Patent Publication Number: US-8983309-B2

Title: Constrained continuous phase modulation and demodulation in an optical communications system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This is the first application filed in respect of the present invention. 
     TECHNICAL FIELD 
     The present application relates generally to optical communications systems, and more specifically to constrained continuous phase modulation and demodulation in an optical communications system. 
     BACKGROUND 
     For the purposes of understanding the present disclosure, it is useful to consider a representation of the total optical E-field E(t) as a vector confined to a plane and emanating from a fixed origin, where the length of the vector gives the amplitude of the E-field at any instant (t), and the direction of the vector gives the phase of the field at any instant (t). Within this construction, we consider two basis sets. The first basis set is a Cartesian coordinate system centered on the E-field origin. In this Cartesian representation, the total E-field E(t) is decomposed along the orthogonal Real (Re) and Imaginary (Im), or, equivalently, In-phase (I) and Quadrature (Q), directions. The second basis set is a polar coordinate system, again sharing its origin with that of the E-field vector. In this polar representation, the E-field is decomposed into vector length (S) and phase angle (φ) relative to the Re-direction. These two basis sets are related by a non-linear transformation, in a manner well known in the art. In each of these representations, the time-sequence of loci of the end-point of the E-field vector may be referred to as a trajectory of the E-field. 
     The present disclosure discusses modulation formats and signals in which the state (e.g. amplitude and/or phase) of the signal at any instant depends on the state of the signal both before and after that instant. For example, the present disclosure discusses a Constrained Continuous Phase Modulation (C-CPM) scheme, in which the state of the modulated signal corresponding to a given symbol depends not only on the value of that symbol, but also on the values of the symbols that precede and follow it. Modulation formats and signals that display this characteristic may be referred to as having “memory”. 
     In the optical communications space, various techniques are used to synthesize an optical communications signal for transmission.  FIG. 1  illustrates a dual-polarization transmitter known in the art, which is designed to transmit two data signals on respective orthogonal polarizations of an optical wavelength channel. The laser  2  generates a narrow-band continuous wave (CW) optical carrier  4  having a desired wavelength. A beam splitter (or power divider)  6  separates the carrier  4  into a pair of linearly polarized CW lights, which are supplied to a respective E/O converter  8  which operates to modulate the amplitude and/or phase of the CW light to generate a respective polarization signal  10  based on one or more drive signals S(t) generated by a driver circuit  12  based on a respective input data signal x(t) and y(t). The two polarizations signals are then combined using a polarization combiner  14  to yield a polarization multiplexed optical communications signal  16  for transmission to a receiver. 
     In the arrangement illustrated in  FIG. 1 , each E/O converter  8  is provided a nested Mach-Zehnder (MZ) modulator known in the art. This arrangement enables the transmitter to utilize a variety of modulation and encoding schemes, to obtain high spectral efficiency and data transmission speeds. For example, transmitters using optical coherent modulation formats such as dual polarization multiplexed Quadrature Phase Shift Keying (DP-QPSK) and a nested Mach Zehnder E/O modulators constructed using Lithium Niobate (or Indium Phosphide) have recently enabled transmission rates exceeding 40 Gbps (Giga-bits-per-second) per optical wavelength channel over geographical distances of several hundred kilometers. A limitation of these transmitters, however, is that the optical components, particularly the E/O converter  8 , are expensive. 
     Techniques that enable high speed communications with low-cost optical components remain highly desirable. 
