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Journal of Electrical and Computer Engineering
Volume 2015, Article ID 736267, 12 pages
http://dx.doi.org/10.1155/2015/736267
Research Article

Maximum Likelihood Sequence Detection Receivers for Nonlinear Optical Channels

1Laboratorio de Comunicaciones Digitales, Universidad Nacional de Córdoba, CONICET, Avenida Vélez Sarsfield 1611, Córdoba X5016GCA, Argentina
2ClariPhy Communications, Inc., 7585 Irvine Center Drive, Suite 100, Irvine, CA 92618, USA

Received 25 July 2014; Revised 12 November 2014; Accepted 14 January 2015

Academic Editor: Martin Haardt

Copyright © 2015 Gabriel N. Maggio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The space-time whitened matched filter (ST-WMF) maximum likelihood sequence detection (MLSD) architecture has been recently proposed (Maggio et al., 2014). Its objective is reducing implementation complexity in transmissions over nonlinear dispersive channels. The ST-WMF-MLSD receiver (i) drastically reduces the number of states of the Viterbi decoder (VD) and (ii) offers a smooth trade-off between performance and complexity. In this work the ST-WMF-MLSD receiver is investigated in detail. We show that the space compression of the nonlinear channel is an instrumental property of the ST-WMF-MLSD which results in a major reduction of the implementation complexity in intensity modulation and direct detection (IM/DD) fiber optic systems. Moreover, we assess the performance of ST-WMF-MLSD in IM/DD optical systems with chromatic dispersion (CD) and polarization mode dispersion (PMD). Numerical results for a 10 Gb/s, 700 km, and IM/DD fiber-optic link with 50 ps differential group delay (DGD) show that the number of states of the VD in ST-WMF-MLSD can be reduced ~4 times compared to an oversampled MLSD. Finally, we analyze the impact of the imperfect channel estimation on the performance of the ST-WMF-MLSD. Our results show that the performance degradation caused by channel estimation inaccuracies is low and similar to that achieved by existing MLSD schemes (~0.2 dB).

1. Introduction

Maximum likelihood sequence detection (MLSD) receivers for nonlinear channels have been extensively investigated in the literature (e.g., [1, 2] and references therein). Their ability to achieve optimal performance in the presence of additive white Gaussian noise (AWGN) has always been of great theoretical and practical interest. The theoretical interest lies in that it provides a performance bound for any reception scheme. The practical interest stems from its ability to actually achieve optimal or nearly optimal performance in transmissions over nonlinear channels.

Much of the early work in this area has been focused on the compensation of nonlinearities in satellite communications [15]. A traditional architecture for the optimal nonlinear receiver consists of a matched-filter bank (MFB) followed by a maximum likelihood sequence detector (MLSD) [1]. Owing to the correlation among the spatial noise components at the MFB outputs, an MLSD with non-Euclidean metrics must be used. The use of oversampling in combination with MLSD (OS-MLSD) has also been proposed to implement the optimal receiver in the presence of nonlinearities (see [2] and references therein). Since the complexity of both MFB and OS MLSD-based receivers grows exponentially with the channel memory, their practical application in transmissions over highly dispersive channels has been limited. Despite this fact, MLSD-based receivers are still preferred over decision feedback equalizers (DFE) in applications such as multigigabit intensity modulation/direct detection (IM/DD) fiber optic systems for the two following reasons.(i)Performance: DFE suffers from severe performance limitations in moderate to high dispersion single-mode fiber links as a result of its inability to compensate nonlinear ISI [6, 7]. Although nonlinear DFE (NL-DFE) structures have also been considered in the literature (e.g., [8, 9]), their performances still degrade significantly at long fiber lengths (e.g., ≥500 km). On the other hand, MLSD can operate with a constant penalty around 3 dB with respect to back-to-back (B2B) at virtually any distance [10, 11].(ii)High-speed implementation: future generation of communication systems will operate at multigigabit per second data rates on highly dispersive channels [12]. In commercial applications, the digital receiver is often implemented as a monolithic chip in CMOS technology [12]. Maximum clock frequency of state-of-the-art complex digital signal processors in 28 nm CMOS technology is limited to frequencies lower than ~1 GHz. Therefore, in order to achieve multigigabit per second data rates, parallel processing techniques are required [12]. Although the complexity of serial implementations of DFE grows linearly with channel memory, all presently known parallel processing implementations require that the bottleneck created by the feedback loop be broken using techniques such as the ones proposed by [1315], whose complexity grows exponentially with the channel memory. Therefore, in high-speed applications, complexity of both DFE and MLSD increases with the channel memory in a similar way.

