1. Introduction

Carrier Recovery (CR) is a critical module in the digital signal processing (DSP) chain of coherent receivers (Rx) for ultra-high speed photonic transmission. Various Carrier Phase (CP) and Estimation and Recovery (E&R) methods have been heretofore introduced for QPSK and QAM transmission, e.g [1–14]. In a recent publication [15] we investigated a promising CR technique for QPSK optically coherent links, based on Multi-Symbol-Delay Detection (MSDD), which is a generalization of delay detection, displaying robust performance, improved phase noise tolerance and low complexity.

This paper is devoted to theoretical and numerical investigation of key extensions of the MSDD CR technique progressing beyond [15], which primarily treated QPSK Carrier Recovery (CR) (though extension from QPSK to QAM was outlined there but not elaborated). This work is devoted the QAM carrier recovery based on MSDD, addressing the following aspects:

(i): Flexible m-QAM operation seamlessly switching between various constellation sizes, e.g. m = 4 (QPSK) to m = 16 (16-QAM). (ii): Accomplishing Carrier Frequency Offset (CFO) estimation and recovery (E&R) in addition to the nominal CP E&R function of the MSDD. Both CP and CFO will be seen to be mitigated within the same MSDD CR module without added hardware, in effect combining the two functionalities in one CR module, progressing beyond conventional single-carrier CFO E&R techniques such as [16–24] (iii): Introducing an alternative improved performance adaptive LMS method for automatically adjusting the MSDD coefficients to track variations in the phase noise channel statistics, as well as enabling automatic tracking of the CFO variations.

(iv): Establishing that the CR also automatically provides, “for free”, Automatic Gain Control (AGC) for the QAM constellation. (v): Investigating further tradeoffs between complexity and performance for the 16-QAM MSDD, for which a non-adaptive non-frequency tracking method attains considerably reduced complexity at the expense of minor performance degradation.

This paper may be considered as a QAM and CFO extension sequel to our QPSK-oriented CP MSDD [15], thus to avoid redundancy we refrain from duplicating the MSDD background material but just briefly review it here, extensively referring to [15]. The paper is structured as follows: Section 2 briefly reviews the U-notU MSDD carrier recovery algorithm used in [15] for QPSK, but also applicable to QAM, and introduces a novel notU-U variant of the MSDD carrier recovery algorithm also applicable to QPSK/QAM.

Section 3 derives an adaptive algorithm for adjusting the notU-U MSDD taps. Section 4 investigates the CFO estimation and recovery property of the MSDD, showing that the same adaptive MSDD module will also automatically correct CFO. Section 5 compares the complexity of the MSDD with that of alternative CR for QAM schemes. In section 6 we introduce a reduced-complexity version of the MSDD CP E&R (without CFO tracking). Section 7 presents QAM numeric simulations of performance and comparisons with two of the leading prior art CR methods: Blind Phase Search (BPS) [7] and BPS + Maximum Likelihood (ML) [12]. Section 8 concludes the paper. Appendix A derives the LMS algorithm for the QAM notU-U MSDD. Appendix B presents a formal proof that the mean-squared-error (MSE) is minimized when the CFO is precisely cancelled. Appendix C collects the relevant abbreviations used in this paper.

2. From QPSK to QAM MSDD

2.1 Brief review of the MSDD carrier recovery algorithm (U-notU original variant for QPSK/QAM)

At this point the readers should review section 2.1 of our precursor paper [15]. While the focus there was on QPSK transmission, the so-called Unimodular-notUnimodular (U-notU) MSDD carrier recovery system was already designed there to be “QAM-ready” in the sense of also being applicable to m-QAM, e.g. 16-QAM (but was only simulated for QPSK in [15]). Adopting the same notation, in the Tx the information symbols ${\underset{˜}{s}}_k$of the QAM constellation alphabet are mapped by a modulus preserving differential precoder (DP) [25] into line symbols, ${\underset{˜}{A}}_k={\underset{˜}{s}}_k{\stackrel{⌣}{\underset{˜}{A}}}_{k-1}$. We also use here the simple phase noise channel comprising additive white Amplified Spontaneous Emission (ASE) ${\underset{˜}{n}}_k$and Laser Phase Noise (LPN) ${\varphi }_k^{LPN}$ with statistics defined in [15]. The channel is fed by the line symbols${\underset{˜}{A}}_k$, and its output is:

The “U-notU” MSDD derived in [15] generates an improved decision variable ${\stackrel{⌢}{\underset{˜}{s}}}_k^{\text{U-notU}}$ (to be input into the slicer to obtain the decisions ${\widehat{\underset{˜}{s}}}_k^{}$), by demodulating ${\underset{˜}{r}}_k$ with the following improved reference:

2.2 Novel notU-U variant of the MSDD carrier recovery algorithm

We now propose a novel MSDD carrier recovery variant applicable for QAM, referred to as “notU-U”, further amenable to adaptability as addressed in the next sub-section. The new structure differs from the “U-notU” original MSDD structure [15] in the positioning of the Uop normalization operation. The notU-U algorithm for generating the improved decision variable, ${\stackrel{⌢}{\underset{˜}{s}}}_k^{\text{notU-U}}$ from the received samples, ${\underset{˜}{r}}_k$ (Fig. 1 ), is as follows:

