Maximum likelihood sequence estimation for optical complex direct modulation

Di Che; Feng Yuan; William Shieh

doi:10.1364/OE.25.008730

1. Introduction

High-speed direct modulation (DM) of semiconductor lasers was one of the most significant breakthroughs for telecommunication in 1980s [1–3]. Since its birth, DM lasers (DML) have been exploited as versatile optical transmitters. (i) When the injection signal is well within the modulation bandwidth, there is an approximately linear relationship between the injection current and the laser output power, which enables the optical intensity modulation (IM). (ii) The injection current perturbs the refractive index of the laser active region, resulting in a wavelength deviation simultaneously with the IM, namely, the frequency chirp [1]. Chirp is no doubt detrimental for the direct detection (DD) based DM systems [4] when the fiber dispersion is present, because it severely expands the optical spectrum, leading to a signal much more sensitive to the dispersion induced inter-symbol interference (ISI) after digital detection. However, from an opposite perspective, chirp sweeps the optical frequency, endowing DMLs the capability of angle modulation [2,3], such as the frequency-shift-keying (FSK) and differential phase-shift-keying (DPSK).

The intensity and frequency responses of DMLs are strongly coupled, as derived by the laser diode rate equation [1]. For very-short-reach interconnect (normally <40 km) where the chirp has little influence on the system performance, the intensity-only DM model can be applied to optical transmitters regardless of the frequency deviation. Owing to the ultra-low-cost feature, high-baud-rate DMLs have become the dominant solution for parallel optical interconnects [5,6]. However, for optical transmission where dispersion cannot be ignored, early DML systems had to utilize either the intensity or the angle modulation separately by restraining the other effect, limited by the direct detection technology. In terms of the conventional DM-DD transmission base on IM, chirp should be minimized, for example, using the chirp-managed lasers (CML) [7]. In contrast, to take advantage of the chirp, the intensity variation must be suppressed [3]. This can either be realized by the small-signal modulation with a bias much higher than the DML threshold; or by short-pulse modulation to change the instantaneous optical frequency. However, the small-signal modulation is strongly limited by the noise of RF devices, which is unrealistic to be scaled to the modern high-baud-rate and high-order modulation; and the short-pulse modulation (namely, the return-to-zero pulse, RZ) suffers from poor spectral efficiency, which has been completely substituted by the non-return-to-zero (NRZ) or even the Nyquist pulse. There emerged recently some efforts of applying the chirp modulation to the low-speed coherent passive optical networks (PON) [8]; but in general, it is not believed DMLs could be utilized to the modern high-speed transmission link.

On the contrary of the conventional 1-D DM above, we demonstrated the DM combined intensity and angle modulation recently [9–11], by virtue of the advanced digital coherent detection. To distinguish this new DM concept, we named it as complex DM (CDM), because it modulates 2-D information on the complex plane. CDM realizes large-signal modulation with the NRZ pulse, similar to the DM transmitter dominant nowadays in very-short-reach optical interconnects. In contrast, using the same transmitter, CDM is capable of fiber transmission over distance beyond 1000 km [9], due to its superior system OSNR sensitivity. CDM successfully rejuvenates the DM as a promising ultra-low-cost optical transmitter for medium-reach transmissions (normally <1000 km), like the metropolitan area networks (MAN). The crucial technique of CDM is the joint demodulation of intensity and differential phase with the maximum likelihood sequence estimation (MLSE) [12,13], enabled by the modern digital signal processing (DSP). In this paper, we further analyze the transition probability calculation for MLSE, and propose an even more powerful MLSE based on the statistical transition probability estimation [14]. This new MLSE not only enhances the accuracy of the demodulation for NRZ-CDM, but more importantly, is generalized to support signal pulses with even higher spectral efficiency than NRZ – the Nyquist pulse. We demonstrate the first single channel 100-Gb/s CDM using 10G class DMLs, over a record distance of 1600 km for PAM-4 with a BER below 20% FEC threshold [14] (corresponding to the net data-rate of 83.3 Gb/s). Owing to the Nyquist pulse shaping, this DM transmitter realizes 8-bit/s/Hz electrical spectral efficiency.

