Abstract:
Methods and systems for decoding a signal include compensating for impairments in a received signal using at least carrier phase estimation, where residual phase error remains after compensation; calculating symbol log-likelihood ratios (LLRs) for symbols in the compensated signal using Monte Carlo integration; demapping the symbols in the compensated signal using the symbol LLRs and extrinsic information from signal decoding to produce one or more estimated codewords; and decoding each estimated codeword with a decoder that generates a decoded codeword and extrinsic information.

Description:
RELATED APPLICATION INFORMATION 
     This application claims priority to provisional application Ser. No. 61/711,287 filed on Oct. 9, 2012, incorporated herein by reference. 
    
    
     BACKGROUND 
     1. Technical Field 
     The present invention relates to optical communications and, in particular, to optical communications using Monte Carlo based log likelihood functions for demodulation. 
     2. Description of the Related Art 
     A 100 Gb/s Ethernet standard has been approved and is already being implemented. This implementation is expected to accelerate in next few years. At these ultra-high data rates, the performance of fiber-optic communication systems is degraded significantly due to presence of various linear and nonlinear impairments. To deal with those channel impairments modulation and detection have been proposed to compensate. 
     For one such compensation technique, carrier phase estimation (CPE), the algorithmic DSP-based approaches are highly popular, and can be categorized into two broad categories: data-aided and non-data-aided. The maximum a posteriori approach is particularly efficient in CPE. However, the complexity of such algorithms grows exponentially with the channel memory. Even upon compensation of chromatic dispersion and nonlinearity phase compensation there will be some residual phase error. 
     SUMMARY 
     A method for includes compensating for impairments in a received signal using at least carrier phase estimation, wherein residual phase error remains after said compensation; calculating symbol log-likelihood ratios (LLRs) for symbols in the compensated signal using Monte Carlo integration; demapping the symbols in the compensated signal using the symbol LLRs and extrinsic information from signal decoding to produce one or more estimated codewords; and decoding each estimated codeword with a decoder that generates a decoded codeword and extrinsic information. 
     A receiver includes a compensation module configured to compensate for impairments in a received signal using at least carrier phase estimation, wherein residual phase error remains after said compensation; a symbol log-likelihood module configured to calculate symbol log-likelihood ratios (LLRs) for symbols in the compensated signal using Monte Carlo integration; a demapper configured to demap the symbols in the compensated signal using the symbol LLRs and extrinsic information from signal decoding to produce one or more estimated codewords; and one or more decoders, each configured to decode an estimated codeword and to generate extrinsic information that is fed back to the demapper. 
     These and other features and advantages will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The disclosure will provide details in the following description of preferred embodiments with reference to the following figures wherein: 
         FIG. 1  is a block diagram of an optical transmission/reception system that uses Monte Carlo methods to calculate symbol log-likelihood ratios (LLRs) in accordance with the present principles; 
         FIG. 2  is a block diagram of an optical transmitter in accordance with the present principles; 
         FIG. 3  is a block diagram of an optical reception system that uses Monte Carlo methods to calculate symbol log-likelihood ratios (LLRs) in accordance with the present principles; and 
         FIG. 4  is a block/flow diagram of a method of signal reception that uses Monte Carlo methods to calculate symbol log-likelihood ratios (LLRs) in accordance with the present principles. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     It has been experimentally verified that, even in beyond-100 Gb/s transmission for non-dispersion managed optical links, the distribution of samples upon compensation of linear and nonlinear impairments is still Gaussian-like with the residual phase error that can properly be modeled as a Markov process. The present principles provide demodulation that calculates symbol likelihoods according to a Monte Carlo method in the presence of residual phase error. 
     The present principles may be applied to conventional and optimized modulation schemes, 2-D and 4-D signaling schemes, to evaluate their efficiency. Optimized modulation schemes, when used in combination with LDPC coding, are more robust in the presence of phase error than conventional low-density parity check (LDPC) coded quadrature amplitude modulation (QAM). Moreover, LDPC-coded 4-D signaling schemes show much better robustness compared to 2-D coded modulation schemes. 
     Referring now to the drawings in which like numerals represent the same or similar elements and initially to  FIG. 1 , an optical communications system is shown that includes a transmitter  100  and a receiver  101 . The transmitter encodes a plurality of data signals at the encoder block  102  and then interleaves those signals at interleaving block  104 . The mapping block  106  then assigns bits of the interleaved signal to a modulation constellation, associating the bits of the interleaved data signals with the points on, e.g., a four-dimensional constellation. It should be noted that any appropriate modulation scheme may be used. 