 

Fig. 1 (top): QAM Tx with modulus-preserving Differential Precoder (DP) as in [15] [25]. (bottom): QAM receiver block diagram detailing the novel “not U-U” MSDD carrier recovery efficient hardware structure with the Uop-1 normalization applied onto the improved reference but not to the partial references. This MSDD realization for QAM differs from the “U-notU” MSDD structure of Fig. 6 in [15] (which is not reproduced here) in the positioning of the Uop-1 normalization. This scheme may also seamlessly “switch-on-the-fly” from QAM to QPSK operation. The coefficients control block implementing an LMS adaptive algorithm is detailed in Fig. 2.

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Unlike the original U-notU MSDD system of Eq. (2) [15], in the new notU-U MSDD variant Eq. (3) the improved reference ${\underset{˜}{R}}^{}{}_{k-1}$is now obtained as a linear combination of un-normalized partial references, ${\underset{˜}{R}}_{k-1}^{(i)}$ (rather than of normalized ones, ${\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{(i)}$ as in the original U-notU version). The partial references are specified in Eq. (3) as ${\underset{˜}{R}}_{k-1}^{(i)}\equiv {\underset{˜}{r}}_{k-i}{\stackrel{⌣}{\widehat{\underset{˜}{s}}}}_{k-i+1}{\stackrel{⌣}{\widehat{\underset{˜}{s}}}}_{k-i+2}\mathrm{..}{\stackrel{⌣}{\widehat{\underset{˜}{s}}}}_{k-1}$, generated by rotating un-normalized past symbols by products of prior decisions. However, it is the improved reference which is now normalized, mapping ${\underset{˜}{R}}^{}{}_{k-1}$ into${\stackrel{⌣}{\underset{˜}{R}}}^{}{}_{k-1}$. The Uop-normalized improved reference and its complex conjugate (cc), are respectively expressed as:

Our final notU-U MSDD estimator (fed into the slicer) is directly expressed in terms of the un-normalized ${\underset{˜}{R}}_{k-1}$as:

Using Eqs. (4),(5), the Mean Squared Error (MSE) is then expressed as:

3. Novel adaptive LMS algorithm for optimizing the taps of the notU-U MSDD carrier recovery variant

In this section we introduce an adaptive LMS algorithm for the notU-U MSDD new variant, generating nearly optimal coefficients approximating the true Wiener coefficients, resulting in improved performance relative to the original U-notU adaptive MSDD structure treated in [15]. This notU-U LMS algorithm will be seen in section 7 to provide 0.4 dB advantage over the U-notU LMS algorithm originally introduced in [15] for 16-QAM coherent transmission at 14 GBaud with 100 KHz linewidth lasers. The mathematical details of the derivation are relegated to Appendix A, working out in Eq. (18) the i-the element of the MSE gradient,

An LMS stochastic gradient algorithm based on the gradient Eq. (7) tends to converge to optimal estimation of the Tx symbol${\underset{˜}{s}}_k^{}$. An alternative form of the MSE Eq. (6) (using${\stackrel{⌢}{\underset{˜}{s}}}_k^{\text{notU-U}}\equiv {\underset{˜}{r}}_k{\stackrel{⌣}{\underset{˜}{R}}}^{*}{}_{k-1}\Rightarrow \left|{\stackrel{⌢}{\underset{˜}{s}}}_k^{\text{notU-U}}\right|=\left|{\underset{˜}{r}}_k\right|$),

This recursion is implemented in Fig. 2 . The “transmitted” symbols ${\underset{˜}{s}}_k^{}$may either originate from decision-feedback ${\widehat{\underset{˜}{s}}}_k^{}$ (decision driven mode) or from a recurring training sequence (data-aided mode), for which pre-Uop-normalized values of these symbols may be used, i.e., Uop-2 may be realized by a trivial lookup-table built into the slicer.

 

Fig. 2 Coherent QAM/QPSK receiver with novel “notU-U” adaptive MSDD, including full detail on the novel LMS coefficients adaptation mechanism introduced in this paper, differing from that of Fig. 8 of [15] (providing 0.4dB performance advantage – see section 7). Notice the presence of two Uop modules, however the one within the slicer does not incur extra complexity, as it may be implemented as a look-up table. The delay line with multipliers at the top of the figure incurs negligible complexity, as multiplication with QAM constellation elements may be implemented as lookup tables.

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Seamless “on-the-fly” switching between QAM levels: A QAM receiver based on this notU-U adaptive MSDD is able to switch “on-the-fly” among various m-QAM constellation sizes, e.g. seamlessly transition back and forth between 16-QAM and QPSK, enabling dynamically adaptive optical networks (in contrast, other carrier recovery systems require separate structures for various m-QAM levels).

