Patent Publication Number: US-2015071395-A1

Title: Complexity reduced feed forward carrier recovery methods for m-qam modulation formats

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of, and claims priority to, co-pending U.S. application Ser. No. 12/968,454, filed Dec. 15, 2010, which is incorporated by reference herein in its entirety. 
    
    
     FIELD 
     The present disclosure relates generally to optical communications, and more particularly to using the carrier phase angles recovered from the coarse phase recovery stage to estimate and remove carrier frequency offset for a carrier system. 
     BRIEF DESCRIPTION OF THE RELATED ART 
     In order to meet growing capacity demands in core optical networks, spectrally efficient techniques, such as digital coherent detection, have attracted recent attention. These techniques allow the use of advanced modulation formats; especially M-ary quadrature amplitude modulation (QAM) modulated systems. However, one major challenge in implementing high-performance coherent detection is in accurate phase and frequency offset recovery, which is caused by intrinsic laser phase noise and signal-local oscillator frequency offset. As a result, for high-order M-QAM modulation formats (where M&gt;4), tolerance to laser phase noise decreases as the modulation level increases, because the Euclidean distance decreases (Yu, X. Zhou and J., “Multi-level, Multi-dimensional Coding for High-Speed and High Spectral-Efficiency Optical Transmission.” to be published in the August issue of J. Light wave Technology, 2009). In particular, while frequency offset is relatively slow-changing, phase drift caused by laser phase noise (characterized by laser linewidth) is fast-changing. Given the small tolerance of high-order M-QAM systems to phase and frequency noise, the quality of phase tracking significantly influences performance of the communication system. 
     Presently, there are three published carrier phase recovery schemes. The first is a decision-directed digital feedback loop (Irshaad Fatadin, David Ives, Seb J. Savory., “Compensation of Frequency Offset for Differentially encoded 16- and 64-QAM in the presence of laser phase noise.”  IEEE Photonics Technology Letters. Feb.  1, 2010, p. 2010; H. Louchet, K. Kuzmin, and A. Richter.,  Improved DSP algorithms for coherent  16- QAM transmission,  Brussels, Belgium: Tu.1.E.6, 2008. Proc. ECOC&#39;08. pp. Sep. 21-25, 2008; A. Tarighat, R. Hsu, A. Sayed, and B. Jalali.,  Digital adaptive phase noise reduction in coherent optical links,  J. Lightw. Technol., vol. 24, no. 3, March 2006, pp. 1269-1276). Since this method relies on negative feedback, its performance depends heavily on the ability of previous samples to be relatively current, which places demands on the sampling frequency. This is especially a problem in parallel and pipeline architectures, in which sampling is both sparse and delayed. 
     The second method uses a classic feed-forward phase correction technique based on an Mth-power Viterbi-Viterbi algorithm, in which the phase quadrant information is deliberately removed to calculate phase error (Seimetz, M., “Laser linewidth limitations for optical systems with high-order modulation employing feed forward digital carrier phase estimation.” San Diego, Calif.: OTuM2, Feb. 24-28, 2008. Proc. OFC/NFOEC). However, this method can only be applied to certain constellation points having equal phase spacing, and therefore only a small subset of incoming signals can be used—this again reduces the linewidth tolerance of the system. 
     A third method proposes using a blind exhaustive phase search to find phase error based on the phase distance to the nearest constellation point, for a collection of points (T. Pfau, S. Hoffmann and R. Noé.,  Hardware - Efficient Coherent Digital Receiver Concept With Feed forward Carrier Recovery for M - QAM Constellations,  Journal of Lightwave Technology, Vol. 27, No. 8, Apr. 15, 2009). While this method is both feed-forward and high-performing, it requires high complexity to process a large collection of points simultaneously. In addition, because of the need to process in parallel, each group of computations required to process the collection of points must be repeated for each parallel branch. Therefore, though this method is high-performing, it is not feasible to implement. 
     SUMMARY 
     The present disclosure includes a method to use a decision-directed digital phase lock loop (PLL), which may be implemented with parallel and pipeline architecture, for coarse phase recovery, and one or more feed forward maximum likelihood (ML) estimators to fine-tune the estimate. Additionally, the use of a weighted ML phase estimator for improved performance is contemplated. For the case with carrier frequency offset between the transmitter laser and the local oscillator laser, the present disclosure proposes to use the carrier phase angles recovered from the coarse phase recovery stage to estimate and remove carrier phase angles recovered from the coarse phase recovery stage to estimate and remove carrier frequency offset. Specifically, the present disclosure proposes a novel time-domain edge detection algorithm to perform carrier frequency recovery prior to the ML phase estimator. The method of the present disclosure performs well even for a highly parallelized system, and the required computational efforts can be reduced by one order of magnitude as compared to the prior art using single-stage based blind phase search method. 
