Patent Publication Number: US-8995834-B2

Title: Blind equalization for polarization-switched QPSK optical communications

Description:
TECHNICAL FIELD 
     This application is directed, in general, to optical communications systems and methods. 
     BACKGROUND 
     This section introduces aspects that may be helpful to facilitating a better understanding of the inventions. Accordingly, the statements of this section are to be read in this light and are not to be understood as admissions about what is in the prior art or what is not in the prior art. 
     Blind equalization is a digital signal processing technique of using signal statistics of a received signal to infer (equalize) the transmitted signal from the received signal, while making use of the transmitted signal statistics. 
     Blind equalization may be viewed as essentially blind de-convolution applied to digital communications. Nonetheless, the emphasis in blind equalization is on online estimation of the equalizer filter (which is the inverse of the channel impulse response) rather than estimation of the channel impulse response itself. This is due in part to the common use of blind equalization in digital communications systems as a way to extract the continuously transmitted signal from the received signal. 
     One modulation format is polarization-switched (PS) Quadrature Phase-Shift Keying (QPSK). Various approaches to blind equalization in PS-QPSK have been investigated, but suffer from various impairments that render PS-QPSK an unsuitable modulation format for some applications, such as long-haul optical fiber communications. 
     SUMMARY 
     One aspect provides an apparatus, e.g. an optical receiver. The apparatus includes an optical front end and an equalizer. The front end is configured to receive an optical signal bearing first and second symbols on respective first and second polarization channels. The equalizer is configured to 1) select a first cost function if the first symbol has greater energy than the second symbol, 2) select a second different cost function if the second symbol has a greater energy than the first symbol, and 3) based on the selected cost function, update coefficients of an adaptive filter configured to demultiplex and equalize the first and second polarization channels. 
     Another aspect provides an apparatus, e.g. an optical receiver. The apparatus includes an optical front end and a phase compensator. The optical front end is configured to receive an optical signal bearing first and second symbols on respective first and second polarization channels. The phase compensator is configured to compute an estimated phase of the optical signal. The estimate is based on the first symbol if the first symbol has a greater energy than the second symbol, and based on the second symbol if the second symbol has a greater energy than the first symbol. 
     Another aspect provides a method, e.g. for forming an optical receiver. The method includes configuring an optical front end and an equalizer. The front end is configured to receive an optical signal bearing first and second symbols on respective first and second polarization channels. The equalizer is configured to 1) select a first cost function if the first symbol has greater energy than the second symbol, 2) select a second different cost function if the second symbol has a greater energy than the first symbol, and 3) based on the selected cost function, update coefficients of an adaptive filter configured to demultiplex and equalize the first and second polarization channels. 
     Yet another aspect provides a method, e.g. for receiving an optical symbol stream by a front end of an optical receiver. A first cost function is selected if a first received symbol on a first polarization channel has greater energy than a second received symbol on a second polarization channel. A second different cost function is selected if the second symbol has a greater energy than the first symbol. Based on the selected cost function, coefficients are updated of an adaptive filter configured to demultiplex and equalize the first and second polarization channels. 
    
    
     
       BRIEF DESCRIPTION 
       Reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a system, e.g. an optical PS-QPSK receiver, according to one illustrative embodiment; 
         FIG. 2  illustrates a multi-stage digital filter configured to, e.g. demultiplex and equalize two polarization signals potentially subject to Polarization Mode Dispersion (PMD) effects; 
         FIG. 3  illustrates a multi-stage tap coefficient update module according to one embodiment and applicable to, e.g. update tap coefficients of an equalizing filter provided in  FIG. 4 ; 
         FIG. 4  illustrates a computational block diagram of a method of compensating the frequency of a received signal, e.g. a PS-QPSK modulated signal, including an illustrative embodiment of an equalizing filter; 
         FIG. 5  illustrates a computational block diagram of a method of feed-forward carrier recovery from a received signal, e.g. a PS-QPSK modulated signal; 
         FIG. 6  illustrates a computational block diagram of a method of correcting for non-linear phase noise in a received signal, e.g. a PS-QPSK modulated signal; 
         FIG. 7  compares simulated bit-error rate characteristics of a conventional PDM-QPSK modulation scheme and a PS-QPSK modulation scheme including various embodiments described by, e.g.  FIGS. 1-6 ; and 
         FIG. 8  presents a method, e.g. for forming an optical PS-QPSK optical receiver including various embodiments as described by, e.g.  FIGS. 1-6 . 
     