     SUMMARY 
     An aspect of the present invention provides a transmitter for use in an optical communications system, the transmitter includes: a digital signal processor for processing a data signal to generate a multi-bit digital sample stream encoding successive symbols in accordance with a constrained phase modulation scheme having an asymmetrical constellation of at least two symbols and in which a modulation phase is constrained to a phase range spanning less than 4π. A digital-to-analog converter for converting the multi-bit digital signal into a corresponding analog drive signal. A finite range phase modulator for modulating a phase of a continuous wavelength channel light in accordance with the analog drive signal, to generate a modulated channel light for transmission through the optical communications system. A corresponding receiver includes an optical stage for detecting phase and amplitude of a modulated channel light received through the optical communications system and generating a corresponding multi-bit digital sample stream, and a digital signal processor for processing the multi-bit digital signal to recover an estimate of each successive symbol of the modulated channel light. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features and advantages of the present invention will become apparent from the following detailed description, taken in combination with the appended drawings, in which: 
         FIG. 1  schematically illustrates principal elements and operations of an optical transmitter known in the art; 
         FIG. 2  illustrates a known quadrature phase modulation scheme in the Re/Im plane; 
         FIG. 3  illustrates a representative constrained continuous phase modulation scheme in accordance with an aspect of the present invention, in the Re/Im plane; 
         FIGS. 4A-4C  schematically illustrate principal elements and operations of constrained continuous phase modulation optical transmitters in accordance with aspects of the present invention; 
         FIGS. 5A and 5B  illustrate respective asymmetrical constellations usable in the transmitters of  FIGS. 4A and 4B ; 
         FIG. 6  is a block diagram schematically illustrating elements of a coherent optical receiver in accordance with aspects of the present invention; 
         FIG. 7  is a block diagram schematically illustrating elements and operations of a carrier recovery block usable in the receiver of  FIG. 6 ; 
         FIG. 8  schematically illustrates a representative window usable in the carrier recovery block of  FIG. 7 ; 
         FIG. 9  is a block diagram schematically illustrating elements and operations of a decoder block usable in the receiver of  FIG. 6 . 
     
    
    
     It will be noted that throughout the appended drawings, like features are identified by like reference numerals. 
     DETAILED DESCRIPTION 
     In very general terms, the present disclosure provides an optical communications system in which encoded symbols are modulated onto the optical carrier  6  using a constrained—continuous phase modulation (C-CPM) scheme. 
     Continuous Phase Modulation (CPM) is known in the field of wireless communications systems. It is well known that the phase evolution in CPM is not bounded. The phase modulation process is modelled as a tree whose outer envelope extends towards infinity over time [See Digital Communications, fourth Edition, John G. Proakis, page 189-190] 
       FIG. 2  illustrates a known quadrature phase modulation scheme, such as Quadrature Phase Shift Keying (QPSK), in the Re/Im plane. The illustrated modulation scheme comprises constellation of four phase states (A-D) symmetrically arranged about the origin, each of which corresponds with a symbol that encodes two bits of data. The optical signal modulation may conveniently be represented as a vector having a constant magnitude and a time-varying phase φ, which transitions between the various phase states in accordance with the modulated data. As is known in the art, the modulation phase φ can increase over time without bound, because any multiples of 4π implicit in the modulation phase φ will not affect detection at the receiver. As such, known quadrature phase modulation schemes also examples of unconstrained phase modulation, because there is no constraint on the maximum value of the modulation phase φ. 
     For the purposes of the present disclosure, constrained—continuous phase modulation (C-CPM) shall be understood to refer to a modulation scheme in which the phase of a continuous wave (CW) carrier can be modulated within a continuous phase range between predefined limits.  FIG. 3  illustrates a possible C-CPM, in the Re/Im plane. As in  FIG. 2 , the illustrated C-CPM scheme comprises constellation of four phase states (A-D) symmetrically arranged about the origin, each of which may correspond with a symbol that encodes two bits of data. The optical signal modulation may also be represented as a vector having a constant magnitude and a time-varying phase φ, which transitions between the various phase states in accordance with the modulated data. However, unlike conventional quadrature phase shift keying, the modulation phase φ is constrained to a phase range θ min  to θ max  spanning less than 4π. 
     In the embodiment of  FIG. 3 , a direct consequence of this arrangement is that a direct (shortest path) transition between phase states A and D (or, equivalently, the corresponding symbols) is not possible. Rather, a phase transition between the phase states (symbols) A and D requires “going the long way around”, via phase states B and C. In any physical phase modulator having a non-zero capacitance, this implies that the time, or power (or both), required to transition between any two states will depend on the phase separation between the involved states. Thus, for example, transitions between phase states (symbols) A and D will take more time, or consume more power (or both) than transitions between phase states A and C and B and D, both of which will take more time and/or consume more power than transitions between phase states A and B; B and C; or C and D. 
