1. Introduction

With the homodyne optical receiver employing phase and polarization diversities, we can detect in-phase and quadrature (IQ) components of an optical electric field of each orthogonal polarization mode [1]. In coherent optical communication systems using such a receiver, a signal electric field can be transmitted as a four-dimensional (4D) vector given by

We can increase the dimension of modulation higher than four using time/frequency slots. For example, when the block containing n time slots is arbitrarily coded, the dimension of modulation is increased up to 4n. The well-designed constellation in the 4n-dimensional vector space can improve the power efficiency and/or the spectral efficiency toward the Shannon limit [2]. In [3, 4] it is shown that the 24-dimensional modulation format using the extended Golay code can achieve 3-dB coding gain at the bit-error rate (BER) =10⁻³ against the dual-polarization quadrature phase-shift keying (DP-QPSK) format. However, as the number of the dimension increases in order to further improve the BER performance, the computational complexity for the maximum likelihood estimation (MLE) of symbols increases exponentially. Therefore, to save the computational cost, we usually employ the forward error-correction code (FEC) for n time slots and decode the symbol by algorithms without relying upon MLE. Thus, it might be a good choice that we optimally design the 4D modulation format using the IQ components and the two polarization modes without wasting the frequency resource and enhance the BER performance with the conventional n-dimensional FEC after that. In addition, 4D optical modulation and demodulation technologies have already been put into practical use by recent developments of optical IQ modulators and digital coherent receivers. For these reasons, this paper concentrates on the BER-performance improvement of the 4D optical modulation format.

Several 4D optical modulation formats have been studied so far. For example, Karlsson and Agrell showed that polarization-switched quadrature phase-shift keying (PS-QPSK) was most power efficient among all of the 4D modulation formats [5, 6]. This is the biorthogonal modulation format, where three bits are transmitted per symbol by using the following eight 4D vectors:

Another type of 4D optical modulation formats has been obtained by set-partitioning the constellation of 4D M-ary quadrature-amplitude-modulation (4D-MQAM) formats [7]. It should be noted that M means the number of modulation levels in the conventional two-dimensional IQ plane and is given as M = 2²ⁿ (n = 1, 2, 3, ···); hence, the number of constellation points in the 4D space is M². Constellation points of a given 4D-MQAM format are partitioned into subsets such that the minimum squared Euclidean distance (MSED) between constellation points in subsets is doubled, resulting in the improvement of the power efficiency at the expense of spectral-efficiency reduction. The PS-QPSK format mentioned above can also be generated by set-partitioning the 4D-4QAM (i.e., 4D-QPSK) constellation. More importantly, the 4D-16QAM format having 256 constellation points in the 4D vector space is partitioned into two subsets of 128-ary set-partitioned (SP) 4D-16QAM. From each subset, the succeeding set-partitioning process generates four subsets of 32-ary SP-4D-16QAM. These SP-4D-MQAM formats have good trade-off between the power efficiency and the spectral efficiency; therefore, their experimental demonstrations and numerical simulations have recently been done extensively [8, 9].

However, as Ungerboeck originally proposed [10], the SP technique can achieve significant coding gain especially when it is combined with the trellis-coded modulation (TCM) scheme. In such a TCM scheme, we use convolutionally-coded bits to select one of the partitioned sets, and the error probability of the coded bits is determined from the free distance on the trellis diagram. On the other hand, uncoded bits choose one of the constellation points in each subset, and the error probability of the uncoded bits is determined from the minimum distance between constellation points in the subsets, which is also recognized as the distance between parallel paths on the trellis diagram. Generally, such parallel paths degrade the overall performance of the TCM system since the distance between parallel paths is often smaller than the free distance. Meanwhile, to eliminate parallel paths, we have to increase the number of states of the convolutional encoder, which results in the exponential increase in computational complexity for Viterbi decoding. Thus, we need to design the optimized TCM system, considering both of the free distance and the parallel path on the trellis diagram.