2. MLSE for CDM

CDM is a 2-D modulation on the complex plane. Optical coherent detection is compulsory to recover this 2-D signal. The frequency can be approximated by the derivative of the phase:

f = \frac{d φ (t)}{d t} = \lim_{t_{1} \to t_{2}} \frac{φ (t_{2}) - φ (t_{1})}{t_{2} - t_{1}}

After digital detection, the smallest time granularity is the sampling period. Therefore, the differential phase between two adjacent sampling points becomes a good characterization of the optical frequency. CDM decoder performs 2-tap MLSE using the Viterbi algorithm, taking the intensity

I_{t}

and differential phase

Δ φ_{t}

as inputs, where t is the discrete time sequence. We define two data sets: (i) the state

{S_{t}}

(each state corresponds to an intensity level); (ii) the transition

{χ_{t} | χ_{t} \underline{\underline{Δ}} (S_{t}, S_{t - 1})}

. The crucial task for MLSE is to calculate the transition probability

P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1})

in order to determine the most reliable transitions.

P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1})

is the conditional probability of observing

(I_{t}, Δ φ_{t})

when the transition

(S_{t}, S_{t - 1})

is sent. To simplify the calculation, we regard

I_{t}

and

Δ φ_{t}

as 2 independent random variables to decompose the 2-D joint distribution into 2 1-D distributions:

P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1}) = P (I_{t} | S_{t}) • P (Δ φ_{t} | S_{t}, S_{t - 1})

It is assumed that the intensity channel has no ISI so that

P (I_{t} | S_{t}, S_{t - 1}) = P (I_{t} | S_{t})

. The distribution

(I_{t} | S_{t})

is simply a linear mapping between the baseband signal and optical intensity; however, a phase model is required to calculate

P (Δ φ_{t} | S_{t}, S_{t - 1})

. The laser rate equation has offered an approximate chirp model only determined by the laser output intensity, which can be converted to the differential phase by time integrals [7]:

Δ f = \frac{α}{4 π} (\frac{d}{d t} \ln I (t) + κ \cdot I (t)) \to Δ φ = \frac{α}{2} (\ln \frac{I (t_{2})}{I (t_{1})} + \int_{t_{1}}^{t_{2}} κ I (t) d t)

where

α / κ

is the laser transient/adiabatic chirp coefficient, and

T = t_{2} - t_{1}

is one symbol period. Equation (3) offers a

Δ φ

model, but the adiabatic part requests continuous intensity information within

[t_{1}, t_{2}]

due to the time integral, instead of the discrete intensity at

t_{1}

and

t_{2}

. Fortunately, the NRZ pulse maintains its intensity as a constant within T. Using the mean value theorem, the integral can be calculated from the two end-values:

Δ φ = \frac{α}{2} (\ln \frac{I (t_{2})}{I (t_{1})} + κ \frac{I (t_{1}) + I (t_{2})}{2} \cdot T)

This is the discrete signal approximation of frequency chirp, namely, the

Δ φ

model required by Eq. (2). By assuming the additive white Gaussian noise (AWGN) channel, the transition probability

P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1})

now can be characterized by the transition distance [7] on the 2-D constellation plane, defined via the intensity difference between

I_{t}

and

S_{t}

, and the phase difference between

Δ φ_{t}

and

Δ φ_{t (e)}

, where

Δ φ_{t (e)}

is the estimated

Δ φ

from Eq. (4).

Equation (4) offers a simple closed-form CDM model. However, the derivation above limits the signal pulse to NRZ only. For the pulse whose transition is not flat within a symbol period, the accuracy of Eq. (4) is questionable. Moreover, Eq. (4) requests the estimation of chirp related parameters, such as $α$ , $κ$ and transmitted IM levels. These estimation errors accumulate in Eq. (4), resulting in transition distances with much lower accuracy. In fact, the most reliable way of calculating $P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1})$ is to acquire the distributions of $(I_{t} | S_{t})$ and $(Δ φ_{t} | S_{t}, S_{t - 1})$ directly through the statistics method [13], and store them into a look-up table. To adapt the DML and channel dynamics, an update strategy can be applied [13]:

D^{(n)} = (1 - λ) D^{(n - 1)} + λ d^{(n)}

where

D

stands for the stored distributions,

n

is the update serial number,

λ

is the step size to control the update speed, and

d

is the current measured distributions. The probability can be calculated in logarithm domain to avoid multiplication. The MLSE only requires 2 statistical histogram tables for

\log P (I_{t} | S_{t})

and

\log P (Δ φ_{t} | S_{t}, S_{t - 1})

. The sophisticated chirp model is hidden behind the statistics of

(Δ φ_{t} | S_{t}, S_{t - 1})

. There is no need to measure any chirp related parameters; and thus, no need to concern the inaccuracy of these coefficients, as well as the chirp model itself.