       FIGS. 4A and 4B  illustrate representative transmitters that may be used to generate a C-CPM optical signal. The transmitter of  FIG. 4A  is configured to encode and modulate an input signal x(t) onto the optical carrier  4  to generate a phase-modulated output signal Eo(t)  10  in the form or either a linearly of circularly polarized signal.  FIG. 4B  illustrates an alternative embodiment, in which respective input signals x(t) and y(t) are encoded and modulated onto respective orthogonal polarizations of the optical carrier  4 . The phase modulated polarizations are then combined to generate the output signal Eo(t)  16  in the form of a dual polarization multiplexed optical signal. In both of the embodiments of  FIGS. 4A and 4B , the E/O converters  8  are provided as voltage-driven optical phase modulators, which impose a phase shift on the respective carrier light in accordance with the applied drive voltage. 
     Voltage-driven phase modulators suitable for use in the transmitters of  FIGS. 4A and 4B  are well known and readily available commercially at comparatively low cost. For the purposes of the present application, two characteristics of such a phase modulator are important, namely: the phase can only be modulated within a finite phase range (that is, the phase range is “constrained”) which is limited by the finite voltage swing of the drive signal S(t); and, within this finite phase range, the phase modulation is continuous (in that it is not limited to a set of discrete phase values) and band limited. 
     The feature of continuous modulation is a product of the analog voltage response of the phase shifter, in that the imposed phase shift is proportional to the applied drive voltage, which can take any analog value within the capabilities of the drive circuitry. The feature of band limited is a function of the physical design of the phase modulator (for example, its capacitance), in that a non-zero amount of time is required to transition between any two discrete phase values, and the amount of transition time will tend to increase with increasing “distance” between the two phase states. 
     Taken together, the above-described characteristics mean that constrained-phase modulation is a non-linear modulation format in the sense that a given symbol modulated onto the carrier light cannot be decomposed to a single pulse shape amplitude modulated by data. By way of comparison, conventional optical modulation formats (such as, for example, OOK, QPSK etc.) are linear modulation formats in which each symbol of the constellation is decomposable into a single pulse shape (that is common to all symbols) with a respective amplitude modulation (either of optical intensity or phase) that represents the value of the symbol, and this fact can be exploited to derive an optimum demodulation method for use in the receiver to detect and recover the transmitted data. 
     Constrained phase modulation may be conceptualized as a phase trellis with a limited phase extent. The phase trellis can then later be exploited in the demodulation design. It may be noted that, in conventional continuous phase modulation (CPM) techniques, such as is known in Radio Frequency (RF) technology, the phase is unconstrained, and so can increase to infinity. In such a case, the phase trellis is represented as a tree which over time extends towards infinite phase extent. In direct contrast, in constrained phase modulation, the phase trellis is limited to a set of N nodes (or phase states). In some embodiments, N=2 m , where m is the number of bits encoded into each symbol. There then exists a unique mapping from one instance of the phase trellis to the next.  FIG. 4C  illustrates a representative mapping between two instances of a four-node trellis, which is applicable to the constellation of  FIG. 3 . In the example of  FIG. 4C , each phase node can be mapped to all of the N nodes in the next instance. As may be seen in  FIG. 4C , this implies that a phase node can be mapped to itself. Again this is another feature not entertained in the continuous phase modulation (CPM) literature, as this mapping arises out of the limited phase excursion constraint. 
     The phase trellis can then be mapped to a modulation method in the following way. A mapping from a phase node in a first instance of the trellis to a phase node in a second instance of the trellis is implemented by a phase waveform which starts and ends at the two sets of phase nodes. For the example of four phase nodes with the mapping described, there are therefore 16 unique phase waveforms between any two instances of the trellis. The phase waveform output by the encoder thus depends on the previous phase node and the new phase in the next instance of the trellis. In this way, the encoder adds memory to the modulation as well as ensuring that the modulation is phase continuous. In some embodiments, an encoded symbol may be represented by the combination of the previous phase node and the current phase waveform, or equivalently, the previous node and the new phase node. 