The TCM scheme with 4D-MQAM constellations for radio communications was proposed and investigated intensively in the 1980s [11–13]. In [12, 13] an eight-state trellis encoder is used for convolutional coding. In the eight-state TCM with 4D-MQAM constellations, number of input bits is 2log₂ M − 1 [bit]. An eight-state convolutional encoder generates three output bits from two input bits. Through double-stage set-partitioning of 4D-MQAM constellations, we have eight subsets having M²/8 constellation points each. The three output bits from the encoder select one of the eight subsets, while uncoded input bits (2log₂ M − 3 [bit]) choose one constellation point in the selected subset. This modulation format has the advantage that the free distance on the trellis diagram is equal to the minimum distance in the partitioned subset. Owing to such a feature inherent to set-partitioning of 4D-MQAM constellations, the TCM enlarges the coding gain effectively because any number of uncoded bits does not limit the overall TCM performance.

In optical communication systems, on the other hand, several experimental demonstrations of TCM have been reported so far [14–16]; however, these works were based on 2D multilevel modulation formats. In this paper, we introduce TCM with 4D QAM constellations into optical communication systems for the first time to our knowledge. More specifically, extending our previous study [17], this paper fully describes the principle of the eight-state optical TCM scheme having 4D-MQAM constellations in a well-organized form. Compared with the SP-4D-MQAM format, the TCM with 4D-MQAM constellations can achieve 3-dB asymptotic coding gain while maintaining the same spectral efficiency. Such theoretical predictions are confirmed through computer simulations on eight-state TCM with constellations of 4D-QPSK and 4D-16QAM. Especially, the eight-state TCM with 4D-QPSK constellations is practically important because of its simple encoder structure and relatively low computational cost for Viterbi decoding. Through BER measurements in the back-to-back state and after 50-km optical fiber transmission, we experimentally demonstrate that its coding gain is 3 dB as high as that of DP-QPSK at BER=10⁻³.

The organization of the paper is as follows: In Sec. 2, we describe the configuration and the principle of operation of the eight-state optical TCM scheme with 4D-MQAM constellations. In Sec. 3, the performance of the TCM is evaluated in terms of the asymptotic power efficiency and the spectral efficiency. Section 4 provides computer simulation results on eight-state TCM with 4D-QPSK and 4D-16QAM constellations. In Sec. 5, we conduct BER measurements of eight-state TCM with 4D-QPSK constellations and verify the significant coding gain against DP-QPSK. Finally, we conclude this paper in Sec. 6.

2. Principle of eight-state trellis-coded modulation with 4D-MQAM constellations

2.1. Set-partitioning of 4D-MQAM constellations

Figure 1 shows the set-partitioning process of the 4D-MQAM constellation. The initial 4D-MQAM constellation has M² points in the 4D vector space, and MSED between constellation points is denoted as $d_0^2$. First-step set-partitioning of the 4D-MQAM constellation generates two subsets, R₀ and R₁. Each subset has M²/2 constellation points, and MSED between them is extended to $2d_0^2$. Next, the second-step set-partitioning process generates eight subsets, S₀, ···, S₇, having M²/8 constellation points whose MSED is $4d_0^2$.

 

Fig. 1 Set-partitioning process for 4D-MQAM constellations. First-step partitioning generates subsets R₀ and R₁, and second-step partitioning does subsets S₀, ···, S₇. Each set-partitioning process doubles MSED between constellation points.

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As an example, we try set-partitioning of the 4D-QPSK constellation. Initial sixteen 4D vectors in the 4D-QPSK format are given as

2.2. Trellis-coded modulation

The encoder structure for eight-state TCM is shown in Fig. 2, where three boxes represent shift registers and ⊕ means the XOR operation. The total number of input bits is I_in = 2log₂ M − 1 [bit]. Two bits out of the input bits are fed into the convolutional encoder with the constraint length K = 4 and the code rate R = 2/3, which is surrounded by broken lines in Fig. 2. Then, we obtain three output bits b₁, b₂, and b₃ including one parity bit. On the other hand, other input bits (I_u = 2log₂ M − 3 [bit]) are remained uncoded. Finally, the mapper assigns a constellation point out of M² points according to the output bits b₁, b₂, b₃, b₄, ··· (I_out = 2log₂ M [bit]).

