a total of 2L+413 full-fledged complex multiplications for the high-end adaptive CR realization of Fig. 8 for our high-end L-taps MSDD system. As FIR taps are inevitable for optimized averaging of the noise, it is apparent that we must invest slightly more than 2 CMs per tap (one for the actual tap multiplication, and a second one for the adaptation) in order to attain the noise reduction with optimized coefficients. The alternative is to settle for the non-optimized version of Fig. 7, in which we just have a single CM overall, but performance is slightly degraded.

7. Polyphase parallelization

Due to its usage of decision–feedback, the MSDD algorithm poses an implementation challenge for coherent optical receivers operating at tens of GBd rates, given that the fastest multipliers currently available with state-of-the-art ASIC technology operate at the rate of 2 to 3 GHz. As shown in [9

9. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009). [CrossRef]

], decision feedback (DF) based algorithms are not directly amenable to parallelization. Indeed, DF creates a dependency between modules, precluding independent parallel operation of identical processing sub-modules. Thus, a polyphase decomposition, i.e., time-parallelization of the processing using identical processing units operating on the polyphase components, would not equivalent to (in fact would have reduced performance relative to) our nominal MSDD, hypothetically operating at the full high rate. Nevertheless, realization-wise we adopt such parallelization strategy as shown in Fig. 9(b)
Fig. 9 (a): Tx implementation for the MSDD polyphase receiver implementation. (b): Polyphase implementation of the MSDD CR. Processing is partitioned into P parallel sub-modules, each acting on a received polyphase. Notice that the clock rates of the DP modules in the Tx and MSDD modules in the Rx are reduced by a factor of P.
, having P MSDD sub-modules, each operating on one of P polyphases, enabling digital hardware processing at P times lower clock rate. Thus, by taking the number P of parallel blocks sufficiently large, the processing clock rate per sub-module is sufficiently reduced to accommodate the available ASIC processing speeds. For this parallelization method, as each of the P parallel units operates at a rate a factor of P slower, the total (low) hardware complexity of the MSDD is essentially retained – the number of CMs per second is still the same. However, this CR system will have somewhat worse performance than the nominal full-speed MSDD.

In order to enable MSDD polyphase operation at the Rx, the Tx is modified to also support a polyphase version of differential precoding, comprising P parallel MP-DPs modules, each operating at reduced rate by a factor of 1/P, as shown in Fig. 9(a). The output of all P units is interleaved in order to create a continuous stream at the high rate. Each DP module essentially accumulates the prior phases in jumps of P samples (operating on a particular polyphase).

7.1 The distant feedback (DF) problem in parallelized MSDD processing

When using the polyphase implementation just introduced, the inputs to each MSDD sub-module are in jumps of P. The larger separation between MSDD input samples does not affect the white ASE noise performance, as there is no correlation between distinct ASE samples of white noise no matter how far apart. However, LPN noise performance is degraded under the polyphase implementation, as samples further away from each other are less correlated, and their relative phase noise is increased. Since the laser phase noise is a Weiner process with independent increments, Ωk=φkφk1, with variance σΩ2=2πΔνT proportional to the time interval T between samples (processing latency), i.e. inversely proportional to the sampling rate, it follows that reduction in sampling rate by a factor of P, due to parallelization, increases the variance of the laser phase noise by a factor of P. This amounts to having an effective laser linewidth P times wider. We refer to this laser phase noise tolerance penalty as the distant feedback effect, exacting a penalty due to the multiple parallel processing paths, which are inevitable at current CMOS clock speeds. Thus, the LPN tolerance will be degraded by a factor of P due to the parallelization, nevertheless, as the normalized phase noise tolerance of the MSDD method is very high to begin with (unless the laser phase noise is dominant relative to the ASE), the penalty will be seen to be small.