     Further, the present disclosure includes a method of carrier phase error removal associated with an optical communication signal. The method includes estimating and removing a first phase angle associated with an information signal using coarse phase recovery, the information symbol being associated with a digital signal, the digital signal representing the optical communication signal; estimating a carrier frequency offset between a receiver light source and a transmitter light source by using the estimated first phase angle, the carrier frequency offset being associated with the information symbol; removing carrier phase error associated with the carrier frequency offset; and estimating and removing a second phase angle associated with the information symbol, the estimated second phase angle being based on the estimated first phase angle and the estimated carrier frequency offset. Estimating and removing the first phase angle may include estimating the first phase angle using a decision-directed phase-locked loop, a decision-aided feedback phase recovery method, or estimating the first phase angle using a coarse blind phase search. Estimating and removing the second phase angle may include performing a maximum likelihood estimate based on the estimated first phase angle and the estimated carrier frequency offset, or estimating an average phase rotation based on the estimated first phase angle and the estimated carrier frequency offset. The method may further include estimating and removing a third phase angle associated with the information symbol, the third phase angle being based on the estimated second phase angle and the estimated carrier frequency offset. Estimating and removing the third phase angle may include performing a maximum likelihood estimate to generate a maximum likelihood estimator used to adjust the estimated carrier frequency offset based on the estimated second phase angle. 
     Additionally, the present invention includes an apparatus for carrier phase error removal associated with an optical communication signal. The apparatus includes a processing device having a processor and a receiver. The receiver receives an information symbol being associated with a digital signal, the digital signal representing the optical communication signal. The processor is configured to estimate and remove a first phase angle associated with an information symbol using coarse phase recovery, an information symbol being associated with a digital signal, the digital signal representing the optical communication signal, the processing device being configured to estimate a carrier frequency offset between a receiver light source and a transmitter light source by using the estimated first phase angle, the carrier frequency offset being associated with the information symbol, the processing device being configure to remove carrier phase error associated with the carrier frequency offset, the processing device being configure to estimate and remove a second phase angle associated with the information symbol, the estimated second phase angle being based on the estimated first phase angle and the estimated carrier frequency offset. 
     Further, the present disclosure includes a non-transitory computer-readable storage medium storing computer instructions that, when executed by a processing device, perform a carrier phase error removal associated with an optical communication signal. The instructions include estimating and removing a first phase angle associated with an information symbol using coarse phase recovery, the information symbol being associated with a digital signal, the digital signal representing the optical communication signal; estimating a carrier frequency offset between a receiver light source and a transmitter light source by using the estimated first phase angle, the carrier frequency offset being associated with the information symbol; removing carrier phase error associated with the carrier frequency offset; and estimating and removing a second phase angle associated with the information symbol, the estimated second phase angle being based on the estimated first phase angle and the estimated carrier frequency offset. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings constitute a part of this specification and include exemplary embodiments, which may be implemented in various forms. It is to be understood that in some instances various aspects may be shown exaggerated or enlarged in the drawings to facilitate understanding of the embodiments. 
         FIG. 1  shows a schematic of a generic optical communications system; 
         FIG. 2  shows a schematic of an optical transmitter; 
         FIG. 3  shows a schematic of an optical receiver; 
         FIG. 4   a  is a block diagram of a second-order decision-directed phase locked loop (PLL). 
         FIG. 4   b  is a block diagram of a second order decision directed phase-locked loop (DD-PLL) with k parallel branches. 
         FIG. 5  is a block diagram of a first embodiment in accordance with the present disclosure. 
         FIG. 6  is a block diagram of a second embodiment in accordance with the present disclosure. 
         FIG. 6   a  is a high-level block diagram of the embodiment of  FIG. 6 . 
         FIG. 6   b  is a high-level block diagram of the embodiment of  FIG. 6 . 
         FIG. 6   c  is a block diagram of a third embodiment of  FIG. 6  for carrier frequency offset estimation purpose. 
         FIG. 6   d  is a block diagram of a fourth embodiment of  FIG. 6  introducing a feedback configuration. 
         FIG. 7  is a plot diagram of phase offset detection of samples with a frequency offset of 1 MHz. 
         FIG. 8  is a block diagram of a multi-branch method for edge detection compatible with wide-range frequency detection. 
         FIG. 9  is a plot diagram of simulated bit error ration (BER) as a function of a number of parallel branches with different phase recovery schemes, where there is no frequency offset in the system. 
         FIG. 10  is a plot diagram of simulated BER as a function of carrier frequency offset by using a three-stage PLL+2ML phase recovery method with a different number of parallel branches. 
         FIG. 11  is a plot diagram of simulated BER versus carrier frequency offset by using both phase (PLL+2ML) and frequency recovery methods. 
     