 

Fig. 2 Encoder structure for eight-state TCM. Bits b₁, b₂, and b₃ are coded by the convolutional process and other input bits b₄, ··· are not coded.

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To perform the optimized constellation mapping, we consider the free distance of the trellis code. Figure 3 shows the trellis diagram of the code generated by the convolutional encoder shown in Fig. 2. Dots in Fig. 3 stand for eight possible states corresponding to logical levels of the three shift registers. Three output bits (b₁, b₂, b₃) are shown on each path. The closest two paths are shown by red lines, and the distance between these paths determines the free distance of the sequence.

 

Fig. 3 Trellis diagram of the code generated by the convolutional encoder shown in Fig. 2. Dots represent the shift-register state of the convolutional encoder. The closest two paths are shown in red lines. Three output bits b₁, b₂, b₃ are shown on each path. The number of selected subsets is also shown on each path.

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Table 1 shows the relation between the output bits (b₁, b₂, b₃) and the assigned subsets S₀, ···, S₇. The number indicated on each path of the trellis diagram given by Fig. 3 represents the selected subset. For example, when the shift-register state changes from 000 to 001, output bits are (001), which in turn select the subset S₁. The subset assignment given in Table 1 can maximize the free distance of the sequence. In fact, the free distance between the closest paths in Fig. 3 can be calculated as

Table 1. Assignment of three output bits to the partitioned subset.

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However, each selected subset still has multiple constellation points; thus, we need to choose one constellation point in the subset using uncoded bits. These uncoded bits cause parallel paths on the trellis diagram. As shown in Fig. 1, MSED between parallel paths is given as

It should be stressed that especially in the case of eight-state TCM with 4D-QPSK constellations, we can perform Viterbi decoding with realistic computational cost, because the number of parallel paths is only two. Together with its simple encoder structure at the transmitter, this scheme seems practically important.

3. Asymptotic power efficiency and spectral efficiency

The power efficiency γ of a modulation format is defined as

Meanwhile, the spectral efficiency SE per polarization [bit/s/Hz/pol] is given as

The power efficiency and the spectral efficiency of eight-state TCM with 4D-MQAM constellations can be calculated as

 

Fig. 4 SE as a function of 1/γ for DP-MQAM, M²/2-SP-QAM and eight-state TCM with 4D-MQAM constellations. Two-dimensional QAM orders are M = 4, 16, and 64.

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Instead of reducing the spectral efficiency by 0.5 [bit/s/Hz/pol], M²/2-SP-QAM becomes more power efficient than DP-MQAM. Furthermore, eight-state TCM with 4D-MQAM constellations is more power efficient than M²/2-SP-QAM by 3 dB without sacrificing the spectral efficiency. In the case of eight-state TCM with 4D-QPSK constellations, improvements of the power efficiency against DP-QPSK and PS-QPSK are 4.8 dB and 3 dB, respectively. On the other hand, in the case of eight-state TCM with 4D-16QAM constellations, improvements of the power efficiency against DP-16QAM and 128-SP-QAM are 5.4 dB and 3 dB, respectively.

4. Simulation results

To verify the performance of eight-state TCM with 4D-MQAM constellations, we conduct computer simulations. When we take computational cost for Viterbi decoding at the receiver into account, the TCM system with M higher than 16 is not realistic; thus, we only focus on modulation formats for M = 4 and 16.

In the case of eight-state TCM with 4D-QPSK constellations, we prepare three random binary sequences. From two bit sequences, three coded bit sequences are generated by using the encoder shown in Fig. 2, whereas one bit sequence is remained uncoded. Thus, we have four bit sequences. In accordance with Table 1, one subset including two 4D vectors (see Eq. (3)) is selected by the three coded bits, and the remaining one uncoded bit determines one of the two 4D vectors in the selected subset. Using the 4D vector thus determined, we modulate the signal complex amplitude four-dimensionally and add white Gaussian noise to control the ratio of energy per bit to noise spectral density E_b/N₀. In the case of eight-state TCM with 4D-16QAM constellations, since each subset has 32 constellation points corresponding to five uncoded bits, coding and mapping are more complicated but can be done in a straightforward manner.