8. Simulation results

The simple channel model of subsection 2.2 is assumed here (Fig. 10
Fig. 10 (a): Tx implementation for the MSDD polyphase receiver implementation. (b): Polyphase implementation of the MSDD CR. Processing is partitioned into P parallel sub-modules, each acting on a received polyphase. Notice that the clock rates of the DP modules in the Tx and MSDD modules in the Rx are reduced by a factor of P.
). This channel model does not address fiber non-linearities, Tx band-limitation, Rx optical and electrical filtering, the effect of the Rx CD equalizer (equalization enhanced phase noise [34

34. W. Shieh and K. P. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16(20), 15718–15727 (2008). [CrossRef] [PubMed]

]) nevertheless this simple model still captures the salient phase noise features, allowing meaningful comparison of the resulting MSDD performance vs. that of the Viterbi&Viterbi M-power CPE of Fig. 11
Fig. 11 Viterbi&Viterbi M-power QPSK CR, against which the MSDD CR is compared.
.

In all Monte-Carlo and LMS simulations we assume a 100G PDM-QPSK system at 28 GBd baudrate per polarization, simulating a single polarization. We also assume a parallelization factor of P = 16, i.e. the DP transmission and MSDD detection is parallelized, as per Fig. 9, into 16 processing sub-modules.

Figure 12
Fig. 12 Viterbi&Viterbi M-power QPSK CR, against which the MSDD CR is compared, for an idealized situation whereby LW = 0, i.e. there is no LPN, just white ASE noise.
simulates the QPSK MSDD system for zero LPN, just in the presence of ASE white noise, for various window sizes. As L is increased, the MSDD performance is seen to approach the so-called coherent HDD limit, namely coherent QPSK with differential precoding and hard (logic) differential decoding, i.e. the final complex-valued decision is expressed as s˜^ks˜^k1*. Notice that the robustness of HDD is much higher than that of soft differential decoding, r˜kr˜k1*, which corresponds to L = 1 (i.e., the window of past samples just includes the last sample), as the HDD hard decision is in error when either of the s˜^k,s˜^k1hard decisions are in error, which occurs with probability double that of either of them being in error. A linear factor of 2 on the BER scale corresponds to about 0.8 dB penalty at BER = 10E-3, which is much smaller than the ~3 dB penalty of differential decoder, as derived in sub-section 2.4. The ~2.2 dB gap between soft and hard differential decoding is bridged over by the MSDD - the higher the window size L, the more the HDD limit is approached. Here, in the absence of LPN, the white-noise performance is monotonic increasing in L.

Figure 13
Fig. 13 Adaptive MSDD performance. (left): Optimizing MSDD over the number of adaptive combining taps for various linewidths (LW). (right): Converged adaptive coefficients (red) vs. the optimal Wiener coefficients (blue).
presents simulated aspects of the adaptive LMS performance. The number L of LMS coefficients is selected to range from 6 and 15 to get most of the benefit, as is apparent from Fig. 13-left. Figure 13-right illustrates that the adaptive coefficients almost perfectly converge onto the optimal Wiener solution (red sticks vs. blue points). It is also apparent that the coefficient amplitudes decay such as to optimize the noise decorrelation due to the LPN.

The final performance attainable with uniform (all equal to 1/L) coefficients, vs. Wiener-optimal and LMS coefficients is shown in Fig. 14
Fig. 14 MSDD CR vs. Viterbi&Viterbi M-power CR BER vs. OSNR performance. The bottom two curves represent theoretical limits corresponding to a purely white noise idealized channel without and with hard differential detection. The top curve corresponds to the Viterbi&Viterbi M-pwr algorithm (worst performance). The MSDD performance is generally better, but successively degrades as linewidth is increased (0.1, 0.5, 1 MHz). At 0.1 MHz, the curves for Wiener-Optimal, LMS and Uniform coefficients coincide. It is then apparent that for systems based on coherent-grade 100 KHz lasers, MSDD OSNR is 1.9 dB better than that of the MPWR system at BER = 10E-3. Moreover, this performance may be attained with the multiplier-free CPE (Uniform coefficients), and there is no need to use the more complex version with optimized coefficients. At 0.5 MHz linewidth, the OSNR gap between uniform and optimized coefficients is just 0.15 dB indicating that even in this case it is still worth adopting the CPE-multipliers-free hardware simplified MSDD. However, at 1 MHz the gap between uniform and optimized coefficients widens to almost 1 dB indicating that in this case the more complex MSDD system with LMS or Wiener optimized coefficients is required for best performance. In all cases the Wiener and LMS coefficients track each other almost perfectly, mutually validating the mathematical analyses leading to the Wiener solution and the LMS update equations.
, for linewidths of 0.1, 0.5 and 1 MHz. Two important effects are apparent: (i): For all linewidth, the performances with LMS and Wiener-optimal coefficients are almost indistinguishable. (ii): For 0.1 MHZ LW, the performance with uniform coefficients (corresponding to multiplier-free CPE) is also indistinguishable from that with LMS and with Wiener-optimal coefficients, whereas for 0.5 MHz linewidth, the penalty due to uniform coefficients is just 0.25 dB. However, for 1 MHz linewidth, the uniform coefficients penalty is ~1 dB. This indicates that the ultra-low complexity multiplier-free CPE (with single full-fledged multiplier required for demodulation but no non-trivial multipliers required for the CPE) is the best choice up to about 0.5 MHz linewidth, as its performance penalty is very small (negligible for coherent-grade 100 KHz lasers), whereas the complexity savings in the multiplier-free version is very large. For linewidths above 0.5 MHz, the coefficients optimization, whether adaptive or Wiener-based, does improve performance, the more so the wider the linewidth, but then one has to invest either 1 CM (for Wiener) or 2 CMs (for LMS) per tap (6 to 9 taps would suffice for optimal performance) to optimize performance. In any case the MSDD performance exceeds that of M-power.