     As may be appreciated, there are many choices available for the phase waveform which traverses between the phase points. One choice is a phase waveform that changes linearly in time between the two phase points. 
     Those skilled in the art will be readily able to design various mappings between data bits (or encoded symbols) and phase nodes or waveforms. Accordingly, a detailed discussion of such mappings is not provided here. 
     Note that the transmitted pass band E-Field signal is in the Cartesian space, where the complex envelope is given by cos(φt))+j sin(φ(t)). In this Cartesian space, the 16 unique phase waveforms are mapped to sixteen unique complex Cartesian signals. 
     As may be appreciated, a constrained phase modulation scheme may be implemented by a driver  10  composed of any suitable combination of hardware and/or software. In some embodiments, the driver  10  may be constructed using a Digital Signal Processor (DSP) Application Specific Integrated Circuit (ASIC) with a high speed Digital-to-Analog Converter (DAC) that supplies the drive signal S(t) via an RF gain stage to an RF electrode of the optical phase modulator. It can be readily shown that a DSP ASIC with DAC can drive the required range of phase waveforms. 
     In the embodiment of  FIG. 3 , the constellation of phase states is symmetrically arranged in the Re/Im plane. However, the inventors of the present application have discovered that improved demodulation performance can be obtained by the use of an asymmetrical constellation.  FIG. 5A  illustrates one possible constellation, in which states A and D have been rotated to increase the phase distance to states B and C, respectively, and thereby provide asymmetry about the Im axis.  FIG. 5B  illustrates another possible constellation, in which all four states have been rotated in such a manner as to provide asymmetry about the Im axis, while also ensuring that the phase distance between any two adjacent states is approximately equal. 
     In all of the constellations illustrated in FIGS.  3  and  5 A-B, the upper and lower limits of the phase modulation range, θ min  and θ max , do not correspond with symbols. This is for simplicity of illustration only. In fact, the upper and lower bounds θ min  and θ max  may correspond with encoded symbols, if desired. The constellations illustrated in  FIGS. 5A-B  extend beyond 270°, which has been found to offer advantageous performance, but is not essential. In fact, the upper and lower bounds θ min  and θ max  of the phase modulation range will normally be determined by the capabilities of the drive circuit and the chosen phase shifter, and the symbol constellation may utilize all or part of the phase range, as desired. 
     All of the constellations illustrated in FIGS.  3  and  5 A-B comprise four symbols. However, this is not essential. More generally, a constellation of at least two symbols can be used. 
     As is known in the art, some commercially available phase shifters exhibit a non-linear phase response. If desired, this phase response may be compensated by predistorting the drive signals in a manner known in the art. 
       FIG. 6  illustrates principle elements of a receiver which may be used to receive and decode a constrained phase modulated signal of the type generated by the transmitter of  FIG. 4B . As may be seen in  FIG. 6 , an inbound dual-polarization multiplexed optical signal is received through an optical link  14 , split into orthogonal received polarizations by a Polarization Beam Splitter  16 , and then mixed with light from a Local Oscillator (LO)  18  by a conventional 90° optical hybrid  20 . The composite optical signals  22  emerging from the optical hybrid  20  are supplied to respective photodetectors  24 , which generate corresponding analog electrical signals  26 . The photodetector signals  26  are sampled by respective Analog-to-Digital (A/D) converters  28  to yield raw multi-bit digital signals  30  corresponding to In-phase (I) and Quadrature (Q) components of each of the received polarizations. 
     The resolution of the A/D converters  28  is a balance between performance and cost. It has been found that a resolution of n=5 or 6 bits provides satisfactory performance, at an acceptable cost. In some embodiments, the sample rate of the A/D converters  28  may be selected to satisfy the Nyquist criterion for the highest anticipated symbol rate of the received optical signal. However, this is not essential. 