Constellation scaling (AGC) capability: Upon designing a receiver for QAM detection, a necessary capability is Automatic Gain Control (AGC) providing proper scaling of the QAM constellation, stretching or shrinking the received signal such that its scale be compatible with the decision boundaries of the QAM slicer. Operating without such constellation AGC capability will result in severe error rate. For a slicer integrated within an adaptive MSDD CR, as described here, it turns out that the constellation AGC functionality is automatically provided by the MSDD, without requiring additional circuitry. To justify this claim, imagine that the LMS system settled on optimal coefficients and we now apply an extra amplitude gain factor g, either in the Tx, within the channel, or in the Rx front-end (preceding the MSDD). The MSDD system then tends to minimize the squared error, which was originally,${\left|{\underset{˜}{s}}_k^{}-{\stackrel{⌢}{\underset{˜}{s}}}_k^{\text{notU-U}}\right|}^2={\left|{\underset{˜}{s}}_k^{}-{\displaystyle {\sum }_{i=1}^L{\overline{c}}_i^{}{\underset{˜}{r}}_k{\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{(i)*}}\right|}^2$, but now, after the transition ${\underset{˜}{r}}_k\to g{\underset{˜}{r}}_k$, in order to minimize the squared error, the LMS adaptive system compensates the increase in the signal amplitude by re-scaling its coefficients according to $c_i\to g^{-1}{c}_i$, such that the product ${\overline{c}}_i^{}{\underset{˜}{r}}_k$remain invariant and the mean squared error be retained at its original minimal value. This argument was also verified by simulation.

U-U MSDD variant: We note that the same LMS recursion hence the same adaptive sub-system as in Fig. 2 will also be applicable in case we elect to normalize all the ${\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{(i)}$, i.e., place a Uop normalizer on ${\underset{˜}{r}}_k$in the upper left corner of the figure while retaining the Uop on ${\underset{˜}{R}}_{k-1}^{}$, prior to demodulation. Evidently the ${\underset{˜}{R}}_{k-1}^{(i)*}$are now replaced by${\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{(i)*}$, however the algorithm with normalizations on both ${\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{(i)*}$and ${\underset{˜}{R}}_{k-1}^{}$will also function with the same structure – however the converged coefficients and performance will evidently be different. We shall not further pursue this U-U MSDD variant, as simulations have shown it to be inferior to either the notU-U or U-notU variants. In fact the novel notU-U MSDD algorithm introduced in this section is the preferred version performance-wise, as will be indicated in the simulations of section 7.

4. CFO estimation and recovery property of the adaptive MSDD

Heretofore, we have ignored the Carrier Frequency Offset (CFO) impairment, having developed the MSDD as a carrier phase E&R system, mitigating phase noise while assuming perfect frequency lock of the LO and Tx laser frequencies, i.e., null CFO. In this section we show that our QAM MSDD CP E&R system, as is, is inherently capable of automatically mitigating CFO, requiring neither modification of the hardware structure, nor additional hardware for the CFO function. Thus CFO E&R is attained “for free” by the MSDD, jointly mitigating both phase noise and CFO by a common MSDD CR structure, the one already presented above. Moreover, the frequency range of our novel CFO cancellation method is in principle infinite, whereas prior CFO E&R methods, e.g [16–24], have a limited capture range, which is just a fraction of the baud-rate.

4.1 Cancelling CFO with the adaptive MSDD carrier recovery system originally designed for carrier phase E&R

Let us first characterize the MSDD response to CFO, which is modeled as modulation by the phase-ramp factor, $e^{j\theta k}$, applied to a phase noisy received signal ${\underset{˜}{r}}_k^o$in turn described as per Eq. (1), assumed to nominally have null CFO, as indicated by the ${}_{}^o$ superscript. The phase ramp slope (phase increment per unit discrete time),$\theta$, is expressed in terms of the frequency offset $\Delta \nu$between the incoming signal and the LO, and the symbol interval, T:

The CFO-affected signal ${\underset{˜}{r}}_k={\underset{˜}{r}}_k^o{e}^{j\theta k}$ (with ${\underset{˜}{r}}_k^o$ the received signal in the absence of CFO) propagates through the MSDD discrete-time delays (Fig. 3(a) ). The i-th delay line output is given by ${\underset{˜}{r}}_{k-i}={\underset{˜}{r}}_{k-i}^o{e}^{j\theta (k-i)}={\underset{˜}{r}}_{k-i}^o{e}^{j\theta k}{e}^{-j\theta i}$. In Fig. 3(b) the fixed phase factors $e^{-j\theta i}$are lumped together with the combining coefficients$c_i$, whereas the discrete-time-dependent phase-ramp factor $e^{j\theta k}$, which is common to all paths, is propagated to the right through the adder. The demodulating reference (prior to Uop-normalization) in the wake of CFO is then:

 

Fig. 3 The notU-U adaptive MSDD in the presence of Carrier Frequency Offset (CFO). (a): First stage of the derivation, propagating the CFO-affected received signal through the delay line. (b): Second stage of the derivation propagating the linear phase factor $e^{j\theta k}$all the way to the demodulator, where it gets cancelled. The resulting system is equivalent to one with no CFO but with coefficients modified by a linear phase taper. A third stage of the derivation, not shown, makes the phase-tapered coefficients, $c_i^{}{e}^{-j\theta i}$, equal to the optimal coefficients, $c_i^o$ (by setting $c_i^{}=c_i^o{e}^{j\theta i}$, i.e. applying an inverse phase taper to the optimal coefficients in the absence of CFO), resulting in cancelling out the CFO.