9. Conclusions

In this paper we introduced the MSDD principle, explaining in detail how a moving window of L prior symbols may be linearly processed in order to generate a cleaner demodulation reference, relative to other carrier-recovery methods. The two MSDD versions presented here (multiplier-free vs. optimized) provide the least complex CR system vs. the best performance, as borne by numeric simulations indicating up to 1.9 dB advantage over the Viterbi&Viterbi algorithms and ultra-low complexity multiplier-free CPE realization.

Moreover, the MSDD features linear (time-varying) processing hence is free of cycle-slips and other phase unwrapping impairments.

The only weakness of MSDD is its reliance on decision-feedback, which exacts a “distant-feedback” linewidth penalty upon polyphases parallelization. Nevertheless, the simulated performance indicates that the resulting degradation is negligible up to ~0.5 MHz linewidth, thus for practical coherent systems, the limited linewidth tolerance may not be an issue – it is the improved resilience in the lower OSNR regime that makes MSDD the preferred scheme.

This work was devoted to coherent QPSK transmission, yet the MSDD CPE method may be extended to higher modulation formats. MSDD QAM operation was previewed in [30

30. N. Sigron, I. Tselniker, M. Nazarathy, A. Gorshtein, D. Sadot, and I. Zelniker, “Ultimate single-carrier recovery for coherent detection,” in OFC’11 Conference on Optical Fiber Communication, OMJ2 (2011).

] [31

31. M. Nazarathy, N. Sigron, and I. Tselniker, “Integrated carrier phase and frequency estimation for coherent detection based on multi-symbol differential detection (MSDD),” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest, Invited paper SPMC1 (2011).

], however this extensive key topic will be fully elaborated in a future publication, covering unique aspects of adaptive MSDD for QAM: consolidation of carrier phase and carrier frequency estimation in a single MSDD system, seamless transition between QAM constellation sizes and automatic adaptive scaling of the received QAM constellation.

Despite the proliferation of CR techniques, e.g. [1

1. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983). [CrossRef]

10

10. K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in ECOC’10 European Conf. of Optical Communication, We.7.A.3 (2010).

], we are convinced that the MSDD approach features the best performance-complexity tradeoffs and will evolve to be increasingly adopted as the carrier recovery method of choice.

Appendix A: Uop and modulus preserving differential encoding math properties

Some Uop properties: The Uop distributes over products, i.e. the Uop of a product is the product of Uops: v˜=z˜w˜v˜=z˜w˜; Uop is an idempotent operation: z˜=z˜. The last two relations lead to v˜=z˜w˜v˜=z˜w˜.

Next, let us evaluate the computational complexity of generating the Uop,
()/||=()(1/||2)=[Re(),Im()][1/[Re()]2+[Im()]2]
(30)
The operational form above indicates that we require a complex-real multiplier (i.e. two real-multipliers (RMs)), a 1/ look-up table (LUT) and the absolute square operation ||2=[Re()]2+[Im()]2 comprising two RMs. Just counting multipliers, resulting overall Uop complexity is 4 RMs. As a single complex multiplier takes three real-multipliers to execute, it is apparent that the Uop complexity essentially amounts to 113 CMs.

