# High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDFE Equalizers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. DQPSK Equalization

_{c}(t), I

_{d}(t), Q

_{c}(t), and Q

_{d}(t), respectively [18]. Although any pair of the combination {I

_{c}(t),I

_{d}(t)} × {Q

_{c}(t),Q

_{d}(t)} may be utilized for the detection of the transmitted sequences, a scheme based on the differential output I(t) = I

_{c}(t) − I

_{d}(t) and Q(t) = Q

_{c}(t) − Q

_{d}(t), known as ‘balanced detection’, is usually used instead, offering in this case a 3 dB OSNR gain compared to the former approach.

_{f}and M

_{b}represent the memory of the FF and the FB part of the equalizer, respectively, and are both related to the amount of distortions that affect the transmission. Signals$\widehat{{I}_{1}}\left(n\right)\triangleq D\left[{u}_{1}\left(n\right)\right]$ and $\widehat{{I}_{2}}\left(n\right)\triangleq D\left[{u}_{2}\left(n\right)\right]$ represent the recovered in-phase and quadrature bit streams, with D denoting the decision device, as it is explained also in [7,9] and [22]. Hence, the VDFE equalizer described by Equation (2), hereafter will be denoted as VDFE[M

_{f},M

_{b}], and its structure is summarized in Figure 2b.

## 3. Proposed Partially-Joint Single-Ended DQPSK Equalization

_{c}, Q

_{c}, Q

_{d}), (I

_{d}, Q

_{c}, Q

_{d}), (I

_{c}, I

_{d}, Q

_{c}), and (I

_{c}, I

_{d}, Q

_{d}). Due to fractionally-spaced sampling, each triplet corresponds to six signals that are digitally processed by the pertinent equalizer. The proposed three-port partially-joint single-ended VDFE equalizers, hereafter denoted by the ignored port, e.g., VDFE[M

_{f}, M

_{b}]-I

_{x}or VDFE[M

_{f}, M

_{b}]-Q

_{x}, offer approximately a 25% reduction in the required circuitry, compared to the full joint processing counterpart. As it will be demonstrated in Section 4, the proposed approach offers a low complexity alternative for electronic equalization, without sacrificing much of the performance if any at all, compared to the fully deployed counterpart. In order to complete the investigation of partially joint equalizer counterparts, also two port partially joint single ended VDFE equalizers are compared to the balanced receiver VDFE equalizers. All of the aforementioned equalization/receiver-related configurations are depicted in Figure 3.

_{c}, Q

_{c}, Q

_{d}), i.e., the destructive port of I channel has been ignored, hence, the six signals (depicted on Figure 4a) fed to the feed-forward part of the equalizer are:

_{f},M

_{b}]-I

_{d}(presented in Figure 4b) is described as:$\text{}\left(l=1,2\right)$.

_{c}, Q

_{c}, Q

_{d}), i.e., the destructive port of the I channel, be ignored. Now the three-port partially-joint single-ended SVDFE equalizers VDFE[M

_{f},M

_{b}]-I

_{d}is described as$:(l=1,2$).

_{f},M

_{b}]-I

_{x}or SVDFE[M

_{f},M

_{b}]-Q

_{x}) are treated similarly. In Figure 5 the differences in structure of the feed-forward part between VDFE (Figure 5a) and SVDFE (Figure 5b) are depicted. While, in this particular block diagram the pruning procedure is presented only for one of the received signals it should be noted that the same technique implies for every different input signal.

_{f}, noting that the contribution of the last term in the summation is rather marginal as $\widehat{{I}_{1}}\left(n\right)$ and $\widehat{{I}_{2}}\left(n\right)$ represent binary digits.

_{c}(t), I

_{d}(t), Q

_{c}(t), and Q

_{d}(t), available at the DQPSK receiver, improves performance by maximizing the signal diversity. MLSE equalizers certainly benefit from this approach, however, in the case of symbol by symbol equalizers, such as DFE and VDFE, some extra attention is required. Following the low-pass equivalent description of a DQPSK link [24], we notice that in the case of ideal identical noise free photodetectors, the four electrical signals available at the receiver, I

_{c}(t), I

_{d}(t), Q

_{c}(t), and Q

_{d}(t) are linearly dependent, as it can be easily shown that I

_{c}(t) + I

_{d}(t) − Q

_{c}(t) − Q

_{d}(t) = 0. In a realistic situation, the presence of noise at the receiver results in (marginally) linear independence, as I

_{c}(t) + I

_{d}(t) − Q

_{c}(t) − Q

_{d}(t) = n(t), with n(t) denoting the contribution of the noise signals from all four photodiodes. Noting that the coefficients of the VDFE equalizer (Equation (2)) are estimated by the solution of a linear system of equations either implicitly or explicitly, the condition number of the associated matrix which is formulated using the available signals (Equations (1)), is of crucial importance concerning the numerical accuracy of the estimated output [22]. In the undesired situation when n(t) is much weaker than the remaining signals, the numerical behavior of the algorithm utilized for the estimation of the equalizer parameters (also known as the linear system solver) will deteriorate, resulting in severe ill-conditioning. A remedy to this problem is to resort to the use of proper regularization, such as the diagonal loading method, requiring extra effort for the handling of this overhead. On the contrary, the proposed partially-joint VDFE and its derivatives, do not suffer from such an effect, as three out of four electrical signals I

_{c}(t), I

_{d}(t), Q

_{c}(t), and Q

_{d}(t), are engaged only. Simulation results indicated that the calculated condition number of the matrices involved into the estimation of the parameters of the equalizers (linear system of equations), vary from 10

^{6}up to 10

^{10}in the case of fully-joint constructive/destructive equalization. This figure is reduced to 10

^{3}in the case of partially-joint equalization.