     From the A/D converters  28 , the respective n-bit I and Q signals  30  of each received polarization are supplied to a respective dispersion compensator  32 , which operates on the raw digital signal(s)  30  to at least partially compensate chromatic dispersion of the received optical signal. If desired, the dispersion compensators  32  may be configured to operate as described in Applicant&#39;s U.S. Pat. No. 7,894,728, which issued Feb. 22, 2011. 
     The complex valued dispersion-compensated digital signals  34  appearing at the output of the dispersion compensators  32  are then supplied to a polarization compensator  36  which operates to compensate polarization impairments in the received optical signal, and thereby produce respective complex valued sample streams  38  corresponding to each transmitted polarization. These sample streams  38  contain both amplitude and phase information of each transmitted polarization, and include phase error due to the frequency offset between the Tx and LO frequencies, laser line width and phase noise. The sample streams  38  appearing at the output of the polarization compensator  36  are then supplied to a carrier recovery block  40  for detection and correction of phase errors. The phase corrected sample streams  42  output from the carrier recovery block  40  are then passed to a decoder block  44  which generates estimates of the symbols modulated on each transmitted polarization. These symbol estimates  46  are then passed to a Forward Error Correction block  48  for data recovery. 
     A specific challenge in designing the polarization compensator  36  is to equalize the optical channel when the modulation format is nonlinear and has memory, as is the case with C-CPM signals. One approach, which is known from unconstrained CPM in the wireless context, is to linearize the equalization by exploiting the fact that a CPM signal can be represented as a parallel set of N unique waveforms (each of which each is linearly modulated by a nonlinear alphabet). In this case, the equalization approach is to build N parallel full through-put equalizers, each of which processes one of the N waveforms. However, in the high speed optical coherent receiver of  FIG. 6 , this approach effectively increases the hardware complexity (by a factor of N) as compared to an equalizer designed for a linear modulation scheme. This additional complexity is cost-prohibitive. 
     In the illustrated embodiments, this problem is overcome by the use of a single equalizer implemented as a 2×2 Multiple-In-Multiple-Out (MIMO) channel equalizer, which uses the sequence of symbols modulated on each polarization as the target. As is known in the art, in a conventional operation of an 2×2 MIMO equalizer, equalization coefficients are calculated using a cost function which operates on the input to the 2×2 MIMO equalizer and an error signal derived from a predetermined target sequence. In some cases, this target sequence may correspond with a known symbol sequence inserted into an overhead of each transmitted polarization. The illustrated embodiment differs from this conventional technique in that the target is not a predetermined symbol sequence stored in a memory, for example, but rather is derived from the actual symbols modulated on each polarization. There are number of different cost functions that may be used, including zero forcing and various methods of Minimum Mean Square like Recursive Least Square or Least Mean Square. A preferred cost function is the Least Mean Square in terms of sufficient dynamic performance for minimum hardware. Note that since the target is the modulated sequence waveforms in each polarization, the MIMO Equalizer needs to operate on a block by block basis which contains sufficient portion of a sequence. In the illustrated embodiment the 2×2 MIMO channel equalizer is steered to a Minimum Mean Square Error (MMSE) solution by a respective set of coefficients computed by a feedback loop  50  comprising a comparator  52  and a coefficient calculator  54 . In general, the comparator  52  operates to compute an error signal between the signals  42  supplied to the data decoder  44 , and the symbol estimates  46  generated by the data decoders  44 . The coefficient calculator  54  then uses the complex valued dispersion-compensated digital signals  34  and the error signal to compute a set of coefficients  58  that, when uploaded to the polarization compensator  36 , minimizes the error computed by the comparator  52 . 
     The use of a 2×2 MIMO channel equalizer is 1/N less hardware complex than conventional methods. In some embodiments, the 2×2 equalizer may be provided as a block frequency domain equalizer, but a time domain equalizer is also possible. In some embodiments, the feedback loop  50  implements a Least Mean Squares (LMS) algorithm, for hardware efficiency. However, other methods may be used if desired, such as recursive least square or block matrix calculation of the MMSE filter. 