     The transmitter  100  then sends the signal to receiver  101  over an optical medium  109 , which may include periodically deployed erbium doped fiber (EDF) amplifiers to maintain the signal strength. Other embodiments include the use of Raman and hybrid Raman/EDF amplifiers. Receiver  101  detects symbols in the constellation generated at block  108 . Upon coherent optical detection  110 , a backpropagation block  112  and equalization block  114  perform carrier phase estimation and, e.g., turbo equalization using a Monte Carlo log likelihood function to compensate for channel impairments such as polarization mode dispersion, chromatic dispersion, and fiber non-linearities. The signals are then de-interleaved and decoded at block  116  to produce the original data signals. 
     The encoders  102  and decoders  116  make use of LDPC codes to provide error correction that brings the transmissions close to the channel capacity. Every communications channel has a channel capacity, defined as the maximum information rate that the communication channel can carry within a given bandwidth. LDPC codes employ iterative belief propagation techniques that enable decoding in time that is proportional to their block length. 
     Referring now to  FIG. 2 , a detailed view of transmitter  100  is shown. m data signals feed into the transmitter  100 . The data streams are encoded at LDPC encoders  202  using different LDPC codes having code rates R=K/N, where K denotes the number of information symbols used in the binary LDPC code and N denotes the codeword length. 
     The m input bit streams from m different information sources pass through identical LDPC encoders  202  that use LDPC codes R. The outputs of the encoders  202  are then written row-wise into an m×n block interleaver  204 . A column of m bits is read out of the interleaver  204  and sent in one bit-stream, in bits at a time instant i, to a 4-D mapper  206 . 
     The 4-D mapper  206  maps each m bits into four-dimensional signal constellation points based on, e.g., a lookup table and produces four streams of symbol coordinates. The four streams represent, e.g., an in-phase and quadrature signal for each of two orthogonal optical polarizations. The mapper  206  assigns constellation points with the mapped coordinates from the mapper  206  being used as the inputs of a  4 -D modulator  208 . It should be understood that the mapper block  206  may include digital to analog conversion and pulse shaping functions. The 4-D modulator may be formed with, e.g., two electro-optical I/Q modulators, one per polarization. 
     A laser  210  produces a laser carrier beam that is split at polarization beam splitter  211  into two orthogonal polarizations. The modulator  208  converts the output of the mapper  206  into the optical domain by modulating the four symbol streams onto the orthogonally polarized carrier beams. The polarized beams are then combined in beam combiner  212  before being transmitted on an optical fiber. Because the combined beams occupy polarizations that are orthogonal with respect to one another, they can be combined without loss of information. 
     Referring now to  FIG. 3 , a detailed view of the receiver  101  is shown. A carrier beam is received from an optical fiber and is split at beam splitter  302  into two orthogonal polarizations. Coherent detectors  304  detect the beams to produce in-phase and quadrature signal estimates by sampling their respective signals. Although detectors  304  are advantageously implemented as coherent detectors, it is contemplated that other sorts of detector might be used. In embodiments that employ coherent detection, a local laser source (not shown) is used to provide the detectors  304  with a local reference that allows them to distinguish between the orthogonal polarizations and extract the information. 
     The in-phase and quadrature signals, for each polarization, produced by the detectors  304  are then passed to the equalization module  306 , where various channel impairments are corrected. In particular, equalizer  306  performs carrier phase estimation (CPE) and compensation of linear and nonlinear effects by, e.g., reduced-complexity digital backpropagation. The data stream is then passed to a symbol log likelihood ratio (LLR) module  308  that uses a Monte Carlo method, described in greater detail below, to find symbol LLRs. Symbol LLR information is used at demapper  310  to demodulate the signals and determine which constellation symbols are present. 
     The demapper  310  produces a set of bit LLRs, based on symbol LLRs and extrinsic information obtained from LDPC decoders  312 , and passes them to m LDPC decoders  312 . To improve bit error rate (BER) performance, extrinsic reliabilities are iterated between the demapper  310  and LDPC decoders  312  in, e.g., turbo equalization fashion until convergence or until a predetermined number of iterations has been reached. The LDPC decoders  312  then produce the reconstructed m data signals as output and feed-back extrinsic LLR information to the demapper  310 . 
     The equivalent channel model for coherent detection, upon compensation of linear and nonlinear impairments and CPE, can be represented as:
 
