## Abstract

Due to the chirp effect, the modulation information is carried by both intensity and phase of the optical signal generated by a directly modulated laser (DML). Compared to the signal recovery based on intensity alone, the receiver sensitivity is significantly improved when phase is also involved for signal recovery. In this paper, we provide thorough analysis of two schemes to recover the directed modulated signal based on the combined information of intensity and phase. The intensity and phase combined recovery for polarization-division-multiplexed (PDM) directly modulated signal is demonstrated for the first time, achieving a data rate of 40 Gb/s over a transmission distance of 160 km. Compared with the conventional intensity-only recovery, the receiver sensitivity is improved by 5.5 dB and 10 dB respectively by using two intensity and phase combined recovery schemes.

© 2015 Optical Society of America

## 1. Introduction

With the rapid increase of data traffic, metro networks are facing the challenges of upgrading its capacity to 100 Gb/s and beyond in the near future. By employing polarization-division-multiplexed quadrature amplitude modulation (PDM-QAM) and coherent detection, long-haul optical networks are able to achieve the high data rate up to 1 Tb/s [1–3]. However, such system is infeasible to be widely deployed in metro networks due to its high cost. Shared by relatively fewer users, the cost becomes a primary consideration for the metro networks. Significant amount of efforts have been made to improve the performance of the metro networks while maintaining low cost [4–9]. In comparison with the in-phase/quadrature (IQ) modulator and external cavity laser (ECL) used for PDM-QAM, directly modulated laser (DML) has the merits of small form factor, low power consumption, and low cost. Therefore, it provides a promising solution for cost-efficient metro networks. In combination with DML, coherent detection is necessary to meet with the chromatic dispersion (CD) requirement in the metro networks [8,9]. 100 Gb/s transmission with pulse-amplitude-modulation (PAM) signal has been demonstrated in [9] using DML and coherent detection.

For the conventional systems with DML, only the intensity modulation is considered during the signal recovery, despite the fact that both intensity and phase are acquired at the coherent receiver [8,9]. The conventional recovery scheme is referred as intensity-only recovery in the following text. Considering the chirp effect, the phase (or frequency) of the optical signal is modulated along with the intensity in DML [10–12]. By utilizing the additional information carried by phase (or frequency), the receiver sensitivity can be significantly improved. With increased OSNR margin, such systems can be upgraded to higher data rate or longer transmission distance. An approach to utilize the frequency modulation has been proposed in [9], which converts the frequency modulation into the intensity modulation with an optical filter. The distance between different modulation levels is enlarged after the conversion, improving the receiver sensitivity. However, the modulation information carried by phase (or frequency) is not analyzed in detail. In this paper, we extend our previous work [13–15] and demonstrate the recovery of polarization-division-multiplexed (PDM) directly modulated signal using combined information of intensity and phase. Detailed analysis will be provided for two intensity and phase combined signal recovery schemes.

Unlike the intensity, the phase is modulated with a more complicated modulation mechanism in DML. Special digital signal processing (DSP) has to be designed to use phase information for signal recovery [13–15]. In this paper, we will introduce two intensity and phase combined signal recovery schemes. The organization of the paper is as follows: the phase modulation mechanism in DML will be analyzed in section 2, derived from the dynamic laser chirp equation. Based on that, the DSP used in the two schemes will be revealed in section 3, including the chirp characterization methods and signal recovery algorithms. The experimental demonstrations of the two schemes will be described in section 4, where the recovery of PDM directly modulated signal using intensity and phase combined recovery scheme is demonstrated for the first time with a data rate of 40 Gb/s over a transmission distance of 160 km. Improving the OSNR sensitivity by 5.5 dB and 10 dB respectively, the two intensity and phase combined recovery schemes show great advantages over the intensity-only recovery.