At a digital coherent receiver, we use the Viterbi decoding algorithm with soft decision. The Euclidean distance between measured and target 4D vectors is used as a metric. Then, the decoded bits are compared with the transmitted bits and BERs are calculated.

Figure 5 shows BERs calculated as a function of E_b/N₀. BERs of eight-state TCM with 4D-QPSK constellations are shown in Fig. 5(a). BERs of uncoded DP-QPSK and uncoded PS-QPSK are also plotted for comparison. We evaluate the coding gain at BER=10⁻³, because the typical FEC code having 7 % overhead assures error-free operation at the uncorrected BER ≤ 10⁻³. We find that the coding gain against DP-QPSK is about 3 dB, and it approaches the asymptotic power efficiency of 4.8 dB as BER decreases. On the other hand, the coding gain against PS-QPSK is about 2 dB, and it approaches the asymptotic value of 3 dB as BER decreases. Figure 5(b) shows the case of eight-state TCM with 4D-16QAM constellations. For comparison, BERs of uncoded DP-16QAM and uncoded 128-SP-QAM are plotted. At BER=10⁻³, coding gains against DP-16QAM and 128-SP-QAM are 2.5 dB and 1.5 dB, respectively. We find that these are much smaller than asymptotic values (5.4 dB and 3 dB as shown in Sec. 3) because of slow asymptotic behaviors of BERs for higher-order QAMs.

 

Fig. 5 BERs as a function of E_b/N₀. (a): Eight-state TCM with 4D-QPSK constellations, uncoded DP-QPSK, and uncoded PS-QPSK. (b): Eight-state TCM with 4D-16QAM constellations, uncoded DP-16QAM, and uncoded 128-SP-QAM.

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5. Experimental demonstration of eight-state TCM with 4D-QPSK constellations

5.1. Experimental setup

We experimentally evaluate BER characteristics of eight-state TCM with 4D-QPSK constellations at 10 Gbaud for the back-to-back state and after 50-km optical fiber transmission. We also measure BERs of DP-QPSK for comparison, using the symbol pattern and the baud rate same as those of TCM, so that the improvement of the receiver sensitivity with TCM can accurately be estimated.

The experimental setup is shown in Fig. 6(a). A distributed-feedback laser diode (DFB-LD) was used as a transmitter. A CW light was split into two polarization components by a polarization beam splitter (PBS). These two components were modulated by two IQ modulators (IQMs). To generate optical 4D signals, a common external clock synchronized two arbitrary waveform generators (AWGs) and an external trigger started them simultaneously. Each AWG drove the IQM and generated the 10-Gbaud QPSK signal for each polarization component. Then, two polarization components were combined by a polarization beam combiner (PBC).

 

Fig. 6 Experimental set up for the BER measurement of eight-state TCM with 4D-QPSK constellations. (a): Experimental system configuration. (b): DSP at the transmitter. (c): DSP at the receiver.

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Figure 6(b) shows the block diagram of digital signal processing (DSP) for generating eight-state TCM with 4D-QPSK constellations. Random three bit sequences were fed into the encoder and one parity bit was appended. Following the constellation mapping procedure discussed in Sec. 4, we determined a 4D vector in AWGs.

The optical signal was transmitted through a 50-km-long standard single-mode fiber (SMF). The launched optical power was set to 0 dBm. At the receiver, the optical power was controlled by a variable optical attenuator (VOA) to measure BERs as a function of the average power before the pre-amplifier. The optical signal was pre-amplified with an erbium-doped fiber amplifier (EDFA) and detected by a homodyne receiver comprising phase and polarization diversities. The local oscillator (LO) was a DFB-LD, whose performance was almost same as that of the signal laser. Outputs from the receiver were sampled at 50 GS/s by analog-to-digital converters (ADCs) embedded in a digital oscilloscope, which had 8-bit resolution and a 16-GHz analog bandwidth.