     The use of a sequence based 2×2 MIMO channel equalizer can cause two problems. The first problem is that the MIMO can color the noise which can degrade the performance of nonlinear sequence detectors in the data decoders  44 . The second problem is that the MIMO channel equalizer can add some additional memory into the nonlinear signal which further complicates and or degrades the sequence detectors. 
     In the illustrated embodiment, these problems are overcome by the use of a programmable target filter  60 , which filters the symbol estimates  46  in order to enable shaping the MIMO equalizer  36  in terms of whitening the noise and minimizing any additional memory being added to the signal. 
     In the illustrated embodiment, the symbol estimates  46  are used to address a look-up table (LUT)  62 , which outputs a sample stream that emulates the complex valued dispersion-compensated digital signals  34  input to the 2×2 MIMO. As such, the output of the LUT  62  is a reconstruction of the single input to the 2×2 MIMO. These sample streams are then supplied to a filter  64 . The preferred frequency response for this filter  64  is to provide additional gain for the high frequency content of the filter function that the feedback loop  50  converges to. This high frequency gain or ‘peaking’ helps to whiten the noise (e.g. makes the statistics of the noise independent from sample to sample) which yields a performance gain to the decoder block  44 . 
     As noted above,  FIGS. 4A and 4B  illustrate respective asymmetrical constellations that may be used in a C-CPM modulation scheme. One characteristic of these modulation schemes is that they have a non-zero mean. Other modulation schemes can also produce a signal with a non-zero mean. One issue with a non zero mean is that is can prevent convergence of convergence of the LMS algorithm to an MMSE solution. In particular, if the LMS error term generated by the comparator  52  has a non zero mean which then is multiplied with the input signal that has a non zero mean, then the product of these two non zero mean terms can prevent convergence of the LMS algorithm. 
     This problem may be overcome by removing the non-zero mean of either the LMS error signal or the dispersion compensated signals  34  supplied to the MIMO channel equalizer  36 . With this arrangement, if either teen in the product is zero, the bias from the non zero mean is removed. 
     There many techniques known in the art that may be used to remove the mean from either the LMS error signal or the dispersion compensated signals  34 . For example, a filter may be used to notch out the non zero mean content. In the case of the dispersion compensated signals  34 , such a filter may be implemented as part of either the dispersion compensators  32  or at the input of the 2×2 MIMO channel equalizer  36 . In the case of the LMS error signal, the filter ma be implemented as part of the coefficient calculator  54 . 
     Co-assigned U.S. Pat. No. 7,606,498, which issued Oct. 20, 2009, describes techniques for carrier recovery in a coherent optical receiver. In these techniques, a phase error Δφ is computed for each symbol estimate, and accumulated to over time to obtain a total phase rotation that is applied to each successive symbol estimate to compensate phase error due to the frequency offset between the Tx and LO frequencies, laser line width and phase noise. For linear modulation schemes (such as QPSK), these techniques are found to be very useful. However, in the case of constrained continuous phase modulation, the per-symbol phase error computation is prone to unacceptably high error. The challenge here is to track signal phase from large linewidth laser and/or cross phase modulation from co-propagating Intensity modulated signal while the signal of interest itself is non-linear and has memory. 
     In order to address this challenge, the carrier recovery block  40  may implement a sliding window Maximum Likelihood Sequence Estimator (MLSE) with a window-length corresponding to Ncr symbols, for determining a phase reference that may be used to estimate the phase error of each symbol.  FIG. 7  illustrates a representative carrier recovery block  40  for the complex-valued X-polarization sample stream X′(n). It will be recognised that the Y-polarization recovery block will be substantially identical. 
     In the embodiment of  FIG. 7 , the carrier recovery block  40  includes a phase detector  66  which processes complex-valued samples X′(n) received from the polarization compensator  36  to calculate a maximum likelihood phase error Δφ X , and a phase rotator  68  which operates to rotate the phase of the received samples X′(n) to generate corresponding rotated samples X′(n)e −jκ(n) . In general terms, the phase detector  66  includes a phase feedback loop  70  based on am Ncr baud sliding window Maximum Likelihood Sequence Estimate (MLSE) phase detector  72 , followed by an anti-causal phase Finite Impulse Response (FIR) filter  74 . The (MLSE) phase detector  72  computes a phase error estimate δφ X , and the anti-causal FIR filter  74  filters the phase error estimate δφ X  obtain the maximum likelihood phase error Δφ X . 
