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Maximum likelihood sequence estimation (MLSE) offers effective equalizations for bandwidth-limited optical signal on mitigation towards inter-symbol-interference (ISI) impairment. In this paper, we provide the first comprehensive comparisons and analysis of three post-compensation algorithms on the same modeling platform for high spectral-efficiency (SE) optical systems employing the spectral prefiltering. Those algorithms include 1-tap constant modulus algorithm (CMA) and 3-tap MLSE, regular CMA and digital filter with 2-tap MLSE, and constant multi-modulus algorithm (CMMA) with 2-tap MLSE. Furthermore, a novel and effective approach is proposed and verified for the generation of both hard value and soft value at the output of MLSE in order to be compatible with the implementation of soft-decision forward error correction (SD-FEC) decoding process.
© 2014 Optical Society of America
1. Introduction
Bandwidth demand of Internet and private line service continues to grow at around 30% per year driven by more and more video streaming and proliferation of cloud computing, big data, social media, and mobile data delivery. This trend combined with the requirement of cost reduction per bit per Hz directly leads to the need of higher spectral efficiency (SE) thus larger capacity of underlying optical transmission systems [1, 2]. Recent progress in digital coherent technologies has opened new horizons for achieving ultra-high SE and higher data rate per channel in long-haul and metropolitan optical links [3, 4]. At present, the 100-Gb/s long-haul systems, whether in development or in deployment, are all based on polarization division multiplexed quadrature phase shift keying (PDM-QPSK) modulation format associated with coherent detection and digital signal processing (DSP) [5]. The achieved SE is 2 bits/s/Hz over conventional 50-GHz optical grid and thus the system capacity has been around 10 Tb/s in fiber C-band transmission window. The optical system with SE beyond 2 bits/s/Hz is now the next step for the optical industry [6]. Besides multi-level modulation or orthogonal multiplexing approaches [7, 8], pulse shaping or narrow prefilterings have also been demonstrated to be another effective way for further improving SE by reaching super-Nyquist bandwidth, where the channel spacing is set to be smaller than the baud rate (Nyquist bandwidth) [9–11]. To achieve the optimum demodulation, a multi-symbol detection scheme that bases its decisions on observation of a sequence of received signals over successive signal intervals is needed. The reason is that such transmitted signal has memory which is introduced by inter-symbol-interference (ISI) or joint ISI and inter-channel-interference (ICI) symbol correlation. Several effective compensation algorithms through the use of multi-symbol detection such as maximum likelihood sequence estimation (MLSE) or maximum a posteriori (MAP) have been developed to counter these impairments in such systems [10, 12, 13]. Compared with high computational complexity of MAP, MLSE shows the unique advantages in terms of performance and efficiency through the search of the minimum Euclidean distance path with trellis of Viterbi algorithm (VA). MLSE uses selective search to approach the performance of exhaustive search of MAP on minimum Euclidean distance [13, 14]. Additionally, some newly designed algorithms with modification of regular constant modulus algorithm (CMA) demonstrate much simpler and more robust capabilities when combined with simplified MLSE algorithms [10, 12]. A series of experiments were demonstrated to investigate transmission performance of high SE WDM systems through tight prefiltering by means of the algorithms with MLSE [12–18]. However, there are no reported results making fair performance comparison among these algorithms to show their principle correlations, ultimate capacity, and spectrum-shaping limitation as well as application scenarios.
In this paper, we provide a comprehensive performance comparison of three typical post-compensation algorithms on the same modeling platform. They include 1-tap CMA and 3-tap MLSE, multi-tap CMA and post digital filter with 2-tap MLSE, and constant multi-modulus algorithm (CMMA) with 2-tap MLSE. The number of tap in MLSE structure refers to the number of neighboring symbols that are considered for the calculation of Euclidean distance. We present the principles and system implementation technologies for PDM-QPSK modulation format and also show that the schemes based on 2-tap MLSE with either digital filter or CMMA process exhibit unique advantages in terms of tolerance of both pure ISI and joint ISI and ICI impairments. To be compatible with channel decoding and take advantage of the benefit of soft-decision forward error correction (SD FEC), we also propose a new approach to generate soft-value output of 2-tap MLSE and verify the effectiveness with 20% turbo-product code (TPC) FEC. As much as 0.6 dB sensitivity benefit is gained from the developed soft-value input to FEC compared with hard-value one.