The statistical transition probability estimation for $(I_{t} | S_{t})$ and $(Δ φ_{t} | S_{t}, S_{t - 1})$ follows some routine procedures below in the Table 1:

Table 1. Procedures for the statistical transition probability estimation

View Table

3. MLSE in NRZ-CDM system

Early CDM experiment demonstrations utilize the modulation format of NRZ PAM. In this section, we reprocess the experiment data in [9,10] to compare the system performance between the MLSE with transition distance (MLSE-TD) and the newly-proposed MLSE with statistical transition probability (MLSE-TP). The baseband 10-Gbaud PAM-4 (or PAM-8) signal is generated by an arbitrary waveform generator (AWG) sampling at 10 GSa/s. This RF signal drives a distributed feedback (DFB) laser working around the wavelength of 1550 nm, with linewidth of 10 MHz. The laser output is fed into a polarization-multiplexing emulator, with one path delayed by 600-m fiber to realize the phase de-correlation. At receiver, an external cavity laser (ECL) with linewidth of 10 kHz serves as the local oscillator (LO). The signal and LO are fed into a dual polarization coherent receiver, whose outputs are sampled by a real-time oscilloscope at 50-GSa/s with 16-GHz bandwidth.

We first reveal the measured distributions in Fig. 1 at the OSNR of 32 dB. The 4 intensity distributions $(I_{t} | S_{t})$ in Fig. 1(a) present Gaussian shapes. Because the signal is modulated by the equal-intensity spacing with the ASE noise added to optical field, PAM level with high intensity suffers from larger noise after the square-law detection. Figure 1(b) illustrates the differential phase distributions $(Δ φ_{t} | S_{t}, S_{t - 1})$ . The differential phase $Δ φ$ behaves regularly as 16 Gaussian distributions, categorised by 4 curve clusters. Each cluster represents the statistics of the current state $S_{t}$ ; within each cluster, the 4 curves stand for 4 previous states, respectively.

Fig. 1 Measured probability density function for NRZ PAM-4 at OSNR of 32 dB (back-to-back measurement) (a) intensity; (b) differential phase.

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Figure 2(a) compares the OSNR sensitivity between the MLSE-TD and MLSE-TP for PAM-4. As expected, MLSE-TP has better accuracy in characterizing the CDM behavior, leading to an OSNR sensitivity advantage of 2 dB at 7% FEC threshold. This gap shrinks to 1 dB at 20% FEC threshold. When the modulation order is increased to PAM-8, the gap between MLSE-TD and MLSE-TP becomes much wider in Fig. 2(b). Even at 20% FEC, MLSE-TP holds 4 dB OSNR advantage over MLSE-TD. For PAM-8, the estimation error of PAM levels increases, leading to more calculation inaccuracy of $Δ φ$ in Eq. (4). In contrast, the statistical estimation avoids such problems by characterizing the chirp as distributions instead of fixed parameters.

Fig. 2 OSNR sensitivity comparison between MLSE-TP and MLSE-TD (back-to-back measurement) (a) PAM-4; (b) PAM-8.

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4. MLSE in duobinary-CDM system

In NRZ-CDM system, the MLSE memory length is only determined by the differential phase, namely, 2 taps for this non-ISI pulse. However, when the channel contains other type of ISI, MLSE equalizer should adopt larger tap number to guarantee its superior decoding gain. In this section, we investigate a typical partial-ISI pulse – the duobinary pulse in CDM system. We use the delay-and-add method to generate the duobinary pulse, in order to guarantee a flat intensity transition within one symbol period [11]. After pulse shaping, a duobinary PAM-4 signal behaves like the NRZ PAM-7. The experiment setup of duobinary-CDM 10-Gbaud PAM-4 is similar to that in section 3, except for the single-polarization configuration.