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Here${\underset{˜}{R}}_{k-1}^{\text{noCFOeff}}$is the demodulation reference effectively obtained if the CFO were turned off but the same level of phase noise were present, and modified effective coefficients $c_i^{\text{CFOeff}}\equiv c_i^{}{e}^{-j\theta i}$were used (with $c_i^{}$ the actual coefficients). The demodulation output, accounting for the Uop applied to the improved reference, is then:

The effect of the ‘linear phase taper’ applied to the optimal CFO-free coefficients in order to make them suitable for cancelling CFO, is amenable to an intuitive frequency-domain interpretation: The application of the CFO phase factor $e^{j\theta k}$onto the CFO-free received component ${\underset{˜}{r}}_k^o$ (Eq. (1)), corresponds to a spectral shift by $\theta$ of the Discrete-Time-Fourier-Transform of ${\underset{˜}{r}}_k$ (which has periodicity $2\pi$). By linear-phase tapering the optimal CFO-free coefficients, $c_i^o$, i.e., by transitioning to modified coefficients $c_i^{\text{CFOeff}}=c_i^o{e}^{j\theta i}$, we also shift the transfer function of the FIR filter representing the weighting of past samples, by the same amount $\theta$:

Thus, the filter transfer function follows the spectral shift of the incoming signal due to the CFO. As the input signal is frequency-shifted, so is the filter also frequency-shifted to track the incoming signal spectral shift, maintaining maximum combining at all times.

4.2 Adaptively cancelling CFO with our MSDD carrier recovery system

The CFO mitigation scheme described above assumes the availability of an estimate of the phase increment, $\theta$. A more practical approach circumvents estimation of $\theta$, but simply runs the LMS adaptive algorithm introduced in section 3, which will be shown to converge by itself onto the correct tap values, automatically steering the MSDD coefficients to acquire correct phase taper $\left\{e^{j\theta i}\right\}$required to derotate away the CFO. This surprizing new property of our MSDD is analytically justified below and is empirically confirmed by simulation (Fig. 4 ).

 

Fig. 4 Simulated complex taps in adaptive cancellation of CFO using the notU-U MSDD CR for 16-QAM at 14 GBaud for an L = 8 window with parallelization factor P = 8, 100 KHz linewidths for the Tx and LO lasers and OSNR = 18.6 dB. The frequency offset is CFO = (0.2154 + 14 n) GHz, where n is any integer. (a). The tap values c_i in the complex (I-Q) plane. The numerals are the indexes i. It is apparent that as the tap index, i, increases, the magnitude | c_i | diminishes while the phase of phase of c_i follows a monotonically increasing progression, which is shown in the figure on the right to be linear. (b): The taps phases as a function of the tap index i. The phases are seen to be linear in i. In both plots ‘LMS taps’ (red) means the coefficients c_i as determined at the end of the training sequence, whereas ‘OPT taps’ (blue) means optimal taps evaluated offline assuming a U-notU MSDD system (the optimal coefficients are not analytically known for the actual notU-U MSDD system used here, but the differences in coefficient values are minute). The match between the phases of the LMS and OPT taps (red and blue) is seen to be nearly perfect.

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The theoretical justification for this rewarding CFO-self-correcting property is that the MSE is precisely minimized by the same coefficient values which cause the CFO to be cancelled - this is analytically proven in Appendix B. Since the LMS algorithm converges onto the MSE solution (actually hovers around it with fluctuating coefficients values staying close to the Wiener-optimal values), it then follows that the LMS solution is assured to provide the best quality estimate, including nearly canceling CFO. Our MSDD then “magically” tunes out CFO, providing CFO E&R “for free” in addition to the phase-noise mitigation original function. Figure 4 presents a numeric simulation illustrating the automatic adaptation of the complex taps to acquire the proper linear phase tilt required to mitigate the CFO.

4.3 Training sequence based operation of the adaptive LMS algorithm for mitigating phase noise and CFO

At the end of each training sequence (TS), assuming that the CFO is stable over the TS duration required for attaining convergence, the converged coefficients will bear the correct phase tilt needed to counteract CFO at that point in time, and will also have acquired correct amplitudes to optimally address the phase-noise statistics. The tap values will be “frozen” over the data-transmission interval until the next TS arrives. If CFO is slowly varying and/or the repetition rate of the TS is not too low, the “frozen” coefficients will then be able to nearly cancel CFO and optimally mitigate phase-noise over the interval prior to the next TS arriving, at which point a new update of the complex taps is generated (we implicitly assumed here that the rate of variation of the phase-noise statistics is even lower than the rate of variation of CFO). Notice that, in principle, the MSDD adaptive LMS algorithm for the taps adjustment could also be run in a decision-directed mode, based on the slicer decisions, however we have found that the error propagation is excessive in this case, and it is preferable to adapt the taps just during the TS intervals, having them frozen in-between (this is not to be confused with the MSDD algorithm itself being decision-feedback directed, i.e. making inherent use of decision symbols for rotating past observations to generate partial references).