    
    
     DETAILED DESCRIPTION 
     There is a need for further reducing the implementation complexity of carrier recovery for high-order M-QAM system. Moreover, the single-stage, blind-phase search algorithm and multi-stage algorithm do not consider carrier frequency offset (the frequency offset between the signal source and the local oscillator). However, in the real world, significant carrier frequency offset (&gt;10 MHz) may occur in many cases, especially for long-haul transmission systems using intradyne detection and coarse automatic frequency tracking techniques (Z. Tao, H. Zhang, A. Isomura, L. Li, T. Hoshida, J. C. Rasmussen, “Simple, Robust, and Wide-Range Frequency Offset Monitor for Automatic Frequency Control in Digital Coherent Receivers,” ECOC 2007, paper 03.5.4). Thus, a carrier recovery method capable of recovering the carrier phase in the presence of carrier frequency offset is also needed. 
       FIG. 1  shows a schematic of a generic optical telecommunications system. Multiple optical transceivers (XCVRs) send and receive lightwave signals via optical transport network  102 . Shown are four representative transceivers, referenced as XCVR 1  104 , XCVR 2  106 , XCVR 3  108 , and XCVR 4  110 , respectively. In some optical telecommunications systems, optical transport network  102  can include all optical components. In other optical telecommunications systems, optical transport network  102  can include a combination of optical and optoelectronic components. The transport medium in optical transport network  102  is typically optical fiber; however, other transport medium (such as air, in the case of free-space optics) can be deployed. 
     Each transceiver has a corresponding transmit wavelength (,T n 1) and a corresponding receive wavelength (,R n 1), where n=1-4. In some optical telecommunications systems, the transmit and receive wavelengths for a specific transceiver are the same. In other optical telecommunications systems, the transmit and receive wavelengths for a specific transceiver are different. In some optical telecommunications systems, the transmit and receive wavelengths for at least two separate transceivers are the same. In other optical telecommunications systems, the transmit and receive wavelengths for any two separate transceivers are different. 
       FIG. 2  shows a schematic of an example of an optical transmitter. Transmit (Tx) laser optical source  202  transmits a continuous wave (CW) optical beam  201  (with wavelength 1) into electro-optical modulator  204 , which is driven by electrical signal  203  generated by electrical signal source  206 . Electrical signal  203  consists of an electrical carrier wave modulated with information symbols (data symbols). The output of electro-optical modulator  204  is carrier optical beam  205 , which consists of a corresponding optical carrier wave modulated with information symbols. In general, the amplitude, frequency, and phase of the optical carrier wave can be modulated with information symbols. Carrier optical beam  205  is transmitted to optical transport network  102  (see  FIG. 1 ). 
       FIG. 3  shows a schematic of an example of an optical receiver. Carrier optical beam  301 , with wavelength 1, is received from optical transport network  102  (see  FIG. 1 ). Carrier optical beam  301  has an optical carrier wave modulated with information symbols. In general, the optical receiver determines the amplitude, frequency, and phase of the modulated optical carrier wave to recover and decode the information symbols. Carrier optical beam  301  is transmitted into optical coherent mixer  302 . Local oscillator laser optical source  304  generates a reference optical beam  303 , with wavelength 1, modulated with an optical reference wave with tunable reference amplitudes, reference frequencies, and reference phases. Reference optical beam  303  is transmitted into optical coherent mixer  302 . 
     Optical coherent mixer  302  splits carrier optical beam  301  into carrier optical beam  301 A and carrier optical beam  301 B. Optical coherent mixer  302  splits reference optical beam  303  into reference optical beam  303 A and reference optical beam  303 B, which is phase-shifted by 90 degrees from reference optical beam  303 A. The four optical beams are transmitted into optoelectronic converter  306 , which contains a pair of photodetectors (not shown). One photodetector receives carrier optical beam  301 A and reference optical beam  303 A to generate analog in-phase electrical signal  307 A. The other photodetector receives carrier optical beam  301 B and reference optical beam  303 B to generate analog quadrature-phase electrical signal  307 B. Analog inphase electrical signal  307 A and analog quadrature-phase electrical signal  307 B are transmitted into analog/digital converter (ADC)  308 . The output of ADC  308 , represented schematically as a single digital stream, digital signal  309 , is transmitted into digital signal processor  310 . Digital signal processor  310  performs multiple operations, including timing synchronization, equalization, carrier frequency recovery, carrier phase recovery, and decoding. 
     The carrier phase recovery refers to estimate and remove the phase error caused by inherent laser phase noise as well as the unknown transmission delay from the information symbol. Carrier frequency recovery refers to estimate and remove the phase error caused by carrier frequency offset (or difference) between the transmitter laser source and the receiver laser source from the information symbol. The information symbol refers to the physical representation of a digital signal such as the binary-modulated signal symbol. QAM-modulated signal symbol in the electrical field form. Coarse phase recovery refers to not very accurate phase recovery. The first phase angle refers to the estimated phase deviation of the received information symbol based on the coarse phase recovery. The second phase angles refers to the phase deviation of the received information symbol based on the second refined phase recovery. 
     An optical signal degrades as it propagates from the optical transmitter to the optical receiver. In particular, laser phase noise introduces some uncertainty in the carrier phase of the received signal relative to the carrier phase of the transmitted signal assuming no laser phase noise. Carrier phase recovery refers to recovery of the correct carrier phase (carrier phase as originally transmitted assuming no laser phase noise) from the received signal. In practice, a best estimate of the carrier phase is determined from the received signal such that a decoded information symbol at the receiver is a best estimate of the corresponding encoded information symbol at the transmitter. Carrier phase recovery determines the phase angle by which an initial decoded information signal is rotated to yield the best estimate of the corresponding encoded information signal. 
       FIGS. 4   a  and  4   b  show block diagrams  10 ,  11  of a decision-directed second-order digital phase lock loop (PLL)  10  and its parallel form  11 , respectively, which are based on a system described in “Compensation of Frequency Offset for Differentially encoded 16- and 64-QAM in the presence of laser phase noise”, by Irshaad Fatadin, David Ives, Seb J. Savor.,  IEEE Photonics Technology Letters.  Feb. 1, 2010, p. 2010. The presently claimed embodiment would benefit these systems as discussed below in detail. 
     The feedback phase error φ error  is calculated as follows: 
     