From the above it is clear that complexity reduction of MLSD is crucial for many practical applications. In IM/DD optical channels, the receiver must compensate the linear fiber dispersion as well as nonlinearities caused by lasers, optical modulators, the fiber Kerr effect, photo-detectors, and other components of the link. Chromatic dispersion (CD) and polarization-mode dispersion (PMD), in combination with the quadratic response of the photo-detector, are major factors that limit the reach and drive the complexity of optical-transmission systems at data rates ≥10 Gb/s [16, 17]. With traditional implementations based on oversampling techniques, an 8192-state MLSD is required to compensate 700 km of fiber at 10 Gb/s [10]. This is prohibitive in current CMOS technology1.

A new MLSD receiver architecture for nonlinear channels has been proposed in [18]. The major breakthrough of this proposal consists in a novel representation of the received signal obtained by a Gram-Schmidt-like orthogonalization of the kernels of a Volterra series expansion of the channel. This procedure yields a special form of space-time whitened matched filter (ST-WMF) [19] whose baud-rate-sampled outputs are sufficient statistics with independent noise components in both space and time. Combined with the minimum phase property of the response of each branch, the ST-WMF provides an effective way to reduce the complexity of MLSD in nonlinear channels. The ST-WMF MLSD technique offers a smooth trade-off between performance and complexity. As complexity is progressively reduced, performance degrades in a graceful manner. Numerical results in [18] demonstrate that the number of states of the VD in ST-WMF-MLSD required on a 10 Gb/s, 700 km, and IM/DD fiber-optic link without PMD can be reduced 8 times compared to an oversampled MLSD.

Further contributions on the ST-WMF-MLSD receiver and its performance on IM/DD optical links are provided in this work. First, the space channel compression achieved by the ST-WMF-MLSD is analyzed in detail. Space channel compression is important because it is the property that enables the major complexity reductions of the proposed architecture. Second, the performance evaluation of the ST-WMF-MLSD receiver is extended by addressing the combined effect of CD and PMD. Numerical results confirm that the ST-WMF-MLSD remains an attractive solution in the presence of these combined impairments. Finally, the impact of channel estimation inaccuracies on the ST-WMF-MLSD performance is assessed. Accuracy and speed of channel estimation are particularly important when tracking nonstationary channels. Nonstationarity in optical channels results from PMD and random rotations of the laser state of polarization. Our results show that the performance degradation caused by an imperfect channel estimation is low and similar to that achieved by existing MLSD schemes (~0.2 dB).

This paper is organized as follows. The nonlinear channel model and the ST-WMF-MLSD architecture are described in Sections 2 and 3, respectively. Performance evaluation of the ST-WMF-MLSD under different channel conditions, as well as its robustness in the presence of imperfect channel knowledge, are presented and discussed in Sections 4 and 5. Finally, conclusions are drawn in Section 6.

2. Nonlinear Channel Model

The noisy received signal is given by where is the noise-free signal and is the noise component, which is assumed to be a white Gaussian process with power spectral density . Component can be expressed in terms of its Volterra-series expansion [20, 21]. For example, neglecting the DC term of the expansion we get where is the linear kernel, with is the th second-order kernel, is the th symbol at the input of the nonlinear channel, is the symbol rate, and is the total number of kernels2.

2.1. Channel Model Orthogonalization

Next we present an alternative representation of the nonlinear signal [18]. Without loss of generality, we consider here the Volterra-series with a dominant linear kernel , and we select the first pivoting response as . Let be the signal space spanned by the set [19]. We assume that signal spaces are Hilbert spaces with inner product defined as , where superscript denotes complex conjugate. From the projection theorem, the nonlinear kernels can be uniquely expressed as where is orthogonal to the signal space ; that is, while is minimum [19], and denotes the set of all integers. We highlight that the first summation in (3) is the projection of onto . Define as the signal space spanned by . For and , from (4) note that ; therefore and are orthogonal signals [19]. Replacing (3) in (2) and operating, we obtain where with operator denoting convolution. Notice that and ; therefore signals and are orthogonal (see (4)). Next we focus on . Without loss of generality we select the second pivoting response as ; then (7) can be rewritten as Similarly to (5)–(7), can be expressed as where with chosen to satisfy Thus, note that is orthogonal to the signal spaces and , spanned by and , respectively. Repeating the processing on (11) and generalizing, we get where is the response of the th channel path and with given by with From (13) and (15) note that signal components from different paths are orthogonal; that is,

Figure 1 shows two multiple input-single output (MISO) representations of the nonlinear channel. Figure 1(a) shows the signal represented by the traditional Volterra model of (2); Figure 1(b) depicts the orthogonal representation given by (13) and (14).
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