Consider now the DP recursion Eq. (3), relating two line symbols separated by one discrete-time unit. More generally by repeated application of Eq. (3), shifted back in time we have
A˜kis˜ki+1=A˜ki+1A˜ki+1s˜ki+2=A˜ki+2....A˜k2s˜k1=A˜k1
(31)
i.e. we have a more general recursion, essentially relating two line symbols which are i time units apart by a complex rotation through the unimodular product s˜ki+1s˜ki+2...s˜k1:
A˜kis˜ki+1s˜ki+2..s˜k1=A˜k1
(32)
It is readily verified that the delay-detection operation, A˜kA˜k1*, undoes DP. Indeed,
A˜kA˜k1*=(s˜kA˜k1)A˜k1*=s˜k(A˜k1A˜k1*)=s˜k
(33)
Thus, s˜k=A˜kA˜k1* and more generally we have the recursion
s˜k=A˜kA˜ki*s˜ki+1*s˜ki+2*...s˜k1*
(34)
which is readily proven using Eq. (32), as follows:

A˜kA˜ki*s˜ki+1*s˜ki+2*...s˜k1*=A˜k(A˜kis˜ki+1s˜ki+2...s˜k1)*=A˜kA˜k1*=s˜k
(35)

Appendix B: Derivation of the Wiener-Hopf equations for the optimal coefficients

vi[Γs˜k,s˜k]i=s˜k(i)s˜k*=|s˜k|2(1+ση˜2δi)eπΔν^|i|Φij[Γs˜]ij=s˜k(i)s˜k(j)*=[1+ση˜2(1+δij)]eπΔν^|ij|
(48)

Appendix C: Abbreviations used in this paper

The two leftmost columns list the 18 abbreviations specific to this paper – the third column contains abbreviations in general use.

CFO = Carrier Frequency Offset
MMSE = Minimal Mean Square Error
ASE = Amplified Spontaneous Emission
CM = Complex Multiplier
MP-DP = Modulus Preserving Diff. Precoder
FIR = Finite Impulse Response
CPE = Carrier Phase Estimation
MSDD =
Multi-Symbol Delay/Differential Detection
QPSK = Quadrature Phase Shift Keying
CR = Carrier Recovery
MSE = Mean Square Error
QAM = Quadrature Amplitude Modulation
DD = Delay/Differential Detector/Demodulator
MSPE = Multi-Symbol Phase Estimation
OSNR – Optical Signal to Noise Ratio
DP = Differential Precoder
PN = Phase Noise
SNR = Signal to Noise Ratio
LMS = Least Mean Squares
SE = Squared Error

LPN = Laser Phase Noise
Uop = Unimodular Normalization (Eq. (2))

LW = LinewidthW-H = Wiener-Hopf (Equations)

Acknowledgments

This work was supported in part by the Israeli Science Foundation (ISF), and by the Chief Scientist Office of the Israeli Ministry of Industry, Trade and Labor within ‘Tera Santa’ consortium.

References and links

1.

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983). [CrossRef]

2.

E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol. 23(12), 4110–4124 (2005). [CrossRef]

3.

R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005). [CrossRef]

4.

M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).

5.

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol. 25(9), 2675–2692 (2007). [CrossRef]

6.

S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett. 20(18), 1569–1571 (2008). [CrossRef]

7.

M. G. Taylor, “Detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009). [CrossRef]

8.

M. G. Taylor, “Algorithms for coherent detection what can we learn from other fields?” in OFC/NFOEC’10, Conf. on Optical Fiber Communication, OThL4 (2010).

9.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009). [CrossRef]

10.

K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in ECOC’10 European Conf. of Optical Communication, We.7.A.3 (2010).

11.

D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun. 38(3), 300–308 (1990). [CrossRef]

12.

F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun. 40(3), 457–460 (1992). [CrossRef]

13.

M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett. 29(15), 1385–1387 (1993). [CrossRef]

14.

F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol. 44(2), 203–210 (1995). [CrossRef]

15.

S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express 17(2), 703–715 (2009). [CrossRef] [PubMed]

16.