    
    
     DETAILED DESCRIPTION 
     Various embodiments provide systems and methods for blindly equalizing polarization-switched (PS) QPSK signals transmitted via, e.g. optical links. Embodiments described herein overcome some of the deficiencies of conventional approaches to PS-QPSK blind equalization, such as the ability to jointly equalize and perform carrier and phase compensation for proper demodulation. In many cases PS-QPSK may provide increased sensitivity relative to polarization-division multiplexed (PDM) QPSK transmission. 
     PS-QPSK is a 4-dimensional constellation method, while PDM-QPSK is a 2-D constellation method. Therefore PS-QPSK requires a different equalizer and symbol detection scheme than does PDM-QPSK. For example, PDM-QPSK employs two 2-D constellations that are independent of one another, which allows for comparatively easier equalization and detection. 
     The PS method may be regarded as more power efficient and as potentially providing longer reach in undersea links and other long-haul optical communications applications than some alternative transmission formants. PS-QPSK typically has about 75% of the spectral efficiency of PDM-QPSK. However, the greater potential power efficiency and transmission reach of PS-QPSK offer sufficient benefits to outweigh the lower spectral efficiency for some applications. PS-QPSK may also be implemented in rate adaptive systems when a lower rate is sufficient. 
     However, approaches that can blindly equalize PS-QPSK signals are relatively unknown, and suffer from various deficiencies. For example, some methods of blind equalization of PS-QPSK signals required a special decorrelation method of signaling in order for a standard constant modulus algorithm (CMA) to work. But such a process can reduce the performance of PS-QPSK, in particular over long-haul links. Such long paths typically use higher instantaneous transmit power than shorter transmission paths, which can lead to a greater noise contribution from nonlinear effects. 
     Moreover, typical methods of blind equalization in CMA communication do not work in the case of PS-QPSK. This is due to, e.g., correlated data among the multiple channels (i.e., two polarizations) of PS-QPSK. Because the light has two polarizations, two symbols (one in each polarization) are received every symbol period. With PS-QPSK, one symbol must be a QPSK constellation point and the other a transmitted 0, and therefore, the polarization “channels” are correlated. 
     It is possible to equalize the channels by using a data-aided approach that uses training data. However, this typically requires extra computational overhead or a decision-feedback approach. The decision-feedback approach suffers from difficulty of implementation due to latency problems with the adaptation. This is because the carrier frequency recovery and phase recovery are performed first before a symbol decision can be fed back. 
     Various embodiments described herein and otherwise within the scope of the disclosure use one or more algorithms that take into account the relative strength of symbols simultaneously received via two polarization channels in a QPSK-modulated signal. For example, some embodiments employ a data-dependent equalizer cost function, data-dependent intermediate frequency compensation, and/or data-dependent carrier recovery. 
     It should be noted that the blind approach used in embodiments of the disclosure could be used after a data-aided approach provides initial convergence of the equalizer. One way of interpreting embodiments of the disclosure is that such embodiments may perform decision-feedback of magnitude only, and thus, phase and carrier recovery are not needed. 
     Two symbols are received in each symbol period, e.g. a QPSK constellation point and a transmitted 0 (sometimes called the “switch symbol”). A probabilistic choice is made as to which symbol is the QPSK bit symbol and which is the switch symbol. The polarization with the greater energy is more likely to be the QPSK symbol and is interpreted as such, while the lower energy polarization is interpreted as the switch symbol. This approach saves energy by not transmitting anything for the switch symbol. This reduces each symbol period to 3 bits (instead of 4 bits with PDM-QPSK), but requires less power than PDM-QPSK for the same bit-error rate transmission of data. 
     Turning now to  FIG. 1 , illustrated is an optical receiver  100  according to one nonlimiting embodiment. The receiver  100  in various embodiments receives a PS-QPSK modulated signal. The receiver includes four analog-to-digital converters (ADCs)  110 - 1 I,  110 - 1 Q,  110 - 2 I and  110 - 2 Q, front-end compensators  120 - 1  and  120 - 2 , chromatic dispersion compensators  130 - 1  and  130 - 2 , and a timing recovery module  140 . Each of the modules  110 - 1 I,  110 - 1 Q,  110 - 2 I.  110 - 2 Q,  120 - 1 ,  120 - 2 ,  130 - 1 ,  130 - 2  and  140  may be conventional or unconventional without limitation thereto. Following the timing recovery module  140  are a polarization tracking and equalization module  150 , a frequency estimation module  160 , a phase estimation module  170 , and a decision module  180 . These modules are described in detail below. 
     The ADC pair  110 - 1 I and  110 - 1 Q respectively receive in-phase (I) and quadrature (Q) signals of a first polarization channel of the received signal. The ADC pair  110 - 2 I and  110 - 2 Q respectively receive I and Q signals of a second polarization channel of a received signal. The first and second polarization channels may correspond to, e.g. horizontal (H) and vertical (V) polarizations of a received optical signal. 
     The front-end compensator  120 - 1  receives symbol streams from each of the ADCs  110 - 1 I and  110 - 1 Q. The front-end compensator  120 - 2  receives symbol streams from each of the ADCs  110 - 2 I and  110 - 2 Q. The compensators  120 - 1  and  120 - 2  provide initial signal processing functions to correct signal imperfections introduced by the receiver front-end, such as by adjusting the levels of one or both of the received signals. 
     The chromatic dispersion compensator  130 - 1  receives the symbol stream from the compensator  120 - 1 , and the chromatic dispersion compensator  130 - 2  receives the symbol stream from the compensator  120 - 2 . The compensators  130 - 1  and  130 - 2  compensate the symbol streams to correct for chromatic dispersion of the received optical signal that may have occurred during transmission. 
     The timing recovery module  140  receives the symbol streams from the dispersion compensators  130 - 1  and  130 - 2  and recovers the symbol timing for the combined symbol stream. 
     The polarization tracking and equalization module  150  receives the symbol stream from the timing recovery module  140 , and inverts the channel and adjusts the polarization of received symbols in the data stream. The frequency estimation module  160  receives the adjusted symbol stream from the equalization module  150 , estimates the intermediate frequency (IF) of the received symbol stream and subtracts it from the symbol stream. The phase estimation module  170  receives the symbol stream from the frequency estimation module  160 , estimates a phase of each received symbol and subtracts it from the symbol. Finally, the decision module  180  receives the symbol stream from the phase estimation module  170  and selects the value of the symbol. 
     The receiver  100  implements a 4-D maximum-likelihood (ML) detection scheme, as described below. This ML approach separates the operation of the cost function in the equalization module  150  from the symbol detections in the decision module  180 . Since the cost function choice does not incorporate phase, it may in some cases choose the switch and QPSK symbols inaccurately due to a low-probability noise realization. After equalization and phase/carrier recovery, respectively by the modules  160  and  170 , a more accurate detection can be achieved by the decision module  180 , resulting in a lower Bit Error Rate (BER). It is noted that such an event is not expected to happen often enough to have a significant impact on the equalizer performance. However, the ML detection scheme can reduce the BER by 5% to 30%, depending on SNR, versus using the switch-symbol decision after the equalizer. 
     The receiver  100  may be implemented with any conventional or novel phase compensator of electronic devices, including without limitation a state machine or microcontroller, combinatorial logic, and a field-programmable gate array (FPGA). Those skilled in the pertinent art are capable of rendering the various embodiments described herein without undue experimentation. 
     The operation of the polarization tracking and equalization module  150 , the frequency estimation module  160 , the phase estimation module  170  and the decision module  180  are now considered in detail in turn below. 
     Polarization Tracking And Equalization Module 
     The polarization tracking and equalization module  150  provides polarization tracking and Polarization Mode Dispersion (PMD) equalization functions. These functions may be performed using a two-in two-out adaptive filter. An adaptive filter can be partitioned into three parts: a filter bank, an error estimator, and an updater for updating the coefficients of the filter bank. 
     The filter bank applies a filter of a form v=Wu, where a symbol pair received by the equalization module  150  is 
               u   =     [           u   1               u   2           ]       ,         
where u 1  is the symbol associated with the first polarization (e.g. H) and u 2  is the symbol associated with the second polarization (e.g. V). A symbol pair output by the equalization module  150  is
 