2. Spectral prefiltering
To enhance SE, it is well known that pulse shaping is necessary to maximize the data rate while minimizing pulses interference to avoid or mitigate the ISI impairment in digital communication systems. Basically, there are three types of pulse shaping filters, sinc, raised-cosine, and Gaussian filters.
A sinc-shaped filter is an ideal filter in the sense that its rectangular spectrum requires minimum Nyquist channel bandwidth without ISI. However, it is not practical because it is infinitely extended in time domain. A raised-cosine filter and its associated root-raised-cosine filter (matched filter version) smoothly approach the frequency stop band. These are practical implementations of ISI-free spectral prefiltering with a controlled rolloff factor for the tradeoff between the number of taps of impulse response and bandwidth occupancy. This raised-cosine filter is basically adopted to reduce channel crosstalk and/or increase the tolerance toward narrow filtering effect with limited achievable SE and without the need of post compensation. A Gaussian filter is another commonly used filter with a smooth transfer function and no zero crossings. One of the most attractive features of Gaussian filter is to significantly reduce the side lobes of signal spectrum and its common availability of real products. To further increase SE, this filter is used to perform spectral prefiltering to lead to super-Nyquist situation, where the channel spacing is smaller than signal occupancy. In the mean time, the introduced severe ISI and/or the ICI can be post compensated with effective algorithms. In this work, we primarily focus on post-compensation algorithms and their performance with pure Gaussian filter or combination with a raised-cosine filter.
One of the implementations of such prefiltered systems is based on regular optical filter, such as wavelength selective switch (WSS), as shown in Fig. 1(a). The optical filter can also perform the optical multiplexing function simultaneously. The filtered spectrum can be smaller than the original signal symbol rate. Another realization is based on a digital-to-analogue converter (DAC) at the transmitter in digital field as shown in Fig. 1(b). In-phase and quadrature-phase components of an optical carrier are modulated in a single IQ modulator driven by two independent electric pulse-shaped signals. Examples of 128-Gb/s PDM-QPSK signals filtered by 3rd-order digital and optical Gaussian filter are also shown in Fig. 1. These two technologies potentially reach the same sensitivity performance with the ideal filter shape.
Fig. 1 Spectral prefiltering in (a) optical and (b) digital domains, O-MUX: optical multiplexing, WSS: wavelength selective switch.
3. Post-compensation schemes
To show the process mechanisms and compare the performance limits of different schemes, a numerical simulation model is firstly set up to emulate various operation situations. Figure 2 shows the system setup of the DSP units and electro and optical components at transmitter (Tx) and receiver (Rx) sides. The tunable optical filter (TOF) is utilized to achieve tight prefiltering modulation at Tx. Basic Rx DSP includes 2x sampling from analog-to-digital converter (ADC), deskew and orthogonality correction, chromatic dispersion (CD) compensation and clock recovery. We then focus on three compensation schemes, Post1: 1-tap CMA and 3-tap MLSE; Post 2: digital filter and 2-tap MLSE, and Post3: 9QAM-like CMMA and 2-tap MLSE, all are followed with frequency offset compensation and carrier phase recovery. All the results in the paper are based on 32G baud rate. The 3-dB bandwidth of the TOF is changed from the range of 22 to 28 GHz for different operation conditions.
Fig. 2 System setup. CW: continuous wave, PBS/C: polarization beam splitter/combiner, LO: local oscillator, O/E: optical to electrical conversion, CD: chromatic dispersion, TOF: tunable optical filter, CMA1: 1-tap CMA, CMA2: multi-tap CMA.