C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express 18(12), 12088–12103 (2010). [CrossRef] [PubMed]

17.

S. Zhang, P. -yuen Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010). [CrossRef]

18.

D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” in OFC’05 Conference on Optical Fiber Communication (2005).

19.

M. Nazarathy and Y. Yadin, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’06 Coherent Optical Technologies and Applications (2006).

20.

X. Liu, “Data-aided multi-symbol phase estimation for receiver sensitivity enhancement in optical DQPSK, CThB4,” in COTA’06 Coherent Optical Techniques and Applications (2006).

21.

M. Nazarathy and Y. Atzmon, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’08 Coherent Optical Techniques and Applications (2008).

22.

X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16(2), 792–803 (2008). [CrossRef] [PubMed]

23.

M. Nazarathy, X. Liu, L. Christen, Y. K. Lize, and A. E. Willner, “Self-coherent multisymbol detection of optical differential phase-shift keying,” J. Lightwave Technol. 26(13), 1921–1934 (2008). [CrossRef]

24.

Y. Takushima, H. Y. Choi, and Y. C. Chung, “Transmission of 108-Gb/s PDM 16ADPSK signal on 25-GHz grid using non-coherent receivers,” Opt. Express 17(16), 13458–13466 (2009). [CrossRef] [PubMed]

25.

J. Li, R. Schmogrow, D. Hillerkuss, M. Lauermann, M. Winter, K. Worms, C. Schubert, C. Koos, W. Freude, and W. J. Leuthold, “Self-coherent receiver for PolMUX coherent signals,” in OFC’11 Conference on Optical Fiber Communication, OWV5 (2011).

26.

N. Kikuchi and S. Sasaki, “Highly sensitive optical multilevel transmission of arbitrary quadrature-amplitude modulation (QAM) signals with direct detection,” J. Lightwave Technol. 28(1), 123–130 (2010). [CrossRef]

27.

N. Kikuchi, “Chromatic dispersion-tolerant higher-order multilevel transmission with optical delay detection,” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest (2011).

28.

S. Adhikari, S. L. Jansen, M. Alfiad, B. Inan, V. A. J. M. Sleiffer, A. Lobato, P. Leoni, and W. Rosenkranz, “Self-coherent optical OFDM : an interesting alternative to direct or coherent detection” in ICTON’11 13th International Conference on Transparent Optical Networks (2011).

29.

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, (Springer, 2011).

30.

N. Sigron, I. Tselniker, M. Nazarathy, A. Gorshtein, D. Sadot, and I. Zelniker, “Ultimate single-carrier recovery for coherent detection,” in OFC’11 Conference on Optical Fiber Communication, OMJ2 (2011).

31.

M. Nazarathy, N. Sigron, and I. Tselniker, “Integrated carrier phase and frequency estimation for coherent detection based on multi-symbol differential detection (MSDD),” in SPPCom’11 Signal Processing in Photonic Communications - OSA Technical Digest, Invited paper SPMC1 (2011).

32.

N. Kikuchi, S. Sasaki, and T. Uda, “Phase-noise tolerant coherent polarization-multiplexed 16QAM Transmission with digital delay-detection, in ECOC’11 European Conference of Optical Communication (ECOC), Tu.3.A (2011).

33.

T. Adali and S. Haykin, Adaptive Signal Processing—Next Generation Solutions (John Wiley, 2010).

34.

W. Shieh and K. P. Ho, “Equalization-enhanced phase noise for coherent-detection systems using electronic digital signal processing,” Opt. Express 16(20), 15718–15727 (2008). [CrossRef] [PubMed]

35.