               v   =     [           v   1               v   2           ]       ,         
where v 1  is the filtered symbol associated with the first polarization and v 2  is the filtered symbol associated with the second polarization. A filter coefficient matrix as shown in Eq. 1 represents an equalizing filter W.
 
     
       
         
           
             
               
                 
                   W 
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                             11 
                           
                         
                         
                           
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     The error estimator implements a cost function, e.g. based on a PS-CMA (constant modulus algorithm) algorithm. The cost function is data-dependent, e.g. is based on the received signal power. If energy in the u 1  polarization, E(u 1 ), is greater than that of the u 2  polarization, E(u 2 ), the PS-CMA algorithm adapts assuming that u 1  contains QPSK data, and u 2  is the switch symbol (0). Conversely, if E(u 2 ) is greater than E(u 1 ) the PS-CMA algorithm adapts assuming that u 2  contains the QPSK data. 
     In either case the PS-CMA algorithm seeks a minimum of the cost function based on a targeted magnitude of each signal. The absolute value of the signal may be used as a proxy for the energy, and a cost function J(W) may be defined for each of two cases. In a first case, at a particular time index k, |u 1 |&gt;|u 2 |, and the cost function may be described by Eq. 2:
 
 J ( W )=(| u   1 | 2 −1) 2 +(| u   2 | 2 ) 2   (2)
 
     On the other hand, if |u 2 |&gt;|u 1 | then the cost function may be described by Eq. 3:
 