The impairments of modulator distortion, frequency offset of 1 GHz, 100-kHz linewidth of Tx laser and LO are induced, errors are counted over 1 million bits, and OSNR is set at 0.1-nm resolution. The tap number in the multi-tap CMA and CMMA is 17 for all simulation results.
3.1 Scheme I: 1-tap CMA and 3-tap MLSE
Figure 3 depicts the detailed structure of this scheme. After the basic DSP modules (as shown in Fig. 2), two steps are adopted. Step one uses the training sequence to generate the ISI induced pattern look-up table through multiple averaging to filter out noise impact at high optical-to-noise ratio (OSNR) situation. Single-tap CMA is employed only to perform the function of polarization demultiplexing while preserving the symbol correlations that are used by the subsequent 3-tap MLSE to boost the performance. The constellations of training look-up table of a 32-Gbaud PDM-NRZ-QPSK signal at different prefiltering bandwidths are shown in Fig. 3. Considering 3 neighboring symbols of PDM-NRZ-QPSK format, the total of 64 elements is in the training table. Moreover, multi-tap CMA can be used firstly to gain more accurate carrier frequency offset and phase estimation that are used for single-tap process, which is shown in Fig. 3 as dotted curve. In the step two, 3-tap MLSE is used for multi-symbol decoding under the case of preserved symbol correlations in aforementioned 1-tap CMA processing. The number of taps in MLSE is selected here for the purpose of making relatively fair complexity comparison with the following 2-tap MLSE schemes. The number of taps needs to be expanded in the conditions of either stronger spectral shaping or the move for higher-order modulation formats. Note that only this scheme requires the generation of ISI pattern through training sequence because of the process of symbol correlations preservation. As shown in Fig. 4, one linear delay-and-add digital filter is used to achieve partial response and simultaneously mitigate the enhanced noise and linear crosstalk in multi-channel
3.2 Scheme II: Digital filter and 2-tap MLSE
condition [16]. MLSE algorithm is still employed to realize symbol decoding and optimal detection but with the reduction of memory length to 2 taps for this digital filter induced memory. The filtering function is performed after the carrier phase estimation in the conventional DSP flow of coherent receiver, which is the major feature of this scheme. Therefore, this scheme is compatible with conventional 100G demodulation through bypassing both the filter and subsequent MLSE in the end of DSP flow in the case of no strong filtering effect. The transfer function of the digital filter is shown in Fig. 4(a). It is noted that the second-tap coefficient of the filter is adjustable for optimization of the overall performance [19]. After the filtering, the enhanced noise and ICI are suppressed. From the constellation point of view, the digital filter enables the change from 4-point QPSK to 9-point quadrature duobinary or 9QAM-like signals. The evolution of the transformation is illustrated in Fig. 4(b). As a result of the delay-and-add effect, the 2-ASK in-phase and quadrature components disappear and independently change into two 3-ASK symbol series. The generation mechanism of 9QAM-like signals can be considered as superposition of two 3-ASK vectors on a complex plane. The size of constellation points represents the relative number of points generated after the digital filter.
3.3 Scheme III: CMMA and 2-tap MLSE
The major DSP flow is shown in Fig. 5 for Scheme III. Multi-tap CMA is replaced with a multi-tap CMMA. The CMMA was recently proposed in optical transmission experiment for PDM-8QAM and PDM-16QAM modulation formats [3, 11]. As the constellations of training table shown in Fig. 3, the filtered signal does not maintain the single constant modulus any more, which is the same situation as 8QAM or 16QAM modulation formats. As shown in Fig. 5(a), multiple reference circles and the corresponding formulas are introduced in cascaded way to minimize the final error coefficient resulting from the stochastic gradient algorithm. Instead of converging to single modulus in conventional CMA, this CMMA leads to converge into three rings for the prefiltered PDM-QPSK signal. The major feature of the algorithm is that one of the rings equals to zero here. Similarly, carrier phase recovery stage is adapted by means of similar concept of QPSK partitioning mechanism. The symbols of two outer rings are used for carrier phase estimation. It is noted that the middle ring needs to be rotated by π/4 and combined with symbols on outermost ring as shown in Fig. 5(b). In this way, the 4th power frequency offset compensation and phase recovery algorithms are the same as conventional PDM-QPSK DSP modules. The maximum likelihood (ML) algorithm can be added to the second stage of phase recovery in order to provide some further improvement. It is also noted that the classic single-ring CMA can be used at the starting stage for pre-convergence with small number of data, and then the system is switched to the CMMA with the inherited filter coefficients at the steady stage.