## 2. Phase modulation mechanism in DML

Controlled by the driving current, the variation of the electron density in DML leads to the modulation of the output signal [10,12]. As the gain increases proportionally with the electron density, the output intensity changes linearly with the modulation level. Due to the linear modulation response, the optical intensity provides a straightforward way to recover the signal in the transmission system with DML [8,9,16,17]. On the other hand, the modulation response of output phase (or frequency) is more complicated. When the refractive index varies with the electron density, the frequency of the output signal is shifted as well. Derived from the rate equations governing the laser dynamics, the frequency chirp $\Delta \nu (t)$ is a function of the output power *P*(*t*), and can be expressed as [10,12]

The phase of the optical signal is an integral of its frequency. Taking the integral of Eq. (1), and considering the impact from the uncompensated frequency offset ${f}_{off}$ and phase noise ${\theta}_{noise}(t)$, the phase$\phi (t)$of the output signal can be written as

For two neighboring symbols in a sequence, i.e., the (*n*-1) and *n*-th (*n* = 0,1,…) symbols, the phase of Eq. (2) can be rewritten as

*i*-th sampling point. The first

*n*-2 symbols lead to the same phase in Eqs. (3) and (4). To remove the accumulated impact of the modulation signal, phase rotation is used instead of the absolute value of phase. Considering the phase noise as a slow-varying parameter, the phase noise in two neighboring symbols is assumed to be constant, ${\theta}_{noise}({t}_{n-1})={\theta}_{noise}({t}_{n})$. Subtracting Eq. (4) with Eq. (3), the phase rotation between the two symbols is given by

By using phase rotation $\Delta \phi $, we obtain two benefits: (1) by removing the accumulated impact of the modulation signal, the phase rotation $\Delta \phi $ is determined by the modulation signal in two neighboring symbols, which makes the signal recovery more manageable, and (2) the laser phase noise is removed, which results in simpler DSP, as opposed to being in need of sophisticated phase locked loop (PLL) algorithm.

## 3. Intensity and phase combined signal recovery

#### 3.1 Characterization of coefficients$\alpha $, $\kappa $

As shown in Eq. (6), the phase rotation $\Delta \phi $ is related to the modulation signal through the coefficients $\alpha $ and $\kappa $. Before the signal recovery, the coefficients, $\alpha $ and $\kappa $, need to be characterized using specially designed pilot symbols. To characterize the coefficient $\alpha $, pilot symbols shown in Fig. 1(a) are used, where the modulation signal switches between high and low levels. For a transition from low to high level, the phase rotation is

*P*and

_{L}*P*are the output power corresponding to the low and high levels respectively. For a transition from low to high level, the phase rotation is

_{H}To characterize the coefficient $\kappa $, pilot symbols shown in Fig. 1(b) are used. For three continuous symbols at time ${t}_{n-2}$, ${t}_{n-1}$ and ${t}_{n}$, the signal is modulated at the same level with output power of ${P}_{1}$. Noting that the first term in Eq. (6) equals to zero when the output power maintains the same, the phase rotation within the three symbols is

#### 3.2 Intensity and phase combined signal recovery

### 3.2.1 Recovery scheme based on complex signal

As shown in Eq. (6), the phase rotation $\Delta \phi ({t}_{n-1},{t}_{n})$ is determined by the modulation signal at time ${t}_{n-1}$ and ${t}_{n}$. The part related to the modulation signal at time ${t}_{n-1}$ can be estimated from the output power at time ${t}_{n-1}$ as

Noting that the amplitude and phase increase simultaneously with the modulation level, the constellation of the complex signal $A({t}_{n})$follows the trajectory in Fig. 2, when the modulation level is increased gradually. By choosing *N* modulation levels properly, we can have *N* constellation points evenly spread on the plane, based on which the signal can be recovered. The described signal recovery scheme is referred as complex recovery in the following text, as it is based on the complex signal of Eq. (14).

When intensity-only recovery is used, the *N* modulation levels correspond to *N* constellation points on the real axis. By converting to complex recovery, the constellation points are rotated and spread on a two dimensional plane. Taking 3-level PAM (3-PAM) signal as an example, the constellations obtained by the two recovery schemes are shown in Figs. 3(a) and 3(b) respectively. By inducing a phase rotation into the signal, the constellation points are separated further away. Therefore, the complex recovery shows a better tolerance to the noise compared to the intensity-only recovery, leading to better performance. However, the performance of complex recovery is limited by two factors: (1) the phase rotation of Eq. (6) is not fully utilized for signal recovery, part of the information is artificially removed for the simplicity of DSP; and (2) additional noise is induced into the phase during the estimation of Eq. (13).