Y. Atzmon and M. Nazarathy, “A Gaussian polar model for error rates of differential phase detection impaired by linear, nonlinear, and laser phase noises,” J. Lightwave Technol. 27(21), 4650–4659 (2009). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems
(060.2920) Fiber optics and optical communications : Homodyning

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: November 14, 2011
Revised Manuscript: December 14, 2011
Manuscript Accepted: December 14, 2011
Published: January 13, 2012

Citation
Netta Sigron, Igor Tselniker, and Moshe Nazarathy, "Carrier phase estimation for optically coherent QPSK based on Wiener-optimal and adaptive Multi-Symbol Delay Detection (MSDD)," Opt. Express 20, 1981-2003 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-1981


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References

  1. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory29(4), 543–551 (1983). [CrossRef]
  2. E. Ip and J. M. Kahn, “Carrier synchronization for 3- and 4-bit-per-symbol optical transmission,” J. Lightwave Technol.23(12), 4110–4124 (2005). [CrossRef]
  3. R. Noé, “PLL-free synchronous QPSK polarization multiplex / diversity receiver concept with digital I & Q baseband processing,” IEEE Photon. Technol. Lett.17(4), 887–889 (2005). [CrossRef]
  4. M. G. Taylor, “Accurate digital phase estimation process for coherent detection using a parallel digital processor,” in ECOC’05 European Conf. of Optical Communication, Tu 4.2.6 (2005).
  5. E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” J. Lightwave Technol.25(9), 2675–2692 (2007). [CrossRef]
  6. S. Hoffmann, S. Bhandare, T. Pfau, O. Adamczyk, C. Wordehoff, R. Peveling, M. Porrmann, and R. Noe, “Frequency and phase estimation for coherent QPSK transmission with unlocked DFB lasers,” IEEE Photon. Technol. Lett.20(18), 1569–1571 (2008). [CrossRef]
  7. M. G. Taylor, “Detection using digital signal processing,” J. Lightwave Technol.27(7), 901–914 (2009). [CrossRef]
  8. M. G. Taylor, “Algorithms for coherent detection what can we learn from other fields?” in OFC/NFOEC’10, Conf. on Optical Fiber Communication, OThL4 (2010).
  9. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol.27(7), 901–914 (2009). [CrossRef]
  10. K. Piyawanno, M. Kuschnerov, B. Spinnler, and B. Lankl, “Low complexity carrier recovery for coherent QAM using superscalar parallelization,” in ECOC’10 European Conf. of Optical Communication, We.7.A.3 (2010).
  11. D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun.38(3), 300–308 (1990). [CrossRef]
  12. F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection,” IEEE Trans. Commun.40(3), 457–460 (1992). [CrossRef]
  13. M. Adachi and F. Sawahashi, “Decision feedback multiple-symbol differential detection for M-ary DPSK,” Electron. Lett.29(15), 1385–1387 (1993). [CrossRef]
  14. F. Adachi and M. Sawahashi, “Decision feedback differential phase detection of M-ary DPSK signals,” IEEE Trans. Vehicular Technol.44(2), 203–210 (1995). [CrossRef]
  15. S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system,” Opt. Express17(2), 703–715 (2009). [CrossRef] [PubMed]
  16. C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express18(12), 12088–12103 (2010). [CrossRef] [PubMed]
  17. S. Zhang, P. -yuen Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol.28(11), 1597–1607 (2010). [CrossRef]
  18. D. van den Borne, S. Calabro, S. L. Jansen, E. Gottwald, G. D. Khoe, and H. de Waardt, “Differential quadrature phase shift keying with close to homodyne performance based on multi-symbol phase estimation,” in OFC’05 Conference on Optical Fiber Communication (2005).
  19. M. Nazarathy and Y. Yadin, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’06 Coherent Optical Technologies and Applications (2006).
  20. X. Liu, “Data-aided multi-symbol phase estimation for receiver sensitivity enhancement in optical DQPSK, CThB4,” in COTA’06 Coherent Optical Techniques and Applications (2006).
  21. M. Nazarathy and Y. Atzmon, “Approaching coherent homodyne performance with direct detection low-complexity advanced modulation formats,” in COTA’08 Coherent Optical Techniques and Applications (2008).
  22. X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express16(2), 792–803 (2008). [CrossRef] [PubMed]
  23. M. Nazarathy, X. Liu, L. Christen, Y. K. Lize, and A. E. Willner, “Self-coherent multisymbol detection of optical differential phase-shift keying,” J. Lightwave Technol.26(13), 1921–1934 (2008). [CrossRef]
  24. Y. Takushima, H. Y. Choi, and Y. C. Chung, “Transmission of 108-Gb/s PDM 16ADPSK signal on 25-GHz grid using non-coherent receivers,” Opt. Express17(16), 13458–13466 (2009). [CrossRef] [PubMed]
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