4.4 Our adaptive MSDD for QAM has unlimited frequency offset capture range

In the analysis above there has been no apparent limitation whatsoever on the amount of frequency offset tolerated by the MSDD, which may be, in principle unlimited. However, in practice, in order to avoid over-specifying the sampling rate of the analog to digital converter (ADC), it is worth relieving the burden off the digital CFO mitigation system by providing some coarse analog control of the LO laser frequency, initially bringing it within $~\pm 1GHz$ of that of the incoming signal.

To see why our CFO capture range is in principle unlimited, we inspect Eq. (12), identifying a CFO aliasing effect, whereby CFO values $\Delta \nu$and $\Delta \nu \pm n{T}^{-1},n\in \mathbb{Z}$ are indistinguishable due to the modulo $2\pi$phase wrap. Thus, it suffices to determine whether CFO can be mitigated over the fundamental spectral range $\Delta \nu \in [-T^{-1}/2,T^{-1}/2]$ of extent equal to the sampling rate, in order to extend the CFO operation, in principle, over arbitrary $\Delta \nu$. Since the derivation of Eqs. (13)-(16) does apply to the range $\theta \in [-\pi ,\pi ]$, the CFO is then seen to be cancelled over the fundamental sampling-rate spectral range (baud-rate spectral interval). Therefore, in principle, the MSDD CR is capable of mitigating an arbitrary amount of frequency offset. This is in contrast to all other CFO methods, e.g., [16–24] which attain at most limited CFO capture ranges, which is just a fraction of the baud-rate. Nevertheless, coping with CFO of more than the order of $~\pm 1GHz$may not be practically necessary, as larger CFO implies larger spectral offset of the baseband signal, which would require opening the ADC anti-aliasing filter wider and providing faster ADC to prevent cutting off the shifted baseband spectrum on one side. Therefore our unlimited capture range is mainly of theoretical interest, just highlighting the robustness of our CFO E&R.

5. Complexity of MSDD QAM vs. alternative CR schemes

There is a proliferation of alternative carrier recovery schemes (e.g [1–14]. for CP E&R and [16–24] for CFO E&R), against which comparison of performance and complexity would be difficult. Here we just consider a comparison of the MSDD complexity vs. that of two leading CP E&R methods, the BPS [7] and the BPS + 2ML [12] along with a conventional CFO mitigation method [19], comparing the relative complexities in this section, while the relative performance of these systems vs. the MSDD is compared in section 7.

Implementation complexities are compared in Table 1 for our MSDD scheme vs. the combination of state-of-the-art CFO E&R [19] and CP E&R [12], which combination replaced by our CR with joint CFO and CP mitigation capability. Our itemized hardware components counts are seen to be lower, establishing that our scheme has the lowest complexity relative to other CP and CFO E&R combinations. Part of our complexity savings is traced to the elimination of the CFO E&R (state-of-the art QAM CFO E&R systems typically involve large sizes FFT blocks, eliminated here). Even considering our MSDD based CP E&R standalone, its complexity is already lower than that of state-of-the-art BPS-based CP E&R systems [7], [12], which are burdened by numerous phase rotations, and multiple comparisons.

Table 1. Complexity Comparisons. CM = complex multiplier; PR = Phase Rotator (may be simpler to realize than standard CM using the CORDIC algorithm); CMeff is an effective CM effectively representing both CP and real multipliers (RM): 3RM = 1CMeff; RAeff = real-adder (effective, corrected due to representing real mult. as CMeff); LUT = lookup table; SLICER = decision unit; CMP = comparators used to find min/max; MUX = multiplexer. Assumed parameters: MSDD: L = 8; BPS + 2ML [12]: N = 6; B = 5(14) for 16(64)QAM, 10bit numbers representation.

View Table

6. Variants of QAM MSDD with further reduced complexity (and slightly reduced performance)

Heretofore we have developed an MSDD carrier recovery system adapting L taps operating onto rotated versions of the L prior received symbols. We have also shown that such CR system is capable of reducing both phase fluctuations as well as carrier frequency offset in the received signal, doubling up as CFO E&R. Nevertheless, the high performance and dual phase and frequency offset mitigation capability are attained at the expense of realization complexity, requiring L complex multipliers at the line-rate and additional L multipliers for adaption. Overall we still feature lower complexity than other CR schemes, however the overall complexity is still high in absolute terms.

Let us now explore a possible tradeoff relaxing the complexity of realization at the expense of giving up some performance, while also relinquishing the ability to simultaneously correct CFO (i.e. re-adopting the more conventional approach of supplying a separate solution for the CFO E&R, rather than an integrated one).

To reduce complexity, we consider MSDD variants with all L taps set to a common value (say equal to $c_0=1/L$). We refer to such a configurations (Figs. 5 , 6 ) as uniform taps MSDD. As all taps are equal, it suffices to use trivial unity taps, i.e. simply sum up the rotated symbols, then apply the common tap value $c_0$ by means of a single multiplier following the summation. Thus the combiner of partial references reduces here to a summer followed by a single multiplier $c_0$ applied to the sum: ${\underset{˜}{R}}_{k-1}={\displaystyle {\sum }_{i=1}^L{c}_0{\underset{˜}{R}}_{k-1}^{(i)}}=c_0{\displaystyle {\sum }_{i=1}^L{\underset{˜}{R}}_{k-1}^{(i)}}$.This saves L-1 expensive complex multipliers for the taps (and additional L multipliers for the LMS taps adaptation).