The duobinary pulse induces 1 more tap of memory at transmitter. Combined with the CDM model, the duobinary-CDM has totally 3-tap memory. The transition now becomes ${χ_{t} | χ_{t} \underline{\underline{Δ}} (S_{t}, S_{t - 1}, S_{t - 2})}$ , and Eq. (2) should be revised as:

P (I_{t}, Δ φ_{t} | S_{t}, S_{t - 1}, S_{t - 2}) = P (I_{t} | S_{t}, S_{t - 1}) • P (Δ φ_{t} | S_{t}, S_{t - 1}, S_{t - 2})

It is noted that the intensity channel now has 2-tap memory due to the duobinary shaping. We can still pick up

I (t), I (t - 1)

and

φ (t)

to calculate the transition distance via Eq. (4), but

S_{t}

and

S_{t - 1}

need to be substituted to

{\bar{S}}_{t} = S_{t} + S_{t - 1}

and

\bar{S_{t - 1}} = S_{t - 1} + S_{t - 2}

every time when calculating

Δ φ_{t (e)}

.

Alternatively, we can perform the MLSE-TP by the statistical transition estimation. Figure 3 shows the measured distributions in the experiment of duobinary PAM-4 at the OSNR of 36 dB. The 16 intensity distributions $(I_{t} | S_{t}, S_{t - 1})$ forms 7 peaks in Fig. 3(a), well coincided with the expectation of the PAM-7 behavior. The phase distributions $(Δ φ_{t} | S_{t}, S_{t - 1}, S_{t - 2})$ contain 64 curves in Fig. 3(b). Figure 3(c) compares the OSNR sensitivity between MLSE-TD and MLSE-TP. At 20% FEC threshold, MLSE-TP presents 4 dB OSNR advantage over MLSE-TD, like the NRZ PAM-8 results shown in Fig. 2(b) (PAM-8 also has 64 $Δ φ$ distributions). For high-order modulation, MLSE-TP offers much higher decoding gain than MLSE-TD from an implementation perspective.

Fig. 3 Measured probability density function for the duobinary PAM-4: (a) intensity; (b) differential phase; (c) OSNR sensitivity comparison between MLSE-TP and MLSE-TD. (Back-to-back measurement).

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5. MLSE in Nyquist-CDM system

In Nyquist-CDM system where the spectrum is restrained inside the Nyquist bandwidth tightly, the chirp model in Eq. (4) becomes inaccurate. In this section, through the statistical transition estimation, we demonstrate the first 100 Gb/s Nyquist-CDM system with 10G-class DML [14], with the setup in Fig. 4. The baseband PAM-4 signal is Nyquist-pulse shaped by a raised cosine filter with roll-off of 0.01. The 25-GBaud signal is resampled to the AWG sampling-rate of 92 GSa/s (In practice the oversampling rate can be less than 1.5). 2 independent 1549.44-nm DFB lasers serve as the light sources of dual polarizations. Their frequency offset is less than 0.02 nm by adjusting the temperature. The 3-dB laser bandwidth is 16 GHz (12.5-GHz bandwidth would be sufficient for this 25-Gbaud signal). Their output spectra are illustrated in the inset (left) of Fig. 4. The 2 outputs are polarization combined and then launched into a recirculating loop consisting of 1-span 80-km standard single mode fiber (SSMF) and an EDFA to compensate the loop loss of 17 dB. The optimum fiber launch power is 2 dB, measured after 640-km fiber transmission [14]. At receiver, an ECL is applied as LO. The signal and LO are fed into a dual-polarization coherent receiver, whose outputs are sampled by a real-time oscilloscope at 80-GSa/s with 33-GHz bandwidth. The receiver offline DSP is shown by the inset (right) of Fig. 4. The polarization demultiplexing is performed by a 40-tap 2 × 2 adaptive equalizer using multi-modulus algorithm (MMA); intensity-only decision is immediately made after this for performance comparison.

Fig. 4 Experiment setup for the 25-Gbaud Nyquist-CDM PAM-4 system. Inset (left) optical spectra of the two lasers (“M” means modulation in the legend); (middle) transmitter DSP; (right) receiver DSP. DAC: digital-to-analog converter; DFB: distributed feedback laser; PBC: polarization beam combiner; SW: optical switch; OF: optical filter; ECL: external cavity laser; I/Q: in-phase/quadrature; x/y: X/Y polarization; TDS: time-domain oscilloscope.