 r   k   =s ( a   k ,θ k )+ z   k ,
 
 r   k   =[r   k   (1)    . . . r   k   (i)   . . . r   k   (N) ] T ,
 
 s (a k ,θ k )= e   θ     k     [a   k   (1)   . . . a   k   (i)    . . . a   k   (N) ] T , and
 
 z   k   =[z   k   (1)    . . . z   k   (i)    . . . z   k   (N) ],
 
where r k   (i)  is the component of an observation vector at the k th  symbol interval, a k   (i)  is the i th  coordinate of the transmitted symbol at the k th  symbol interval, and z k  is the corresponding noise vector with a Gaussian-like distribution of components. θ k  denotes the residual phase error at the k th  time instance due to laser phase noise, nonlinear phase noise, and imperfect CPE. In polarization-division multiplexing (PDM), these equations apply to each polarization state. In 4-D signaling, the components above represent projections along in-phase and quadrature basis functions corresponding to the x- and y-polarizations. This model is applicable to few-mode fiber applications as well.
 
     For example, to describe the laser phase noise and imperfect CPE, the Wiener phase noise model can be used:
 
θ k =(θ k-1 +Δθ k )mod2π,
 
where Δθ k  is a zero-mean Gaussian process with variance σ Δθ   2 =2πΔfT s , with T s  denoting the symbol duration and Δƒ denoting either linewidth or frequency offset. The cyclic slips can also be modeled by a Markov-like process of certain memory. The probability density function (PDF) of the phase increment above is given as:
 
     
       
         
           
             
               
                 
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     where p(0,σ Δθ   2 ,Δθ k −n2π) denotes the Gaussian PDF of zero-mean, variance σ Δθ   2 , and argument Δθ k −n2π. The resulting noise process is Gaussian-like, with the power spectral density of N 0 , so that the corresponding conditional probability function is given by: 
     
       
         
           
             
               
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     For non-Gaussian channels, the method of histograms may be used instead to estimate the conditional probability density function p R (r|a k ,θ k ). 
     The likelihood function may be defined as: 
     
       
         
           
             
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     If the sequence of L=T/T S  statistically independent symbols, a=[a 1  . . . a L ] T , is transmitted, the corresponding likelihood function will be 
     
       
         
           
             
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     To avoid numerical overflow problems, the log-likelihood function should be used instead, producing:
 
 l ( a,θ )=log( L ( a,θ)).  
 
     A maximum likelihood approach would lead to exponential increase in complexity as sequence length L increased. Other potential approaches have included factor graphs, expectation maximization, and blind turbo equalization. According to the present principles, however, a Monte Carlo method is used. In particular, the log likelihood function is calculated using the following numerical integration:
 
 l ( a )=log(∫ . . . ∫ e   l(a,θ)   pΘ (θ) d θ).
 