       
         
           
             
               
                 
                   
                     
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     where k is the time index, y k  is the kth received sample (one sample per symbol, after equalization), â* k  is the conjugate of the kth decided bit, and Δ{circumflex over (φ)}(k) is the kth estimated phase offset. Here, y k e −jΔ{circumflex over (φ)}(k)  is the received sample with phase correction based on an estimated phase offset. By multiplying y k e −jΔ{circumflex over (φ)}(k)  with â* k , any phase information encoded in the sample is removed, and the symbol is rotated to the x-axis. Any deviation of this result from the x-axis is, therefore, representative of the error in phase estimation. For small angles, φ≈sin(φ). Thus, to simplify calculations, the angle is estimated by measuring the imaginary component, and normalizing for magnitude scaling. This normalization is not present in the prior art (i.e., Irshaad Fatadin, David Ives, Seb J. Savory., “Compensation of Frequency Offset for Differentially encoded 16- and 64-QAM in the presence of laser phase noise.”  IEEE Photonics Technology Letters. Feb.  1, 2010, p. 2010). However, this normalization of the presently claimed embodiment significantly improves the performance of the phase-locked loop (PLL). 
     The remaining PLL equations are as follows: 
       φ i ( k )=φ i ( k− 1)+ g   i φ error ( k )   (2)
 
       φ B ( k+ 1)= g   p φ error ( k )+φ i ( k )   (3)
 
       Δ{circumflex over (φ)}( k+ 1)=Δ{circumflex over (φ)}( k )+φ δ ( k+ 1)   (4)
 
         â   k   =y   k   e   −jΔ{circumflex over (φ)}(k)    (5)
 
     The PLL can only be used for coarse phase recovery. This is because the PLL performance relies heavily on negative feedback, and therefore used for dense sampling. However, in order to support high optical data rates (greater than 10 Gbaud), electronic processing has to be parallelized, and therefore must rely on sparse sampling. 
       FIGS. 5  and  FIG. 6  describe PLL-based phase recovery method. PLL is the preferable coarse phase recovery method in accordance with the present disclosure due to its hardware efficiency. Specifically,  FIG. 5  shows the basic scheme  20  of the present disclosure, which relies on a series of maximum likelihood (ML) phase estimators  22  to adjust the phase offset and improve the bit error ration (BER). Because the ML phase estimator  22  is a feed-forward method, its performance does not degrade due to system parallelization. 
     The ML phase estimator  22  is implemented as follows (Proakis, J. G.,  Digital Communications,  4th edition, Chapter 6, pp. 348): 
     
       
         
           
             
               
                 
                   
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     Here, Y(k), and {circumflex over (X)} old (k) denote the kth received sample  24  (before carrier recovery) and the decided symbol from the previous coarse phase recovery stage  26 , respectively, and {circumflex over (X)} new (k) and Δ{circumflex over (φ)} MLE (k) are the newly decided symbol and phase offset, respectively. For each ML phase estimator  22 , the decided symbols of the previous stage are used as a reference. Through simulation, this cascade works best with two or fewer stages, as performance quickly reaches a BER floor. 
     The performance of the ML phase estimator  22  can be improved by weighting the phase estimates inversely to the magnitudes of the symbols, due to the fact that symbols further from the origin have a higher probability of error, and therefore contribute more to phase error. A weighted ML phase estimator is implemented by substituting H(k) in with 
     
       
         
           
             
               
                 
                   
                     