4. Performance comparison
Figure 6 shows the signal constellation evolutions before and after polarization demultiplexing, after carrier recovery, and before the MLSE decoding at the OSNR of 30 dB with prefiltering bandwidth of 25 GHz of 3rd-order Gaussian optical filter. The foremost thousand data points are used for illustration purpose in Fig. 6.
Single-tap CMA only performs the function of polarization demultiplexing and is not capable of providing matched filter function. It is clearly seen in the 3-tap MLSE case that converged ring is much broader than the other two cases because of the existence of unequalized ISI impairment. In this 1-tap CMA case, symbol correlations induced by energy exchange through ISI are preserved after carrier recovery for the subsequent 3-tap MLSE process. In the second scheme, DSP flow is compatible with the conventional PDM-QPSK until the delay-and-add digital filter turns the four-point constellations into 9-point constellations right before MLSE input. Different from the above two cases, three rings are converged at CMMA case and one ring is at the origin. The 9-point constellation is shaped right after carrier recovery without the need of digital filter. It is also seen that the converged rings are much cleaner compared with single-tap CMA case. The reason is that linear equalizer of multi-tap CMA or CMMA equalizes ISI distortion by prefiltering through the matched filter response.
The scenario of pure ISI impairment is mainly for the compensation of bandwidth limitation from devices or filter narrowing effect of individual channel. Figures 7(a)–7(c) show the effectiveness of these algorithms in single-channel optical spectral prefiltering case. These 3rd-order optical filters for the TOF in Fig. 2 are selected in order to emulate the commonly used optical devices or the impairment of cascade narrowing effect.
Fig. 7 BER curves at (a) 22-GHz, (b) 25-GHz, and (c) 28-GHz 3rd-order Gaussian filter in single-channel case.
In Fig. 7(a), the improvement as much as 5 dB at BER = 1E-2 is obtained from digital filter and CMMA post-compensation algorithms when compared with no post-compensation system. Only 1-dB benefit is obtained from 1-tap CMA, which is because the memory length of 3-tap MLSE is not long enough when more neighboring symbols are involved in the correlations in such tight prefiltering process. The addition of memory length or symbol states from 43 = 64 of 3-tap MLSE to 45 = 1024 of 5-tap MLSE or even higher can further improve the performance of 1-tap CMA scheme, however, the complexity of this algorithm increases exponentially which prohibits the real implementations. In the relaxed prefiltering cases, the benefit difference among these algorithms decreases, as small as 0.25 dB at BER = 1E-2 is observed in the case of 25-GHz prefiltering as shown in Fig. 7(b). Another noticeable feature in Fig. 7(a) is that worse performance occurs at very low OSNR condition for 1-tap CMA scheme becausethe more noisy situation maintains though the whole DSP flow. Figure 7(a) also shows that the scheme based on CMMA and 2-tap MLSE demonstrates the noticeable performance advantage over the digital filter and 2-tap MLSE scheme. The reason lies in two facts. The first one is that fully adaptive finite impulse response (FIR) filters with three modulus (especially one of modulus is zero) in CMMA scheme is compared with fixed post digital filter and one modulus. The second one is that the carrier recovery unit (especially phase recovery) in the scheme of CMMA is in less noisy condition than digital filter one.