### 3.2.2 Recovery scheme based on Viterbi algorithm

To avoid the drawbacks in complex recovery, Viterbi algorithm is applied to facilitate a more advanced signal recovery. For Viterbi algorithm, the possible modulations in a signal sequence are represented by a trellis shown in Fig. 4 [18,19]. The *L* symbols in the sequence are denoted by the *L* nodes in the trellis, and *N* modulation levels are denoted by the *N* states (${S}_{1},{S}_{2},\cdots ,{S}_{N}$) at each node. Using a branch in the trellis to represent a modulation transition, an input sequence corresponds to a unique path (red line in Fig. 4) through the trellis. Viterbi algorithm searches the most likely path based on two metrics, the branch metric and the state metric [18].

The branch metric $m\left[{S}_{i}(K-1),{S}_{j}(K)\right]$($i,j=1,2,\cdots ,N$) is a measurement of the likelihood for the transition from the *i*-th state at (*K*-1)-th node to the *j*-th state at *K*-th node. It is usually characterized by the distance between the ideal outputs and the actual observations at the receiver. For the transition $\left[S(K-1),S(K)\right]$, we have three observations, the intensity for the two symbols, $P(K-1)$ and $P(K)$, and the phase rotation between the two symbols $\Delta \phi (K-1,K)$. The branch metric is then defined as

The state metric, $M\left[{S}_{j}(K)\right]$, records the minimum path weight over all the possible paths leading to the *j*-th state at *K*-th node. To minimize the path weight over *K* nodes, it is equivalent to minimize the path weight leading to each state at the first *K*-1 node $M\left[{S}_{i}(K-1)\right]$($i=1,2,\cdots ,N$), and then minimize the sum of $M\left[{S}_{i}(K-1)\right]$ and the corresponding branch metric $m\left[{S}_{i}(K-1),{S}_{j}(K)\right]$ to the *j*-th state at the *K*-th node. Then, the state metric for the (*K* + 1)-th node can be calculated from the *K*-th node. The iteration process is known as Viterbi algorithm, and can be expressed as [18]

*i*. The signal recovery scheme using Viterbi algorithm is referred as Viterbi recovery. The procedures of Viterbi recovery can be summarized as follows. The branch metrics for all the transitions in the trellis are calculated using Eq. (15). Starting from the given state ${S}_{i}(1)$ at the first node, the state metrics are initialized at the second node with the branch metrics $m\left[{S}_{i}(1),{S}_{j}(2)\right]$. After that, the state metrics at each node are calculated using Eq. (16). The paths corresponding to the state metrics are also recorded. The survivor path is the one with the minimum state metric at the last node, from which the signal is recovered.

Compared to the complex recovery, the Viterbi recovery can provide the following advantages: (1) the phase information in Eq. (6) is fully utilized by the Viterbi recovery; and (2) the phase noise induced by the phase estimation of Eq. (13) can be avoid using the Viterbi recovery. Therefore, the system performance can be further improved by the Viterbi recovery.

## 4. Experiments and results

Two intensity and phase combined recovery schemes are experimentally demonstrated in both single-polarization and dual-polarization systems. The advantages of the two schemes are verified by comparing their performances with the conventional intensity-only recovery.

The experiment setup for single-polarization system is shown in Fig. 5(a). A directly modulated distributed feedback (DFB) laser is used as transmitter with output wavelength of 1550 nm and 3-dB modulation bandwidth of 10 GHz. Biased by a current source, the DFB laser is driven by the amplified multi-level PAM signal, produced by an arbitrary waveform generator (AWG) working at the sampling rate of 10 GBaud/s. For the single-polarization transmission, a data rate of 15.88 Gb/s is achieved by using 3-PAM signal, which carries log_{2}(3) = 1.588 bits per symbol, and a data rate of 20 Gb/s is achieved by using 4-PAM signal, which carries log_{2}(4) = 2 bits per symbol. For the two recovery schemes using the phase information, pilot symbols depicted in Figs. 1(a) and 1(b) are inserted before the transmitted signal for the characterization of the coefficients $\alpha $ and $\kappa $. The characterization only needs to be done for one time, and the signal can be transmitted without pilots after that. Therefore, the data rate will not be reduced by the pilot symbols. The drive signals are optimized separately for intensity-only and complexity recovery schemes. For the intensity-only recovery and Viterbi recovery schemes, bias current of 70 mA and electrical signal with Vpp of 4V are used to drive the DML. The drive voltage of each modulation level is adjusted to generate optical signals with equally spaced amplitude. For the complex recovery scheme, bias current is increased to 90 mA to enlarge the radius of the constellation circle. Phase rotation from the first to the last modulation level is controlled by the Vpp of the electrical signal. For 3-PAM and 4-PAM signal, Vpp of 2.9 V and 3.4 V are used respectively. The drive voltage of each modulation level is adjusted to have evenly spread constellations. The output signal from the DFB laser is transmitted over 160 km standard single mode fiber (SSMF). For the single-polarization transmission, the launch power is set to be −3 dBm with an optical attenuator inserted before the SSMF.