     A standard MLSE detector with Viterbi decoding, which uses a long trace back to decode a sequence, is too slow to be usable in the phase feedback loop  70 , because the time delay in the Viterbi decoder slows the tracking speed and so inhibits the ability of the phase feedback loop  70  to compensate phase transients due to laser line width, for example. The illustrated sliding window MLSE detector  72 , on the other hand, determines the most likely data symbol sequence in an Ncr baud window (encompassing Ncr+1 samples), with minimal processing delay. In a preferred implementation, a pair of Viterbi algorithms are used in parallel, namely: a forwards Viterbi algorithm which computes a most likely sequence starting at one end of the window; and a backwards Viterbi algorithm which computes a most likely sequence starting at the opposite end of the window. Each Viterbi algorithm processes approximately half of the window, with an overlap region in the middle of the window spanning one baud in the case of Ncr odd, or two baud in the case of Ncr even. This arrangement approximately halves the processing time since the forwards and backwards Viterbi algorithms can operate simultaneously. 
       FIG. 8  illustrates a representative window  76  spanning Ncr=5 baud and a symbol constellation of 4 symbols. In this case, each of the forwards and backwards Viterbi algorithms processes three baud, overlapping at baud  3 . For this 5-baud window  76 , there are 4 6 =4096 possible sequences. At each baud, a Viterbi algorithm determines the 4×4=16 branch metrics and then each node selects the path with the minimum Euclidean distance. Hence after the branch metric calculations there are only 4 paths that remain of interest for each Viterbi algorithm. This means that each Viterbi algorithm will calculate a respective set of four paths (sequences), which will overlap at baud  3 . Within this overlap region, there are 16 possible “links” between the four paths computed by the forwards Viterbi algorithm and the four paths computed by the backwards Viterbi algorithm. To determine the most likely sequence as measured from the middle of the window (baud  3  in  FIG. 8 ), these 16 possible links are examined, and the path with the minimum Euclidean distance is selected. 
     Once the most likely data symbol sequence has been determined, the phase difference φ1 between the most likely data symbol sequence and the received samples X′(n) can be readily determined. In some embodiments, the phase difference φ1 can be used as the phase error estimate δφ X , and supplied to the anti-causal FIR  74 . However, improved performance can be obtained by determining a second phase difference φ2, between the second most likely data symbol sequence and the received samples X′(n), and then calculate the phase error estimate δφ X  as an approximate Minimum Mean Square Estimate (MMSE) by using a weighted average of φ1 and φ2. The second most likely data symbol sequence can be determined by again examining the 16 possible links in the middle of the window and selecting the path with the second smallest Euclidean distance. For convenience, the Euclidean distance of the most likely data symbol sequence may be referred to as E1, and the Euclidean distance of the second most likely data symbol sequence may be referred to as E2. One way to implement this weighted average is to use an equation of the form: 
     
       
         
           
             
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     Where diff is proportional to (E2−E1), and may be used as an index into a look up table that outputs the value “scale”. In this case, the Sliding window MLSE phase detector  72  computes the values of φ1, φ2, E1 and E2, and a Phase loop filter block  78  implements the function 
     
       
         
           
             
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     The embodiment illustrated in  FIG. 7  also exploits the phase trellis structure of the C-CPM modulation format to achieve a further reduction in hardware size, and processing delay in the Viterbi decoding operation. A measure of the size of the hardware is the number of multiplications required per baud. For a dual polarization optical signal and an Ncr baud window, one would need 2×2×(4 Ncr ) multiplications to compute the 2×(4 Ncr ) Euclidean distances for the 2×(4 Ncr ) possible data sequences. Using the forwards and backwards Viterbi decoder described above, only 2×2×(16)=64 multiplications are required, even with an Ncr=5 baud window. However, it would be preferable not use any multiplications in the Euclidean distance calculations, since multiplications add to the loop delay and hence reduce the bandwidth of the phase feedback loop. 