It is also valuable to mention that the 2-tap MLSE schemes bring penalty when the filter bandwidth is larger than 28 GHz in comparison with no post-compensation process as shown in Fig. 7(c). The reason is that both the CMMA and digital filter shows worse performance compared with conventional multi-tap CMA in terms of balancing the equalization of ISI impairment and enhanced high-frequency noise components. Also shown in Fig. 7(c), the scheme of 3-tap MLSE provides 0.25-dB OSNR improvement at BER = 1E-2, which demonstrates that this scheme is more suitable for the relaxed filtering cases.
The scenario of joint ISI and ICI impairments is mainly for the compensation of both ISI and ICI impairments of WDM channels or multiplexed superchannels. As shown in Fig. 8, three PDM-QPSK channels with 3rd-order 25-GHz prefiltering are emulated with 28-GHz channel spacing. As much as 1-dB OSNR penalty appears at BER = 1E-3 for 1-tap CMA scheme because of severe ICI impairment brought by the crosstalk of high-frequency components. The other two schemes have less than 0.3-dB penalty, which exhibits much better ICI tolerance capability.
Figure 9 shows the comparison of ICI tolerance capability between two post-compensation schemes for 22-GHz filtered signal and Nyquist pulse shaping with a rolloff factor of 0.01. The selection of 22 GHz of 3-dB bandwidth is to shape the optical spectrum for 100-GHz grid occupancy of 400G data rate per superchannel. Thus the SE can be increased to 4 bits/s/Hz. As shown in Fig. 9, Q-value improvement as much as 2.2-dB at 28-GHz spacing is achieved. It is seen that the SE of 4 bits/s/Hz is only achievable for the post-compensation schemes. It is also observed that the performance of Nyquist pulse shaping drops quickly as channel spacing narrows down with more severe ICI impairment. The optical spectra are also inserted in Fig. 9 as insets.
Fig. 9 Performance comparison vs. channel spacing between Nyquist pulse shaping and post-compensation schemes.
5. Soft-decision MLSE for soft-decision FEC
As verified above, MLSE is an effective approach on compensating ISI or joint ISI and ICI which can be a severe impairment in high-SE systems with strong filtering effect. On the other hand, soft-decision forward error correction (SD FEC) is a powerful channel encoding/decoding technique and has established their indispensable roles in high-SE and ultra-long-haul optical transport to further increase the receiver sensitivity or lower required optical signal to noise ratio (ROSNR) [20]. A SD FEC decoder achieves its best performance by taking as input multiple bit “soft” information that represents a confidence level or reliability of the received data (i.e., whether a bit is very likely one, likely one, likely zero, or very likely zero). To achieve both the receiver sensitivity and SE goals by implementing both SD-FEC decoding and MLSE, we devise an algorithm that generates soft values in addition to the conventional hard decision values at the output of MLSE. The soft MLSE decisions are fed to the input of SD FEC decoder to improve decoding performance. The reliability calculation in the MLSE process is based on the calculated path metrics with maximum probability criterion in the Viterbi algorithm.
Figure 10 depicts the calculation of soft values in MLSE trellis structure. The data used in this illustration example is generated in the context of a spectral prefiltering for the 128-Gb/s PDM-QPSK signals with digital filter and 2-tap MLSE processing at 20dB OSNR. The example structure has a memory length of 2, alphabet set of {-1, 1} and, thus, 2 states and 4 possible transitions between two consecutive states. Each trellis branch indicates a possible state transition labeled with a calculated path metric. The path metric represents the likelihood of the corresponding state transition. In the example, a smaller value of the path metric represents a higher likelihood of the corresponding state transition. The MLSE process is to find the surviving path of a giving trellis that corresponds to the sequence estimation with maximum likelihood. The soft value in this proposed approach is calculated base on the 4 branch metrics (D1, D2, D3, D4) between each two consecutive states as
After finding a surviving path, the corresponding states along the surviving path are the MLSE hard decisions. Combining the hard decisions and the corresponding soft values, we can generate MLSE soft decisions asNote that convergence state, in which the surviving paths are converged to the same trellis path, needs to be evaluated before exporting the soft value to the output.As an example of 3rd-order 25-GHz bandwidth constrained 128-Gb/s PDM-QPSK signal, Fig. 11 shows complex constellations before and after digital filter and soft-decision MLSE recovery. It is clearly seen that the number of points at the neighboring regions of each quadrant is much less compared with the one without soft process of MLSE. It is also observed that the statistical distribution is changed into non-Gausssian distribution.