The data rate is doubled when polarization-division-multiplexed (PDM) signal is transmitted. For the dual-polarization system, the PAM signal is generated using the same method as in the single-polarization system. The generated signal is split to two paths, with one path delayed by 600 m fiber, as shown in Fig. 5(b), so that the signal from the DFB laser with a linewidth of ~10 MHz is phase-de-correlated for the two paths. The two de-correlated paths are multiplexed by a polarization beam combiner (PBC) into two orthogonal polarizations. The launch power is adjusted to 0 dBm for the PDM signal. With polarization-division-multiplexing, the data rate is increased to 31.76 Gb/s when 3-PAM signal is used, and 40 Gb/s when 4-PAM signal is used.

At the receiver, the signal is detected by a standard coherent receiver, where the local oscillator (LO) is provided by a tunable laser with a linewidth of 11 kHz. To reduce the electrical bandwidth requirement at the receiver, the wavelength of the LO is tuned to 1550 nm, close to the central wavelength of the optical signal. For single-polarization transmission as shown in Fig. 5(a), a polarization controller (PC) is used to align the polarization of the LO with the received signal before mixing them in a 90° optical hybrid. The *I* and *Q* components of the received signal are provided by a pair balanced detectors (BDs) with 40 GHz bandwidth. The detected data is collected by a digital sampling oscilloscope working at the sampling rate of 50 GSa/s, and processed offline. After the chromatic dispersion compensation, the intensity and the phase of the received signal are separated for different processing. After a Bessel filter, the intensity information is down-sampled to 10 GSa/s. For intensity-only recovery, the symbol decision is made directly after that. For other two recovery schemes using phase information, the frequency offset needs to be compensated. After the Bessel filter and down-sampling, the phase rotation is calculated for the neighboring symbols. Coefficients $\alpha $ and $\kappa $ are estimated from the pilot symbols using Eqs. (9) and (12). For the complex recovery, the phase of Eq. (13) is estimated and removed from the phase rotation. The residual phase is then combined with the intensity to construct the complex signal, based on which the symbol decision is made. For the Viterbi recovery, the branch metric is calculated using Eq. (15) with the combined information from phase and intensity. The state metric is then calculated to search the minimum path weight, based on which the signal is recovered. The channel memory for the Viterbi algorithm is 1, and the number of states is 3 for 3-PAM signal and 4 for 4-PAM signal. Good performance can be reached by storing 20 sequences of data.

For dual-polarization transmission, the received signal is separated into X and Y polarizations using a polarization beam splitter (PBS) as shown in Fig. 5(b). Two paths of LO are aligned with the X and Y polarizations respectively. The two polarizations are mixed with the LO in a 90° optical hybrid followed by two pairs of BDs, by which *I* and *Q* components of the two polarizations are provided. After the CD compensation, digital polarization demultiplexing is implemented using an adaptive butterfly equalizer with 41 taps [20]. From that, the DSP for each polarization follows the same procedures for single-polarization transmission.