     Instead of numerical integration, the present principles estimate the log likelihood function l as:
 
 l ( a )=log( E   θ ( e   l(a,θ) )),
 
where the expectation averaging E θ  is performed for different phase noise realizations. This is particularly simple for memoryless phase noise processes, Wiener phase noise process, and cyclic slip phase noise processes described as Markov processes of reasonable memory. Expectation averaging is performed by generating a phase noise sample by a Monte Carlo method, calculating log-likelihood functions, and by averaging the likelihood function with respect to different phase noise realizations.
 
     It can be shown that complexity of this method is O((m 2 +L)N r ), where m is the channel memory, L is the sequence length, and N r  is the number of phase noise realizations. Compared to the maximum likelihood method, which has a complexity of O(M L ), where M is the signal constellation size, the complexity of the present Monte Carlo method is significantly lower for long sequences. The Monte Carlo method uses the knowledge of Markov phase noise process, which can be characterized by training. In particular, for the Wiener phase noise process and the memoryless phase noise process, only the Gaussian noise generator is needed. 
     The present embodiments are directed specifically toward four-dimensional signaling, but it should be understood that other signaling schemes, such as few-mode fiber and PDM applications. 
     Referring now to  FIG. 4 , a method for decoding received signals in the presence of residual CPE and imperfectly compensated channel impairments. Block  402  receives a signal as described above. It is specifically contemplated that the received signal may be an optical signal, but it should be understood that the present principles apply with equal force to other forms of information transmission. Block  404  compensates for linear and non-linear impairments in the channel. In the case of optical transmission, such impairments may include polarization mode dispersion, chromatic dispersion, and non-linear properties of the transmission fiber. Block  406  performs carrier phase estimation using any appropriate method, whether data- or non-data-aided. Upon compensation of these impairments and the performance of carrier phase estimation, however, there will be some residual phase error. 
     To address the residual phase error, block  408  calculates symbol LLRs to aid in demapping. Instead of using the standard numerical integration of log likelihoods, block  408  uses a Monte Carlo method, calculating the log likelihood function of a symbol a as l(a)=log(E θ (e l(a,θ) )), where the expectation averaging E θ  is performed for different phase noise realizations and the function l(a,θ) is the logarithm of a likelihood function for the symbol a and the residual phase error θ. 
     Block  410  uses the calculated symbol LLRs to demap the symbols  410  with, e.g., a four-dimensional constellation. Block  410  generates bit LLRs from the symbols, which block  412  decodes using an LDPC code. Block  412  determines extrinsic decoding information associated with the LDPC decoding and iterates that information back to block  410  to be used in reducing future decoding/demapping errors. 
     Embodiments described herein may be entirely hardware, entirely software or including both hardware and software elements. In a preferred embodiment, the present invention is implemented in software, which includes but is not limited to firmware, resident software, microcode, etc. 
     Embodiments may include a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. A computer-usable or computer readable medium may include any apparatus that stores, communicates, propagates, or transports the program for use by or in connection with the instruction execution system, apparatus, or device. The medium can be magnetic, optical, electronic, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. The medium may include a computer-readable storage medium such as a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk, etc. 
     A data processing system suitable for storing and/or executing program code may include at least one processor coupled directly or indirectly to memory elements through a system bus. The memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code to reduce the number of times code is retrieved from bulk storage during execution. Input/output or I/O devices (including but not limited to keyboards, displays, pointing devices, etc.) may be coupled to the system either directly or through intervening I/O controllers. 
     Network adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks. Modems, cable modem and Ethernet cards are just a few of the currently available types of network adapters. 
     Having described preferred embodiments of a system and method for LDPC coded modulation for optical transport in the presence of phase noise (which are intended to be illustrative and not limiting), it is noted that modifications and variations can be made by persons skilled in the art in light of the above teachings. It is therefore to be understood that changes may be made in the particular embodiments disclosed which are within the scope of the invention as outlined by the appended claims. Having thus described aspects of the invention, with the details and particularity required by the patent laws, what is claimed and desired protected by Letters Patent is set forth in the appended claims.