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The measured distributions for the Nyquist signal are shown in Fig. 5 as reference. Figure 6(a) shows the back-to-back OSNR sensitivity. This 100-Gb/s CDM signal requires 24-dB OSNR at 7% hard-decision FEC (HD-FEC) and 19-dB OSNR at 20% soft-decision FEC (SD-FEC) using MLSE-TP. MLSE decoding achieves about 7 dB OSNR gain over the intensity-only hard decoding, a bit less than our previous 40-Gb/s system (9 dB) [9]. For high-baud-rate signal, the adiabatic chirp induced $Δ φ$ would decrease, as indicated via Eq. (2), leading to the reduction of the dynamic range of differential phase. In Fig. 5(b), the two $Δ φ$ extrema have less than 3.5-radian gap (only considering the peak values); in contrast, this gap is more than 6-radian in Fig. 1(b). The lack of adiabatic chirp severely degrades the MLSE decoding gain. Figure 6(b) shows the BER versus SSMF distance. The hard decoding achieves <400-km reachable distance for 20% SD-FEC, similar to the early 100-Gb/s PAM-4 DM-COHD experiment [15]. In contrast, using MLSE, the CDM 100-Gb/s signal is error free at 0 km; the BER reaches 7% HD-FEC threshold (4e-3) after 640 km SSMF, 20% HD-FEC (1.5e-2) after 1280 km, and 20% SD-FEC (2.4e-2) after 1600 km.

Fig. 5 Measured probability density function for Nyquist-pulse-shaped PAM-4.

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Fig. 6 100 Gb/s PAM-4 system. (a) Back-to-back OSNR sensitivity; (b) BER versus SSMF transmission distance. 7%-HD/20%-SD FEC threshold: 4e-3/2.4e-2; ‘Hard’ stands for intensity-only decision.

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6. Summary

The MLSE significantly increases the OSNR sensitivity of the DM coherent system through the CDM model. This paper proposes a statistical method for the MLSE transition probability estimation. The method is generalized to any transmitted pulse shaping, and further enhances the accuracy of CDM models, especially for higher-order modulation systems like PAM-8. The paper demonstrates the first 100-Gb/s CDM PAM-4 using only 10G-class DMLs, with a record electrical spectral efficiency of 8 bits/s/Hz and a record SSMF transmission distance of 1600 km for DM PAM transmitters. Using DM lasers with customized chirp parameters, the OSNR sensitivity advantage of the MLSE could be >10 dB compared with the intensity-only hard-decision. As such, high baud-rate CDM systems have great potential to be the ultra-low-cost solution for medium-reach optical transmissions.

References and links

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4. B. W. Hakki, “Evaluation of transmission characteristics of chirped DFB lasers in dispersive optical fiber,” J. Lightwave Technol. 10(7), 964–970 (1992). [CrossRef]

5. 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 (2013), paper OM3H.1. [CrossRef]

6. Y. Matsui, T. Pham, W. Ling, R. Schatz, G. Carey, H. Daghighian, T. Sudo, and C. Roxlo, “55-GHz bandwidth short-cavity distributed reflector laser and its application to 112-Gb/s PAM-4,” in Optical Fiber Communication Conference (2016), paper Th5B.4. [CrossRef]

7. Y. Matsui, D. Mahgerefteh, X. Zheng, C. Liao, Z. F. Fan, K. McCallion, and P. Tayebati, “Chirp-managed directly modulated laser,” IEEE Photonics Technol. Lett. 18(2), 385–387 (2006). [CrossRef]

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Procedure	Description
Linear ISI elimination	The dispersion compensation and adaptive channel equalization should be performed to eliminate the linear ISI. It guarantees the MLSE only deals with the channel memory induced by the differential phase.
Down-sampling	Signals should be down-sampled to the symbol rate before estimate the distributions.
Pattern recognition	As the training signals are known, their patterns should be recognized and categorized. In terms of PAM-4, there are 4 pattern bins for the intensity, and 16 bins for the differential phase.
Histogram generation	The measured $I_{t}$ are thrown to intensity bins based on their transmitted pattern $S_{t}$ ; $Δ φ_{t}$ are thrown to phase bins based on the pattern $(S_{t}, S_{t - 1})$ . Each bin generates a histogram uniquely associated with a MLSE transition in the trellis diagram.

Maximum likelihood sequence estimation for optical complex direct modulation

Abstract

1. Introduction

2. MLSE for CDM

3. MLSE in NRZ-CDM system

4. MLSE in duobinary-CDM system

5. MLSE in Nyquist-CDM system

6. Summary

References and links

Cited By

Figures (6)

Tables (1)

Equations (6)

Optics Express