 

Fig. 5 Low complexity notU-U MSDD for QAM reception with uniform taps and the common multiplier adaptively adjusted for AGC capability.

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Fig. 6 Ultra-low complexity notU-U MSDD for QAM reception with uniform taps and fixed common multiplier c₀ = 1/L, which may be discarded, absorbed within the slicer scaling.Thus, the CP estimation is performed multiplier-free, and just a single full-fledged complex multiplier, the one used for demodulation, is required– all the other multipliers at the top, marked green, are trivial (multiplications by QAM constellation points are realized as lookup tables).

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Such uniform taps MSDD, albeit simpler to realize, would be sub-optimal, i.e., it would experience some performance degradation relative to a Wiener or LMS adaptive solution which individually optimizes the L complex taps. Nevertheless, for relatively narrow linewidth lasers (100 KHz) and for 16-QAM transmission, the performance penalty will be seen to be small, in particular when narrow linewidth lasers are used for transmission and LO.

Four versions of such reduced complexity uniform taps MSDD are possible, namely U-not U vs. notU-U and adaptive vs. non-adaptive. The notU-U adaptive and non-adaptive structures are shown in Fig. 5 and Fig. 6 respectively. For brevity the remaining U-notU (none) adaptive versions, which are similarly structured, will not be presented.

Figure 5 presents the adaptive uniform taps “notU-U” MSDD variant for QAM. As usual, “notU-U” implies that the Uop normalization is applied onto the improved reference, but not onto the received signal (hence neither is it applied onto the partial references). The common tap $c_0$ (replacing the L taps of the full-fledged MSDD version) is made time-varying, $c_0[k]$, adjusted by means of an LMS algorithm as illustrated, acting on the estimation error, ${\left|{\epsilon }_k\right|}^2={\left|c_0[k]{\underset{˜}{q}}_k^{}-{\underset{˜}{s}}_k^{}\right|}^2$which is in turn evaluated by sourcing ${\underset{˜}{s}}_k^{}$from either a training sequence or from the slicer the decisions ${\widehat{\underset{˜}{s}}}_k^{}$. Adapting $c_0[k]$ essentially provides AGC capability, scaling the noisy received constellation demodulator output,${\underset{˜}{q}}_k^{}\equiv {\underset{˜}{r}}_k^{}{\stackrel{⌣}{\underset{˜}{R}}}_{k-1}^{*}$, such as to match the fixed decision boundaries of the slicer.

Notice that when another adaptive LMS sub-system precedes the MSDD, such as an adaptive LMS polarization demultiplexer (MIMO 2x2) algorithm, then the AGC capability is no longer required - A non-adaptive variant may just be adequate, as the LMS adaptive polarization system would automatically adjust its gain to correctly scale the received signal correctly relative to the slicer – the properly scaled received constellation would be the optimal state towards which the LMS system would automatically evolve in its quest to minimize the squared error.

Figure 6 presents the non-adaptive version of the uniform taps “notU-U” MSDD for QAM, which is our preferred variant (provided LMS adaptation is used in the polarization demux preceding stage). The adaptation of the common tap is now eliminated. In fact the common fixed tap itself may be eliminated, absorbed in the slicer decision boundaries. For 16-QAM transmission, the multipliers used in generating the partial references (in the top part of the figure) may be simply realized by lookup tables, thus this non-adaptive version exhibits remarkably low complexity. An itemization of the hardware complexity for this scheme reveals a total of 7 real-multipliers and 5 simple lookup-table multipliers (out of the 7 real-multipliers, 4 are needed in the Uop, whereas the demodulator complex multiplier is realized using the remaining 3 real-multipliers).

7. QAM numeric simulations of performance and comparisons

Figure 7 considers several alternative variants of our MSDD system for 16-QAM, comparing their performance with each other and with that of two exemplary state-of-the-art CR conventional systems.

 

Fig. 7 Comparison of BER vs. OSNR performance for three carrier recovery systems, our MSDD or BPS [9] and BPS + 2ML [12]. The last two conventional systems correspond to the lowest two curves. The poor performance top curve is a naïve delay detector (corresponding to an MSDD with an L = 1 window). From the top down we generally progress through increasingly larger window sizes, L, for the MSDD, selecting either notU-U vs. notU-U structures, and uniform fixed or AGC-ed taps, vs. adaptive taps. Key conclusions are that the notU-U variant generally performs better than the U-notU variant. Our best system is an adaptive notU-U MSDD with L = 8, performing only 0.3 dB worse than the BPS + 2ML, but being less complex. MSDD complexity may be significantly further reduced by using a uniform taps structure with fixed taps, falling just 0.15 dB behind our MSDD adaptive “leader”.