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     where, d is a very small number to prevent division by 0. For systems with only one ML phase estimator, this normalization step will improve performance. However, for cascaded ML systems, the improvement may not be significant. 
     For a highly parallelized system of  FIG. 5 , the above discussed PLL  28  and ML phase estimator  22  multistage solution works well only for the case with no carrier frequency offset or with very small carry frequency offset. Its performance degrades when frequency offset is present. This is because the ML phase estimator  22  acts as a smoothing filter, and will attempt to remove the slope in Δ{circumflex over (φ)} caused by the frequency offset Δf. To address this problem, estimation and removal of this frequency offset is proposed prior to the ML phase estimation  22  by using the estimated phase offset  29  from the first PLL coarse phase recovery stage  28 . A time-domain edge detection algorithm of the present disclosure performs such carrier frequency offset estimation. This frequency offset estimation method in general can also be used in the case that the first phase recovery stage employs different coarse phase recovery methods (e.g. the coarse blind phase search method). In addition, although the proposed time-domain edge detection based frequency detection method has the advantage of simple implementation, more complicated fast-fourier-transform (FFT) based method or time-domain phase slope based method may also be employed to detect the carrier frequency offset by using the recovered carrier phase angles obtained from the first coarse phase recovery stage. 
     Although phase offset changes quickly, frequency offset changes slowly, and can be corrected on a separate time scale than phase offset. This is an additional advantage, in that frequency estimation can be performed on a much slower timescale, thereby decreasing complexity. The schematic illustration for the proposed carrier recovery 30 method with the presence of carrier frequency offset is shown in  FIG. 6 . 
     The coarse phase recovery shown in  FIGS. 6   a - 6   d  is not limited to PLL, other phase estimation methods such as the blind phase search based methods or the decision-aided feedback phase recovery method reported in IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 21, NO. 19, Oct. 1, 2009, entitled “Parallel Implementation of Decision-Aided Maximum-Likelihood Phase Estimation in Coherentary Phase-Shift Keying Systems,” by S. Zhang, C. Yu, Member, P. Y. Kam, and J. Chen, may also be used as the coarse phase recovery method. The ML estimator shown in  FIG. 6  corresponds to 2nd-stage refined phase recovery in  FIG. 6   a - 6   d.  Although ML estimator is preferable choice for the 2nd-stage refined phase recovery, some other phase estimators such as the known phase-range constrained blind phase search method may also be used in the second stage phase recovery. 
       FIG. 6   a  shows a high-level block diagram  60  of the embodiment shown in  FIG. 6 . In this embodiment, the carrier phase recovered from the first coarse phase recovery stage  61  is used as the input of a carrier frequency offset detection/estimation circuit  62 , where the carrier frequency offset  63  between the incoming signal source  64  and the local oscillator light source  65  is estimated by either the proposed time-domain edge detection method or some other known carrier frequency offset detection methods such as the fast Fourier transform (FFT)-based frequency domain methods or time-domain phase slope detection based methods. The frequency offset of a copy of the original signal can then be removed  66  and the frequency offset-removed signal  67  along with the phase recovered signal  68  from the first coarse phase recovery stage  61  are then used as two inputs of the second-stage refined carrier recovery circuit  69  where a maximum likelihood (ML) based carrier phase estimation method or phase-constrained blind phase search method may be applied to do a more accurate carrier phase recovery. The phase recovered signal  70  from the second phase recovery stage  69  and a copy of the frequency-offset removed signal  67  can be further used as two inputs of the third carrier phase recovery circuit  71  to further refine the carrier phase recovery. 
       FIG. 6   b  shows another embodiment of a high-level block diagram of  FIG. 6 . The second embodiment  72  is similar to embodiment  60  of  FIG. 6   a  except that the phase-recovered signal  68  resulted from the first coarse phase recovery stage  61  is made to pass through a frequency offset removal circuit  73  before entering into the second phase recovery stage  69 . The second embodiment  72  may achieve better performance than the first embodiment  60  if the coarse phase is estimated in a block-by-block basis (i.e the carrier phase is assumed identical for all the symbols within the same block). For this case, carrier frequency offset  74  introduces an additional phase error  75  if without performing carrier frequency offset removal. 
       FIG. 6   c  shows the third embodiment  80  of the present disclosure. For third embodiment  80 , the coarse phase recovery circuits serve only for carrier frequency offset estimation purpose. The estimated carrier phase angles  62  from the coarse phase recovery circuit  61  is used for carrier frequency offset detection  76  but the coarse phase-recovered signal  77  does not pass to the second phase recovery stage  69 , instead, the carrier frequency offset removal operation  66  is applied to the original signal  81  and then the frequency offset-removed signal  67  goes to the second phase recovery stage  69  and then the third phase recovery stage  71  for carrier phase recovery. This embodiment may achieve better performance than the first embodiment  60  and the second embodiment  72  when the frequency offset is significant. 
       FIG. 6   d  shows that the fourth embodiment  82  is an improvement of the third embodiment  80  of  FIG. 6   c  by introducing a feedback configuration  83  such that the phase recovered signal  62  from the first coarse phase recovery stage  61  can be used along signal  84  by the second phase recovery circuit  69  even for large carrier frequency offset. Note that if phase locked loop (DD-PLL) based methods are employed for the coarse phase recovery, the PLL may fail to lock for a large frequency offset. For this case, scanning of the frequency offset at the system acquisition stage (i.e. the starting stage) with a frequency step smaller than the maximum tolerable frequency offset of the PLL is done. Once the PLL can lock (i.e. the test frequency is within its locking range), the system can switch to the normal operation state. 
     The basic idea for the frequency offset estimation is to exploit the periodicity of the measured phase offset Δ{circumflex over (φ)} caused by range-limiting the value to [−π,π), eg: 
       Δ{circumflex over (φ)}≈mod(2π(Δft+Δφ(t)), 2π)   (11)
 