Since the phase is random in the intensity-only recovery, a ring is obtained for the constellation of each modulation level. Figures 6(a) and 6(b) are the constellations of 3-PAM signal when intensity-only recovery is used. In Fig. 6(a), three clearly separated rings can be found when OSNR equals to 25 dB. However, the boundary of each level becomes blurred in Fig. 6(b) when OSNR decreases to 19 dB. For complex recovery, the constellation of each modulation level is rotated by a fixed angle. Figures 6(c) and 6(d) are the constellations of 3-PAM signal when complex recovery is used. At OSNR of 25 dB, three ‘clouds’ can be found distantly separated from each other, as shown in Fig. 6(c). Though the ‘clouds’ become noisier when OSNR decreases to 19 dB, the three modulation levels are still clearly separated, as shown in Fig. 6(d). The phase rotation leads to an enlarged distance between different modulation levels, resulting in a better performance when complex recovery is used. Figures 7(a) and 7(b) are the constellations of 4-PAM signal at OSNR of 28 dB and 22 dB, when intensity-only recovery is used. Figures 7(c) and 7(d) are the constellations of 4-PAM signal at OSNR of 28 dB and 22 dB, when complex recovery is used. Enlarged distance can also be observed for 4-PAM signal using complex recovery. For Viterbi recovery, the phase information is more accurate, by avoiding the phase estimation in complex recovery, leading to an even better performance. Total of 4x10^{5} bits have been collected for BER calculation.

Figure 8 shows the measured BER vs. OSNR for the single-polarization transmission using (a) 3-PAM and (b) 4-PAM signal. The system performances, using three different recovery schemes, are compared for both back-to-back and 160 km transmission. Compared with intensity-only recovery, the OSNR sensitivity is improved by around 5.5 dB by complex recovery and by around 10 dB by Viterbi recovery for both 3-PAM and 4-PAM signal. It is noted that the BER performance of Viterbi recovery is slightly improved, compared to our previous measurement of [14] for single-polarization transmission using 4-PAM signal. The improvement of around 0.5 dB is achieved by an optimization of DSP associated with the estimation of coefficients *α* and *κ*. The recovery of polarization-division-multiplexed signal using our proposed schemes is demonstrated for the first time. The measured BER vs. OSNR for dual-polarization transmission with 3-PAM and 4-PAM signal are shown in Figs. 9(a) and 9(b) respectively. The OSNR sensitivity of dual-polarization transmission is around 3.5 dB worse than single-polarization transmission. The OSNR improvements due to complex recovery and Viterbi recovery are similar to the case of single-polarization transmission. For the dual-polarization transmission, intensity-only recovery can only be used to recover 3-PAM signal, achieving the threshold of 20% FEC, and fails to recover 4-PAM signal. However, the 4-PAM signal can be successfully recovered by both complex recovery and Viterbi recovery schemes. By achieving the threshold of 7% FEC, signal with a net data rate of 37.38 Gb/s can be transmitted. The better performance of the Viterbi recovery scheme makes it more suitable for the transmission with higher data rate or longer distance. By using a transmitter with higher electrical bandwidth, the intensity and phase combined recovery schemes can be readily extended to the application over 100 Gb/s.

## 5. Conclusion

We have demonstrated two recovery schemes which use combined information of intensity and phase in polarization-division-multiplexed system with DML and coherent detection. Besides the intensity, the output phase of DML is also modulated through the chirp effect. By exploiting the information carried by phase, the receiver sensitivity is greatly improved when the signal is recovered based on combined information of intensity and phase. For the scheme of complex recovery, the constellations of different modulation levels are distinctly separated due to the induced phase rotation, which makes the system more resistant to the noise. The system performance can be further enhanced using the scheme of Viterbi recovery, where the phase information is utilized more efficiently, and the noise induced by the phase estimation is avoided. Compared to the conventional intensity-only recovery, the receiver sensitivity is improved by 5.5 dB using complex recovery and 10 dB using Viterbi recovery. With intensity and phase combined signal recovery, 40 Gb/s PDM-4-PAM signal can be transmitted over 160 km fiber with 7% FEC overhead. Considering its superior performance, higher data rate at longer distance can be achieved by means of Viterbi recovery if higher baud-rate DMLs are used.