     The main function of the phase loop filter  78  is to center the MLSE phase detector  72  in order to optimize the phase error estimate δφ X . If this phase reference is beyond the phase observation range of the MLSE phase detector  72 , then there will be corresponding degradation in the phase estimate from the MLSE phase detector. Given the bandwidth of the phase impairments of laser linewidth and XPM, fast tracking by the phase feedback loop  70  is required. 
     Accordingly, Euclidean distances may be computed in the phase domain only. With this arrangement, the Euclidean distance is computed between two vectors, where the amplitude of each vector is assumed to be the same. Under this assumption, the Euclidean distance is proportional to 1−cos(φ) where φ is the difference in phases between the two vectors. Thus, for example, the Euclidean distances E1 and E2 are proportional to 1−cos(φ1) and 1−cos(φ2), respectively. The function 1−cos(φ) can easily be implemented with a look up table. Also because of the symmetries in this function, it is possible to reduce φ to a small unsigned integer which may be used as an index into this table. Hence this table can be quite small. This method of computing Euclidean distance is dramatically smaller and faster than the 64 high resolution multiplications required by the method described above. An additional advantage of working in just the phase domain is that the rotation of the received data samples X′(n) is a simple subtraction of the phase estimate from the phase feedback loop. In the Cartesian domain, this rotation would require a high resolution complex multiplication which again would add time and size. 
     In the embodiment of  FIG. 7 , a phase detector  80  (such as, for example, a CORDIC converter) is used to obtain the respective phase φ of the Cartesian received data samples X′(n). This phase value θ is then subtracted (at  82 ) from the output of the Phase loop filter  78 , to obtain corresponding rotations that are supplied to the sliding window MLSE phase detector  72 , so that the phase feedback loop  70  can be implemented entirely in the phase domain. For the C-CPM modulation format, where the transmitted signal has close to a constant amplitude, implementation of the MLSE phase detector  72  in the just phase domain causes only a very small reduction in accuracy of the phase error estimate δφ X . 
     The second stage of this carrier recovery is to extract the phase error estimate δφ X  from the phase feedback loop  70  and then filter this unfiltered phase estimate δφ X  with an anti causal FIR  74 . Anti causal phase filtering has the benefit of using information from the past (causal) and the future (anti causal) to estimate present phase value. Typically the improvement in phase estimate with anti causal filtering is 3 dB. Note the phase feedback loop is just a causal filter. 
     There are many options for structuring the anti-causal FIR  74 . The key requirement is that the maximum likelihood phase error Δφ X  estimate is obtained from filtering both the past and future unfiltered phase estimates δφ X . One hardware efficient approach which eliminates multiplications is to set all of the FIR coefficients to one. In this case, only one multiplication is needed. It is known that there can be different amounts of correlation between XPM between the X polarization and Y polarization. This correlation can be exploited in the Carrier Recovery by linearly combining the estimates from X and Y polarization. One method to do this combining is to add a low resolution 2×2 multiply  84  at the output of the anti-causal FIR  74 . 
       FIG. 9  schematically illustrates a representative decoder block  44  for computing an estimated value for each symbol modulated on the X-polarization at the transmitter. In the embodiment of  FIG. 9 , the decoder block  44  is configured as a sliding window Maximum Likelihood Sequence Estimator (MLSE)  86  with a window-length corresponding to N baud (N+1 symbols). In some embodiments, N&gt;Ncr. In the illustrated embodiment, a respective shift register  88  is provided for receiving the phase rotated sample stream  42  from the carrier recovery block  40 . As the sample stream  42  is latched through the shift registers  88 , the MLSE block  86  may implement forward and reverse Viterbi algorithms in a manner directly analogous to those described above with reference to  FIGS. 7 and 8 , in order to compute a maximum likelihood estimate of the symbol X(n) corresponding to the sample(s) at the center of the window. 
     The embodiments described above are intended to be illustrative only. The scope of the invention is therefore intended to be limited solely by the scope of the appended claims.