A TPC SD-FEC code is used to verify the effectiveness of the proposed approach of generating soft values from MLSE. The code rate is 0.8308 and the length of coded bits is 110592. Half million data points are used to investigate the water-fall region of FEC decoding algorithm. Figure 12(a) shows the BER performance comparisons of soft and hard output of MLSE with SD FEC as well as the one without both MLSE and FEC in the case of spectral prefiltering with 3rd-order 22-GHz optical Gaussian filter. It can be clearly seen that around 0.6-dB OSNR improvement has been achieved with the use of soft value that is generated from the proposed algorithm as the input to the SD FEC. It is also seen that the system employing both soft output of MLSE and SD FEC provides the best BER performance and enables the possibility of implementations in such spectral-narrowing condition.
Fig. 12 BER performances of soft and hard output of MLSE w/ SD FEC as a function of OSNR. Spectral prefiltering with (a) 3rd-order 22-GHz optical Gaussian filter; (b) 3rd-order 25-GHz optical Gaussian filter.
Note that the scheme of digital filter and 2-tap MLSE is used in the modeling here. As shown in Fig. 12(b) with the bandwidth of prefiltering relaxed to 25 GHz, it is observed that as much as 0.45-dB OSNR benefit is obtained in this relatively weaker ISI impairment. Note that the output signal samples of a MLSE module as shown in Fig. 11(b) have a non-Gaussian statistic distribution. Therefore, it is expected that an optimal FEC performance can be potentially achieved through a statistic-adaptive soft-decision FEC for such non-Gaussian statistical distribution [21].
6. Conclusions
We discussed super-Nyquist pulse generation in optical or digital domains through tight prefiltering for high-SE coherent optical transmission systems. On the same modeling platform, we compared three compensation schemes of the bandwidth-constraint signals through different kinds of multi-symbol decoding mechanisms to take advantage of prefiltering induced symbol correlations. Our results show that the schemes based on the 2-tap MLSE with either a digital filter or CMMA process exhibit some advantages in terms of tolerance of both pure ISI and joint ISI and ICI impairments in the context of single-channel or WDM systems. We verified that as much as 0.25 dB advantage at BER = 1E-2 is gained from the 2-tap MLSE algorithms at 25-GHz prefiltering case when compared with 3-tap MLSE scheme. For the even narrower spectral shaping condition, the scheme of 3-tap MLSE provides much limited capability. Therefore, larger number of memory taps is needed with the cost of exponentially increased complexity for this approach. In the application scenario with tighter spectral shaping (like less than 25 GHz for 32-Gbaud signal), the schemes of 2-tap MLSE are preferable with the facts that the digital-filter based algorithm offers seamless compatibility with conventional PDM-QPSK DSP module while the CMMA based one shows a little better performance advantage.
Meanwhile, to take advantage of the both benefits from SD FEC and MLSE in high-SE systems employing spectral prefiltering, we also propose a new approach to generate soft-value output of 2-tap MLSE and verify the effectiveness with 20% TPC SD FEC code. As much as 0.6-dB OSNR benefit is gained from the developed soft-value input to FEC compared with hard-value one at the condition of 22-GHz optical spectral shaping. The proposed approach demonstrates practical feasibility of the high-SE WDM transmission system that employs both MLSE and SD FEC.
Acknowledgments
This work was partially supported by China 863 project under grant numbers 2012AA011303 and 2013AA010501.
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