     Since Δft&gt;&gt;Δφ(t) for most values of t, this modulo operation will cause the detected Δ{circumflex over (φ)} to resemble a nearly perfect sawtooth wave with frequency≈Δf. Additionally, because of the sharp edges in a sawtooth wave, a simple edge detection technique can be used to find the average period of Δ{circumflex over (φ)}, thereby estimating Δf. 
       FIG. 7  is an example plot of Δ{circumflex over (φ)} for Δf=1 MHz. The measured period is very close to 1/Δf; however, in instances with high jitter, one edge may manifest itself as several edges, which can be corrected. An example of a simple edge detection scheme is shown below, where K is the set of sample indices closest to each edge. 
       δ[ k]=Δ{circumflex over (φ)}[k]−Δ{circumflex over (φ)}[k− 1]  (12)
 
         K={k:|δ[k]|&gt;π}   (13)
 
     Because Δft&gt;Δφ, the sawtooth gives very sharp edges, making these frequency estimates very precise. However, as  FIG. 7  also shows, there can sometimes be jitter in the system, causing one sawtooth edge to appear as a cluster of edges. If this is not corrected, the distance between edges within the cluster can skew the estimate dramatically. One way to identify edges caused by jitter is that these edges will be followed by an edge going the opposite direction. For example, an edge going from π to −π caused by jitter will be soon followed by an edge going from −π to π. Therefore, of all the edges detected in K, the edges actually pertaining to the sawtooth, K c , can be found as: 
         K   c   ={k ∈ K:δ[k]+δ[k− 1]&gt;2π}  (14)
 
     Then, the average frequency magnitude can be found as 
     
       
         
           
             
               
                 
                   d 
                   = 
                   
                     mean 
                      
                     
                         
                     
                      
                     spacing 
                      
                     
                         
                     
                      
                     in 
                      
                     
                         
                     
                      
                     
                       K 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                      
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         f 
                         ^ 
                       
                     
                      
                   
                   = 
                   
                     d 
                     
                       F 
                       s 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where F s  is the sampling frequency. The sign of the frequency can then be found by summing the slopes of the edges, or 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     sign 
                      
                     
                       ( 
                       
                         
                           ∑ 
                           k 
                         
                          
                         
                           δ 
                            
                           
                             [ 
                             
                               k 
                               ∈ 
                               K 
                             
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Frequency offset can then be estimated and removed: 
       Δ {circumflex over (f)}=s·|Δ{circumflex over (f)}| 
 
         x   B   =xe   −2πΔft    (18)
 
     where x B  is the baseband transmit signal. 
     Since this method includes additions and thresholding, the only major complexity comes from the number of samples L needed for good frequency estimation. For a 64 QAM system with 26 dB open-source cognitive radio (OSNR) or higher, it is possible to achieve a frequency precision of 1 MHz for the entire dynamic range with L=1000 samples by downsampling at the appropriate rate. This rate must be chosen such that the sampling window is wide enough to include two or more sawtooth periods, while also sampling dense enough that edges are detected closely. Since systems with more parallel branches can only sync to a smaller range of frequencies, a highly parallelized system can achieve good frequency estimation for the entire effective frequency range with a very small number of samples. For example, in a 64-QAM system and 100 kHz laser linewidth, with a PLL with 4 or more parallel branches, a sampling rate of 300 MHz for 1000 samples works well for all the possible frequency offsets in which the PLL will sync (≦20 MHz), and can consistently predict Δf to within 1 MHz error. A PLL with fewer parallel branches can sync to higher frequencies, making it difficult to achieve frequency precision for the entire dynamic range with a small number of samples. In this case, a multi-branched method is used in which the estimation is done with multiple L-sample banks, all sampling the data at different rates.  FIG. 8  shows a multi-branch method for edge detection compatible with wide-range frequency detection  50 . Each branch  51 , 52  and  53  then performs edge detection (Equations(12)-(14)) in parallel, and the best window  54  produces K c  for Equation (15). In this case, the best window is the one with the densest sampling but includes more than T sawtooth edges (≈3). This method grows in complexity, in that it requires L×M samples, where M is the number of branches. However, in most cases, M is very small. In 64-QAM, for example, for fewer than 4 PLL branches, M=2 is sufficient to acquire 1 MHz accuracy in all effective frequency ranges (≦100 MHz). Here, the preferred choice for the two sampling rates is 300 MHz and 4.75 GHz. 
     The frequency offset estimate Δ{circumflex over (f)} is removed from Y as follows: 
         Y   B   =Y·e   −jΔft    (19)
 