## 4. Optical Layer Simulation

_{f},M

_{b}] are presented accompanied with the total number of coefficients used for each case. It has become evident that, for both VDFE and SVDFE, large numbers of M

_{f}and M

_{b}(e.g., [13,7] and [11,6]) result in a dramatic increase of the number of coefficients but does not result in a notable increase in performance compared to the equalizers with smaller values of [M

_{f},M

_{b}] (e.g., [9,5]).

_{f},M

_{b}] with [M

_{f},M

_{b}] = [5,3] and [9,5] are sufficient [2]. As a result, the performance of the optical communications system can be significantly improved, in the sense of reaching an extended transmission distance, or by improving the bit error rate (BER) of the received signal. In order to provide the whole picture about the suggested electronic equalization solution, their efficiency is evaluated by means of two different numerical modeling sets of an optical link that operates at 40 Gb/s using a DQPSK modulation format (Figure 7).

_{0}= 12.5 GHz is used to calculate the OSNR value. On the receiver side a bandpass optical filter with 40 GHz bandwidth is used to reduce the ASE noise that enters the MZDI and subsequently the PIN.

_{2}has a value of 2.6 × 10

^{−20}m

^{2}/W while PMD is not considered. Moreover, a two-stage amplification process is used, modelled by two separate amplifiers with a 5 dB noise figure, each one of them utilized to compensate either the SMF or the DCF losses. For both of the aforementioned set up scenarios the transmitter/receiver configuration is considered identical. At the transmitter side, the DQPSK signal is generated via two different MZM (one per channel) which operate in a push-pull mode and have an extinction ratio of 35 dB each. In terms of electrical filtering two low pass third-order Bessel filters are used with a cut-off frequency of 40 GHz. For the PRBS, the modified Wichman-Hill generator [26] is used with a mark probability of 0.5. The transmitter operates at the optical frequency of 193.1 THz with 0 dBm output power. On the receiver side, an optical filter of 40 GHz bandwidth with a third-order Gaussian frequency response is utilized. In order to demodulate the DQPSK signal, two different MZDI are used. After each MZDI output port a photodiode of 1 A/W responsivity and a fourth-order Bessel frequency response electrical filter is used in order detect the optical signal. The input power is chosen to be well above the noise limit and below the nonlinear limit of the system. It is noted, however, that in a realistic system where WDM is used and the number of channels increases, the optical power and crosstalk will be dominant, and the parameters will be re-evaluated.

^{6}bits are used for BER computation, in order to achieve higher accuracy.

_{f},M

_{b}] = [5,3] and [9,5]. The specific sets of [M

_{f},M

_{b}] exhibit sufficient compensation capability, and the equalizer complexity makes it feasible in terms of implementation [2]. The required OSNR with respect to accumulated dispersion is plotted for all equalizer cases discussed in Section 2 and Section 3, along with the performance of the optical system when dispersion compensation is performed only by optical means (without the equalizer case). Apart from the single-ended joint version, where all four output ports are utilized (denoted in figures as 4I), each equalizer is compared with all three partially-joint input counterparts (denoted in figures as 3I). Those partially-joint single-ended configurations differentiate themselves according to the number of ports (e.g., 3I three input) and the specified disregarded input port (e.g., 3I-VDFE[5,3]-I

_{d}). Depending on the output that is not utilized, every configuration exhibits slightly different performance. All of the different configurations seem to offer a significant improvement in the performance of the system under investigation, compared to the case where no electronic equalization is used (without equalizer). Assuming that the rOSNR should be approximately 18 dB, 4Ι equalization schemes seem to increase the amount of tolerable CD up to 1200 ps/nm (SVDFE[5,3]) and 1325 ps/nm (VDFE[5,3]), corresponding to 70 km and 80 km of uncompensated fiber, respectively. The increase of the tolerable amount of CD is even greater when equalizers with [M

_{f},M

_{b}] = [9,5] are deployed. When no equalization is used the tolerable amount of chromatic dispersion reaches up to 300 ps/nm (corresponding to 17 km of uncompensated CD), hence, utilizing equalization can extend the uncompensated distance approximately up to 60–80 km (depending on the equalizer type and the set of [M

_{f},M

_{b}]). This improvement on the CD tolerance of the system comes at the expense of increased required OSNR, since as the amount of residual dispersion increases, the value of rOSNR increase also. The rOSNR deterioration is saturated at 600 ps/nm, indicating that the equalizer is slowly reaching its full potential in terms of alleviating the residual CD. Although every equalizer succeeds in ensuring the proper operation of the system (by achieving a BER of 10

^{−3}), for a certain amount of additional dispersion there is point where the increase in the required OSNR becomes dramatic and the performance is deteriorated. From Figure 8 it becomes evident that although the best efficiency is achieved by the four-input joint single-ended VDFE and SVDFE, all three-input partially-joint alternatives exhibit marginal differences in performance.