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Our numeric simulations of MSDD performance assume the simple channel model of Eq. (1), coinciding with that used in [15], which simulated 100G QPSK transmission over a 28 GBaud coherent link. Here, for 16-QAM, we retain the same 100G bitrate, signaling now with double spectral efficiency at a 14 GBaud rate, requiring a parallelization factor of P = 8 for ASIC realization (bringing the clock-rate down to 14/8 = 1.75 GHz), thus splitting the received signal into 8 polyphases to be MSDD processed in parallel, and also having the transmitted samples split into P = 8 polyphases each of which is differentially encoded in the Tx at 1/P of the baudrate (an exception is Fig. 8(b) below which pertains to a PON system at 3.125 GBaud and P = 2). All plots assume a linewidth of 100 KHz for the Tx and LO lasers, except for Fig. 8(a) (which features variable linewidth).

 

Fig. 8 Comparison of BER vs. OSNR performance for three CR systems, adopting as CP E&R either our QAM MSDD or BPS [7] or BPS + 2ML [12] with the prior-art systems either preceded or not by the CFO E&R of [20], for either 16-QAM or 64-QAM; Tx and LO laser are fixed at LW = 100KHz in (b,c,d); the baud-rates are 14 GBaud and the parallelization factors are P = 8 in (a,c,d). (a): BER vs. normalized linewidth (LW/baudrate) at OSNR = 19 dB and P = 1: MSDD and BPS + 2ML closely track, BPS has best performance but has prohibitive complexity. Generally, due to the hardware parallelization, the effective LW of MSDD is enhanced by P, whereas the LW of the feedforward BPS( + 2ML) is not. For low-baudrate applications, such as PON, P is small, hence the parallelization penalty is negligible. (b): BER vs. OSNR for a Coherent PON 16-QAM system: At the PON lower baudrate, P is as low as 2. The BPS + 2ML and MSDD performances are then virtually identical (but the MSDD is less complex (Table 1)). (c,d): BER vs. OSNR for long-haul 16/64-QAM without and with CFO E&R: At the higher baudrate, P is increased to 8 and MSDD lags BPS + 2ML by 0.3 dB for 16-QAM and by 2 dB for 64-QAM. However, this is for an ideal situation with no CFO E&R; For 3 GHz CFO, when the BPS( + 2ML) CP E&R are preceded by a practical CFO E&R such as the coarse CFO mitigation stage of [19], due to enhanced phase noise and residual CFO from the CFO E&R stage, the performance of the CP E&R stage [7] [12] is severely degraded. In contrast, our MSDD is virtually unaffected by CFO in the range 0...8GHz (and even for arbitrarily large CFO values, tested but not shown). Thus, for 16-QAM, on an end-to-end system basis the MSDD CP + CFO E&R significantly outperforms BPS + 2ML in performance and has lower complexity.

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In Fig. 7 the comparisons are carried out solely for CP E&R, assuming no CFO is present. The top seven curves pertain to nonU-U MSDD structure with uniform taps based on adapting a single common tap performing the AGC function (as in Fig. 5). The trend is that performance is improved (BER is reduced) upon successively increasing the MSDD window size (L = 1,2,...,8). The next two curves respectively correspond to the U-notU and notU-U MSDD structures for L = 8 uniform taps with but with a non-adaptive common coefficient set to the fixed value$c_0=1/L$. It is apparent that the notU-U adaptive MSDD attains better performance, by about 0.4 dB, relative to the U-notU system introduced in our prior work [15]. Again, all these nine graphs refer to the uniform taps structure, i.e. feature low complexity. Performance may be improved by providing non-uniform adaptive taps, at the expense of requiring L multipliers but circumventing the need for a separate CFO mitigation sub-system. The 11th and 12th curve represent adaptive taps structure for the U-notU and notU-U respective cases. Here we see that a full-fledged notU-U adaptive system provides 0.4 dB performance advantage over its U-notU counterpart, vindicating our introduction of the notU-U MSDD variant in this paper. Moreover, interestingly, the notU-U adaptive LMS system displays about the same performance as the U-notU Wiener solution with optimized fixed coefficients (we recall that we haven’t been able to derive a corresponding Wiener optimal solution for the notU-U system, but had such a solution been available, it would then provide slightly better performance than the notU-U adaptive system, resulting in the lowest BER curve relative to the multiple MSDD variants plotted in Fig. 7.

The only distinct disadvantage of our decision-feedback based MSDD data-aided CP E&R scheme is a reduction in laser linewidth tolerance relative to the BPS based CR schemes which operate in forward mode hence do not incur a performance penalty due to DSP-parallelization. Nevertheless, assuming standard coherent-grade 100KHz linewidth (LW) ECL light sources, and comparing CP E&R performance for 16-QAM while ignoring CFO, it is seen in Fig. 7 that the parallelization penalty incurred by our full-fledged MSDD scheme is quite small, just 0.3 dB, relative to using a state-of-the-art long-haul CR system consisting of BPS + 2ML [12]. However, but in exchange for the slight performance disadvantage the MSDD requires substantially less complexity than the alternative BPS + 2ML CR scheme.