     After this, Y B , a baseband version of Y, is fed into the ML phase estimation. For a sampling speed of 300 MHz and window of 1000 samples, the time needed to acquire each window is about 3.3 μs. This means that this method will consistently remove frequency offset correctly if the frequency offset changes at a rate of 300 kHz per microsecond or less. Typically, frequency offset changes much more slowly, in which case the frequency estimation can be applied less frequently. 
     In the above, a time domain edge detection solution  55  is used to decide both the magnitude and sign of the carrier frequency offset  56 . Time-domain based solution  55  such as the one described in ‘Frequency Estimation in Intradyne Reception”, by A. Leven, N. Kaneda, U. V. Koc, Y. K. Chen, (See  IEEE Photonics Technology Letters, Vol.  19, No. 6, Mar. 15, 2007) and an FFT based solution as described in “Frequency Estimation for Optical Coherent MPSK System Without Removing Modulated Data Phase,” by Y. Cao, S. Yu, J. Shen, W. Gu, Y. Ji (See  IEEE Photonics Technology Letters,  Vol. 22, No. 10, May 15, 2010) can also be used for carrier frequency offset estimation by using the phase offset output from the first stage PLL. The method of the present disclosure provides more accurate frequency offset estimation than the alternative time-domain based solution, which cannot provide good estimation for M-QAM based systems, where M&gt;4. Additionally, the method of the present disclosure is much less complex than the FFT-based solutions, which requires at least a 2048-length FFT to achieve our performance level. In contrast, the method of the present disclosure requires 1000 or 2000 samples (depending on the number of PLL branches), and uses only additions and thresholding. 
     Although the above discussed time-domain edge detection method used the PLL to remove the data modulation and find the phase offset. Other data modulation-removing methods (such as the well known Mth-power algorithm used for phase-shifting key (M-PSK system) can also be used to find the phase offset. Once phase offset is found, the time-domain edge detection method can then be used to estimate the carrier frequency offset. 
     The effectiveness of these methods has been verified by numerical simulations of resulting BER for back-to-back transmission. In all cases, the following assumptions were made: the symbols constituted a square 64 QAM constellation, transmitted at a baud rate of 38 Gsym/s, with an optical signal to noise ratio (OSNR) at 0.1 nm noise bandwidth to be 28 dB, and a laser linewidth of 100 kHz for both the signal source and local oscillator. At the receiver, the 3 dB electrical receiver bandwidth is 0.55×baud rate, sampling performed at 38 Gb/s (1 sample per symbol, after equalization), and receiver filtering effects is equalized by using a cascaded multi-modulus algorithm based adaptive equalizer (X. Zhou, J. Yu., 200- Gb/s PDM -16  QAM generation using a new synthesizing method.  s.1.: paper 10.3.5, 2009. ECOC) prior to carrier recovery. The resulting BER for these schemes are shown in  FIGS. 9-11 . 
       FIG. 9  shows the simulated BER performance versus the number of parallel branches for a cascaded PLL and ML system as illustrated in  FIG. 5  for the case that there is no frequency offset in the system. Dashed lines are systems where ML phase estimators use weighted estimates.  FIG. 9  shows that the BER performance of the PLL decreases as the number of parallel branches increase, showing that negative feedback systems do not perform well with many parallel branches. However, this degradation in performance is not apparent after two ML phase estimators, showing that our cascaded system works well with parallel systems despite the negative feedback PLL. The third ML phase estimator does not improve performance, showing that two ML phase estimators is optimal. Additionally, though a weighted ML phase estimator improves performance for one ML, the improvement is negligible when two or more ML phase estimators are cascaded. In the remaining figures, our system consists of a PLL, followed by two unweighted ML phase estimators. 
     In  FIG. 10 , the embodiment in  FIG. 5  is used in a receiver with phase and frequency offset.  FIG. 10  shows the simulated BER versus carrier frequency offset by using three-stage PLL+2ML phase recovery method with different number of parallel branches. The labels show number of parallel PLL branches used. Here, the performance decreases for increasing frequency offset, for two reasons. For lower frequency offsets, the performance of the ML phase estimator degrades, resulting in a slowly rising BER. This results from the block implementation of the ML phase estimator, which attempts to filter out the frequency offset component of the phase detection. Additionally, for Δf too high, Δ{circumflex over (φ)} changes too quickly, and the PLL cannot sync, causing the BER to jump to maximum error. This is especially a problem as the number of parallel branches increases, as the feedback delay is greater. 
     In  FIG. 11 , the embodiment presented in  FIG. 6  is used to mitigate frequency offset. By removing the frequency offset after the PLL, the performance degradation caused by the ML phase estimator smoothing effect is eliminated. However, because frequency removal depends on data from the PLL, it cannot help with the PLL&#39;s inability to sync to high-frequency systems.  FIG. 11  shows simulated BER versus carrier frequency offset by using both phase (PLL+2ML) and frequency recovery method. The labels show number of parallel PLL branches used. 
     Overall, these plots show that the multistage PLL, frequency removal, and ML phase estimator can successfully perform phase recovery for low frequency offset. Although a highly parallelized PLL cannot sync to high frequency offsets, it is possible to use a PLL with fewer branches for systems with smaller frequency offsets, in which the performance is just as good as if there is no frequency offset at all. For a large frequency offset, an independent frequency recovery should be performed prior to the above discussed PLL/ML multi-stage phase recovery. 
     Additionally, the ML phase estimator stages can be replaced by other phase estimate methods. For example, the carrier phase can be estimated by directly calculating the average phase rotation of the original signal relative to the decoded signal obtained from the previous stage. In addition, the blind phase search method with refined/reduced phase scan range may also be used in the second and the third stages. 
     High-order M-QAM is the most promising modulation formats to realize high-spectral efficiency optical transmission at a data rate beyond 100-Gb/s. Because high-order M-QAM is very sensitive to laser phase noise, laser linewidth-tolerant and hardware-efficient feedforward carrier recovery method is critical important for practical implementation of these high-order modulation formats. So far, among all the published carrier recovery algorithms, only single-stage blind phase search method can achieve high laser line-width tolerance. But the required computational efforts are very high for this method, especially for highly paralleled systems. The multi-stage method of the present disclosure can achieve high laser line-width tolerance (comparable to the blind phase search method because the method not only use feedforward configuration but also use the most current symbol for the phase estimate) with substantially lower implementation complexity. The reason that the method of the present disclosure has lower implementation complexity is due to the fact that the method of the present disclosure requires only one phase estimate in each of the two or three stages, while the blind phase search (BPS) method require many phase estimates. 
     For example, using 64 QAM, the single stage BPS requires testing of more than 64 different phase angles (T. Pfau, S. Hoffmann and R. Noé., “Hardware-Efficient Coherent Digital Receiver Concept With Feed-forward Carrier Recovery for M-QAM Constellations.” JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 8, Apr. 15, 2009). Table 1 shows the hardware complexity required for one block of 2N symbols. In order to find one phase estimate, 64 such blocks are required. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 BPS complexity for 64 QAM system, for a single block of 2N symbols. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Complex 
                 Real 
                 Real 
                   