^{−3}.

_{d}can offer the best trade-off between complexity and efficiency. On the contrary, all equalizers that ignore Q

_{c}output seems to experience the worst performance although its difference in efficiency is still marginal. Similarly, regarding the I channel of information, ignoring the I

_{d}port instead of the I

_{c}port presents better performance. Τhis feature can be justified by noting that the constructive output port is resembles a duobinary (DB) signal, while the destructive output port resembles an alternating-mark-inversion (AMI) signal [27,28]. The intrinsic high tolerance of the DB modulation format in chromatic dispersion originates from its narrow spectrum [29] and can explain the better performance of every equalizer when the constructive, rather than the destructive, port (of each channel) is maintained to undergo the process of equalization [30].

_{f},M

_{b}]. However, the superiority of balanced equalization becomes more evident as the values of M

_{f}and M

_{b}increase. It should be highlighted that the balanced VDFE[9,5] (i.e., B-VDFE[9,5]) is able to alleviate up to 630 ps/nm of residual dispersion and, thus, almost doubles the dispersion tolerance of the optical system, compared to the case where the compensation is performed only with optical means, whereas the equivalent two-input partially-joint equalizer (2I-VDFE[9,5]-IdQd) that carries the same complexity can reach dispersion tolerance values up to ~500 ps/nm. The performances of four-input and three-input equalizers exceed, by far, those offered by the two-input partially-joint and balanced equalizers and are presented here only for completeness.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Block diagram of a NRZ-DQPSK set exhibiting N transmission spans and a single wavelength transmitter together with a typical single-ended receiver.

**Figure 2.**(

**a**) Schematic illustration of utilized signals sampled with fractional spacing f and (

**b**) signal flow graph or the DQPSK single-ended joint VDFE equalizer.

**Figure 3.**Schematic explanation of different equalizer configurations depending on the receiver type (single-ended or balanced) and number of inputs engaged in the equalization process: (

**a**) four-input joint single-ended equalization; (

**b**) three-input partially-joint single-ended equalization; (

**c**) two-input partially-joint single-ended equalization; and (

**d**) balanced equalization.

**Figure 4.**(

**a**) Schematic illustration of utilized signals and (

**b**) signal flow graph for the proposed partially-joint single-ended DQPSK VDFE[M

_{f},M

_{b}]-I

_{d}.

**Figure 5.**The differences in FF part structure for an equalizer of M

_{f}= 4 between (

**a**) VDFE and (

**b**) SVDFE.

**Figure 7.**Simulation generic setup for (

**a**) a noise-loading technique and (

**b**) multi-span scenarios with eight spans.

**Figure 8.**Required OSNR performance of comparison between the joint single-ended configurations (4I-) and the partially-joint three-input ones (3I) for the cases of (

**a**) SVDFE[5,3]; (

**b**) VDFE[9,5], (

**c**) SVDFE[9,5]; and (

**d**) VDFE[9,5].

**Figure 9.**Estimated BER of optical links that utilize (

**a**) SVDFE[5,3]; (

**b**) VDFE[5,3]; (

**c**) SVDFE[9,5]; and (

**d**) VDFE[9,5] equalizers depending on the ignored input signal (I

_{c},I

_{d}, Q

_{c}, Q

_{d}). 4I and 3I denote the number of inputs.

**Figure 10.**Estimated BER of optical links that utilize (

**a**) SVDFE[5,3]; (

**b**) VDFE[5,3]; (

**c**) SVDFE[9,5] and (

**d**) VDFE[9,5] for 2I-, 3I, and balanced equalization.

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## Share and Cite

**MDPI and ACS Style**

Nanou, M.; Politi, C.; Stavdas, A.; Georgoulakis, K.; Glentis, G.-O.
High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDFE Equalizers. *Photonics* **2017**, *4*, 13.
https://doi.org/10.3390/photonics4010013

**AMA Style**

Nanou M, Politi C, Stavdas A, Georgoulakis K, Glentis G-O.
High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDFE Equalizers. *Photonics*. 2017; 4(1):13.
https://doi.org/10.3390/photonics4010013

**Chicago/Turabian Style**

Nanou, Maki, Christina (Tanya) Politi, Alexandros Stavdas, Kristina Georgoulakis, and George-Othon Glentis.
2017. "High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDFE Equalizers" *Photonics* 4, no. 1: 13.
https://doi.org/10.3390/photonics4010013