## References and links

**1. **S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express **15**(5), 2120–2126 (2007). [CrossRef] [PubMed]

**2. **C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, G.-D. Khoe, and H. D. Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. **26**(1), 64–72 (2008). [CrossRef]

**3. **W. Shieh, Q. Yang, and Y. Ma, “107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing,” Opt. Express **16**(9), 6378–6386 (2008). [CrossRef] [PubMed]

**4. **W. R. Peng, X. Wu, K. M. Feng, V. R. Arbab, B. Shamee, J. Y. Yang, L. C. Christen, A. E. Willner, and S. Chi, “Spectrally efficient direct-detected OFDM transmission employing an iterative estimation and cancellation technique,” Opt. Express **17**(11), 9099–9111 (2009). [CrossRef] [PubMed]

**5. **B. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental demonstrations of electronic dispersion compensation for long-haul transmission using direct-detection optical OFDM,” J. Lightwave Technol. **26**(1), 196–203 (2008). [CrossRef]

**6. **D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in *Optical Fiber Communication Conference 2014* (OSA, 2014), paper Th5C.7. [CrossRef]

**7. **Q. Hu, D. Che, Y. Wang, and W. Shieh, “Advanced modulation formats for high-performance short-reach optical interconnects,” Opt. Express **23**(3), 3245–3259 (2015). [CrossRef] [PubMed]

**8. **C. Xie, P. Dong, P. Winzer, C. Gréus, M. Ortsiefer, C. Neumeyr, S. Spiga, M. Müller, and M. C. Amann, “960-km SSMF transmission of 105.7-Gb/s PDM 3-PAM using directly modulated VCSELs and coherent detection,” Opt. Express **21**(9), 11585–11589 (2013). [CrossRef] [PubMed]

**9. **C. Xie, S. Spiga, P. Dong, P. Winzer, A. Gnauck, C. Gréus, C. Neumeyr, M. Ortsiefer, M. Müller, and M. C. Amann, “Generation and transmission of 100-Gb/s 4-PAM using directly modulated VCSELs and coherent detection,” in *Optical Fiber Communication Conference 2014* (OSA, 2014), paper Th3K.7. [CrossRef]

**10. **R. S. Tucker, “High-speed modulation of semiconductor laser,” IEEE Trans. Electron. Dev. **32**(12), 2572–2584 (1985). [CrossRef]

**11. **G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” J. Quantum Electron. **25**(11), 2297–2306 (1989). [CrossRef]

**12. **T. L. Koch and J. E. Bowers, “Factors affecting wavelength chirping in directly modulated semiconductor lasers,” in Proceedings Conference Laser and Electrooptics (IEEE, 1985), pp. 72–74.

**13. **Q. Hu, D. Che, Y. Wang, A. Li, J. Fang, and W. Shieh, “Beyond amplitude-only detection for digital coherent system using directly modulated laser,” Opt. Lett. **40**(12), 2762–2765 (2015). [CrossRef] [PubMed]

**14. **D. Che, Q. Hu, F. Yuan, Q. Yang, and W. Shieh, “Enabling complex modulation and reception of directly modulated signals using laser frequency chirp,” IEEE Photonics Technol. Lett. **27**(22), 2407–2410 (2015). [CrossRef]

**15. **D. Che, Qian Hu, Feng Yuan, and W. Shieh, “Enabling complex modulation using the frequency chirp of directly modulated lasers,” in *European Conference on Optical Communication 2015* (ECOC, 2015), paper Mo.4.5.3.

**16. **A. Larsson, P. Westbergh, J. Gustavsson, A. Haglund, and B. Kögel, “High-speed VCSELs for short reach communication,” Semicond. Sci. Technol. **26**(1), 014017 (2011). [CrossRef]

**17. **W. Yan, T. Tanaka, B. Liu, M. Nishihara, L. Li, T. Takahara, Z. Tao, J. C. Rasmussen, and T. Drenski, “100 Gb/s optical IM-DD transmission with 10G-Class devices enabled by 65 GSamples/s CMOS DAC core,” in *Optical Fiber Communication Conference* (OSA, 2013), paper OM3H.1. [CrossRef]

**18. **G. D. Forney, “The Viterbi algorithm,” Proc. IEEE **61**(3), 268–278 (1973). [CrossRef]

**19. **G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory **18**(3), 363–378 (1972). [CrossRef]

**20. **S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express **16**(2), 804–817 (2008). [CrossRef] [PubMed]