                   
               
               
                   
                 multiplier 
                 multiplier 
                 adder 
                 Slicer 
                 Other 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 BPS (×64) 
                  2N 
                 0 
                  2N + 1 
                  2N 
                 1 comparator 
               
               
                   
                   
                   
                   
                   
                 2N selectors 
               
               
                 Total 
                 128N 
                 0 
                 128N + 64 
                 128N 
                 64 comparator 
               
               
                   
                   
                   
                   
                   
                 128N selectors 
               
               
                   
               
            
           
         
       
     
     In contrast, the multistage DD-PLL and ML scheme requires more hardware per 2N symbols (Table 2), but requires only one calculation per phase estimate. In this respect, this scheme has a complexity reduction of almost 10 times. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 DD-PLL with 2 ML phase estimators complexity for a 64 QAM system, 
               
               
                 for a single block of 2N symbols in phase recovery, and 2L symbols in a 
               
               
                 2-branch frequency recovery. 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Complex 
                 Real 
                 Real 
                   
                   
               
               
                   
                 multiplier 
                 multiplier 
                 adder 
                 Slicer 
                 Other 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 PLL 
                 2N 
                 4N 
                 6N 
                 2N 
                   
               
               
                 ML (×2) 
                 4N 
                 2N 
                 4N 
                 2N 
                 Arctangent 
               
               
                 Freq. 
                   
                   
                 &lt;8L 
                 2L 
               
               
                 removal 
               
               
                 Total 
                 10N  
                 8N 
                 10N + 8L 
                 4N + 2L 
                 Arctangent 
               
               
                   
               
            
           
         
       
     
     In addition to the complexity reduction for carrier phase recovery, the present disclosure also proposes a complexity-reduced frequency recovery method that essentially enables accurate carrier phase recovery in the presence of carrier frequency offset for the first time. 
     The present disclosure includes a new laser linewidth-tolerant multi-stage feed-forward carrier phase recovery algorithm for arbitrary M-QAM modulation formats. As compared to the prior art, it is shown that the proposed new algorithm can significantly reduce the required computational efforts for high-order modulation formats. We also propose an edge detection frequency recovery method that essentially enables us to perform accurate carrier phase recovery even with carrier frequency offset. 
     The Abstract is provided to comply with 37 C.F.R. §1.72(b) and will allow the reader to quickly ascertain the nature and gist of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims. In addition, in the foregoing Detailed Description, it can be seen that various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive subject matter lies in less than all features of a single disclosed embodiment. Thus the following claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separately claimed subject matter.