The digitized signals were sent to the offline DSP circuit shown in Fig. 6(c). The signals were down-sampled by 2/5 and equalized with half-symbol-spaced adaptive finite-impulse-response (FIR) filters in the butterfly configuration. To avoid the polarization singularity problem, the decision-driven least-mean-square (DD-LMS) algorithm with the training mode was used in the initial tap-adaptation procedure [18]. After tap coefficients were converged, the adaptation algorithm was switched to the constant modulus algorithm (CMA) [19]. After the signal equalization, we performed carrier phase estimation using the 4-th power algorithm, which was aided by the pilot sequence to eliminate the phase ambiguity of π/2. Finally, decoding was done by the Viterbi algorithm. The decoded bits were compared with the transmitted bits and bit errors were counted.

5.2. Experimental results

Figure 7 shows simulation and experimental results of BERs as a function of E_b/N₀. Curves, dots, and triangles represent simulation results, back-to-back BERs, BERs measured after optical transmission through a 50-km-long SMF, respectively. BERs for eight-state TCM with 4D-QPSK constellations are shown in the red color, whereas those for DP-QPSK are shown in the black color. The simulation results are the same as shown in Sec. 4.

 

Fig. 7 BERs of eight-state TCM with 4D-QPSK constellations (red color) and DP-QPSK (black color) at 10 Gbaud as a function of E_b/N₀. Curves, dots, and triangles represent simulation results, back-to-back BERs, and BERs after 50-km transmission, respectively.

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In this experiment, we measured BERs as a function of the received optical power before the pre-amplifier; however, we convert the received power into E_b/N₀, taking the signal bit rates (i.e., 30 Gbit/s for TCM and 40 Gbit/s for DP-QPSK), the noise figure of the pre-amplifier, and the bandwidth of the signals into account. Hence, the coding gain of TCM against DP-QPSK is directly read from Fig. 7.

For both of TCM and DP-QPSK, experimental BER curves after 50-km transmission have 1-dB power penalty at BER=10⁻³ from back-to-back BER results. This power penalty may be due to the imperfect compensation for group-velocity dispersion (GVD) of the 50-km-long SMF.

In addition, the back-to-back BERs have also power penalties from the simulation results for TCM and DP-QPSK. These penalties may stem from imperfect bias points set at IQMs at the transmitter and IQ imbalance at the receiver. However, we can experimentally confirm that at BER=10⁻³, the eight-state TCM offers about 3-dB coding gain against DP-QPSK both in the back-to-back case and after 50-km transmission. Such an experimental coding gain is in good agreement with the simulation result.

6. Conclusions

We have discussed eight-state TCM with 4D-MQAM constellations and shown that the free distance on the trellis diagram is the same as the minimum distance between constellation points in the partitioned subset. This fact enables us to avoid coding-gain reduction due to parallel paths on the trellis diagram. Compared with SP-4D-QAM, the eight-state TCM with 4D-MQAM constellations can achieve the asymptotic improvement of the power efficiency by 3 dB without sacrificing the spectral efficiency.

We have applied such a TCM scheme to optical communication systems for the first time. From the point of view of the computational cost, we have focused on eight-state TCM with 4D-QPSK and 4D-16QAM constellations. BER performances of these modulation formats are analyzed by computer simulations. At BER=10⁻³, coding gains of eight-state TCM with 4D-QPSK constellations are 3 dB and 2 dB against DP-QPSK and PS-QPSK, respectively. On the other hand, coding gains of eight-state TCM with 4D-16QAM constellations are 2.5 dB and 1.5 dB against 16QAM and 128-SP-QAM, respectively.

We have experimentally evaluated BER characteristics of eight-state TCM with 4D-QPSK constellations in the back-to-back case and after 50-km optical fiber transmission. Comparing the BER performance of TCM with that of DP-QPSK, we confirm that at BER=10⁻³, the coding gain of TCM against DP-QPSK is about 3 dB, which is in good agreement with the computer simulation result.