It is then apparent that the various MSDD alternatives provide the best performance-complexity trade-offs for 16-QAM with 100 KHz standard coherent-grade lasers. If we are willing to tolerate an extra 0.15 dB of 16-QAM performance penalty in order to gain simplicity, we may transition from a fully adaptive L-taps MSDD system to the “uniform taps” MSDD structures of Figs. 5,6 further reducing complexity significantly for the CP E&R part by eliminating lots of multipliers (unfortunately we now must provide a separate solution for the CFO E&R part, whereas the full-fledged adaptive MSDD also automatically provided CFO mitigation capability over the same hardware).

Comparison of complexity vs. performance tradeoffs for CP + CFO E&R CP systems

Heretofore we have compared CP E&R performance. To further carry out a fair comparison of our CR system vs. other CRs, we must model the full CR chain including both CP and CFO mitigation. Figure 8 includes plots modeling the interactions between the CFO and CP E&R sub-systems for state-of-the-art CR conventional systems, revealing substantial degradation due to the residual CFO and phase noise left after the CFO E&R stage, which propagates on to the CP E&R stage. Hence, once the deterioration of the prior-art systems due to the CFP and CP interaction is accounted for, the performance comparison overwhelmingly comes out in our favor (Fig. 2(b,c)), at least for the two competitive systems shown in Fig. 8(c,d), for which undesired interactions between CP and CFO mitigation stages degrade overall performance as indicated. Evidently, we have only investigated two exemplary serial configurations of CFO and CP E&R modules, however we are not aware of research systematically addressing interactions between the serial CFO and CP mitigation stages for other types of CR systems. In contrast, such detrimental interactions are entirely eliminated in our joint CFO + CP E&R MSDD CR system, which is both simple and robust, attaining improved overall BER performance in the wake of arbitrarily large CFO.

Therefore, we conclude that MSDD is the preferred CR approach for 16-QAM transmission. Unfortunately, for 64-QAM (Fig. 8(d)) MSDD system performance worsens, incurring 2 dB penalty relative to the competitive BPS + 2ML scheme. We do not fully understand the sources of degradation upon transitioning from 16-QAM to 64-QAM. We also simulated a next generation coherent PON system operating at the baud rate (Fig. 8(b)) of 3.125 GBaud, for which the parallelization factor may be reduced down to P = 2. The results for such lower-speed PON Rx indicate negligible parallelization penalty. Virtually identical performance is attained with our CR as with the BPS + 2ML CR described in [12], which track each other, whereas the BPS CR attains the best performance but is far too complex to be practically suitable for low-cost energy efficient PON optical network unit (ONU) home terminals.

8. Discussion and conclusions

Our simple MSDD-based multi-tap adaptive CR CP E&R doubles up as CFO E&R, achieving unprecedented robustness to LO frequency drifts. For any m-QAM format (QPSK included), the MSDD CR hardware is fixed and is substantially simpler than that of state-of-the-art CFO and CP E&R systems, yet attaining comparable end-to-end performance. The MSDD CR is further endowed with a key feature essential for the next generation of dynamic optical networks, wherein transmission rate is to be rapidly traded off for OSNR when link routes and conditions change: Our novel CR structure introduced here is capable of seamlessly accommodating either QPSK or 16-QAM with best OSNR performance or 64-QAM with degraded performance. In contrast, other CR systems require distinct hardware structures for each of the m-QAM formats and for QPSK, therefore, in those conventional schemes hardware would have to be inefficiently replicated for “on-the-fly” adaptation of m-QAM/QPSK.

However, one of the most striking features of our proposed system is in what it has not. Our novel CR system for QAM lacks a dedicated CFO E&R stage, yet is totally immune to arbitrarily large frequency offset. Surprisingly, our CP E&R hardware is able to automatically mitigate CFO E&R “for free”, i.e. at no additional hardware cost, in addition to its original phase noise mitigation role. Both roles are achieved simply and robustly by turning the original CP E&R MSDD structure into a novel adaptive one (never disclosed before in the wireless literature – it might also be applicable there). The MSDD Wiener combining coefficients automatically adjust to track and cancel arbitrary frequency offset, in addition to nicely adapting to the slowly time-varying statistics of the various phase noise sources. Moreover, our CFO capture range is the largest reported - we are able to withstand arbitrarily large frequency offsets up to and exceeding the baud-rate - although this large a CFO does not arise in practice, yet this property underscores the robustness of our CFO E&R. By disclosing here a CFO mitigation system with unlimited capture range we may contribute to tempering the relentless stream of publications on CFO mitigation systems boasting incrementally larger CFO capture ranges, especially in light of our system enjoying the benefit of not even requiring separate hardware for the CFO, but rather reusing the same MSDD CP E&R adaptive hardware “for free”.

Future previewed work in the MSDD domain will include an evaluation of performance over more realistic optical channels comprising chromatic dispersion (equalization enhanced phase noise), polarization and non-linear phase noise effects, optimizing the multi-level multi-phase transmission constellations (the QAM format is quite non-optimal), and combining the MSDD with other CR and detection methods.

The novel proposed MSDD CR emerges as key contender for implementing the carrier recovery function in the next generation of QAM coherent systems.