 J ( W )=(| u   1 | 2 ) 2 +(| u   2 | 2 −1) 2   (3)
 
     For the case that |u 1 |=|u 2 | the cost function may be pseudo-randomly selected from between Equations 1 and 2. 
     The updater updates the coefficients of the W matrix after the filter processes a kth symbol, with the updated filter coefficients applied to the k+1 st  symbol. This update is expressed as
 
 W   i+1   =W   i   −μ∇J ( W )  (4)
 
     where ∇J(W) is the gradient of the cost function (e.g. Eq. 2 or 3) with respect to W, and μ is a weighting coefficient determinable by one skilled in the art for a particular application. 
     The cost function gradient is determined for each of two cases represented by Equations 1 and 2. Eq. 5 shows the first case, in which u 1  is taken to be QPSK-modulated, and u 2  is taken to be the switch symbol (0) based on the relative energies of u 1  and u 2 , e.g. |u 1 |&gt;|u 2 |. Eq. 6 shows the second case, in which u 2  is taken to be QPSK-modulated, and u 1  is taken to be 0, e.g. |u 2 |&gt;|u 1 |. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     where u* signifies the complex conjugate of u. 
     The PS-CMA adaptation may be extended to the case of polarization mode dispersion (PMD). In this case the filter applies a set of L matrixes that represent four digital filters. The l th  coefficient matrix is denoted W(l). Then for the case that |u 1 |&gt;|u 2 |, 
                       ∇     J   ⁡     (     W   ⁡     (   l   )       )         =     [                 2   ·     (                v   1     ⁡     (   i   )            2     -   1     )     ·                   v   1     ⁡     (   i   )       ·       u   1   *     ⁡     (     i   -   l     )                           2   ·     (                v   1     ⁡     (   i   )            2     -   1     )     ·                   v   1     ⁡     (   i   )       ·       u   2   *     ⁡     (     i   -   l     )                         2   ·              v   2     ⁡     (   i   )            2     ·       v   2     ⁡     (   i   )       ·       u   1   *     ⁡     (     i   -   l     )               2   ·              v   2     ⁡     (   i   )            2     ·       v   2     ⁡     (   i   )       ·       u   2   *     ⁡     (     i   -   l     )               ]       ,           (   7   )               
and for the case that |u 2 |&gt;|u 1 |,
 
     
       
         
           
             
               
                 
                   
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                   ) 
                 
               
             
           
         
       
     
     In the rare case that |v 1 (k)|=|v 2 (k)|, Eqs. 7 and 8 may be selected pseudo-randomly. 
       FIG. 2  illustrates a method, e.g. an equalizing filter  200 , configured to implement the PMD case. The filter  200  includes L stages, wherein each stage receives a corresponding filter matrix W, e.g. W( 0 ),W( 1 ) . . . W(L−1). Inputs to the filter  200  include u 1 (k) and u 2 (k), which are the received polarization signals at sample k. Outputs v 1 (k) and v 2 (k) are provided as outputs from the filter  200 . 
     Multipliers  205 ,  210  respectively multiply u 1 (k) by W 11 ( 0 ) and W 21 ( 0 ), where l=0 for the first filter stage. Multipliers  215 ,  220  respectively multiply u 2 (k) by W 12 ( 0 ) and W 22 ( 0 ). An adder  225  sums the outputs of the multipliers  205  and  215 , and an adder  230  sums the outputs of the multipliers  210  and  220 . Delays  235  and  240  respectively delay u 1 (k) and u 2 (k) by one period of the filter clock for use by the second filter stage at the next filter clock period. 
     The second stage (not shown) receives the delayed u 1 (k) and u 2 (k), and also respectively receives the outputs of the adders  225 ,  230  at adders  245  and  250 . The second filter stage applies filter coefficients W 11 ( 1 ), W 21 ( 1 ), W 12 ( 1 ) and W 22 ( 1 ) and again delays u 1 (k) and u 2 (k). The delayed u 1 (k) and u 2 (k) and each intermediate stage output step with each period of the filter clock through the remaining L−2 filter stages to the L th  stage. Adders  255 ,  260  provide the output of the penultimate filter stage to the L th  stage. Multipliers  265  and  270  receive the delayed u 1 (k) from a delay  275 . Multipliers  280  and  285  receive the delayed u 1 (k) from a delay  290 . An adder  295  sums the outputs of the adder  255  and the multipliers  265  and  280  to produce the filter output v 1 (k). An adder  299  sums the outputs of the adder  260  and the multipliers  270  and  285  to produce the filter output v 2 (k). 
       FIG. 3  illustrates a filter updater  300  according to one embodiment, e.g. a multistage equalizer tap update module. The updater  300  may be used to update tap coefficients to the tap coefficient inputs of the filter  200 . Only the 0 th  stage is explicitly shown in  FIG. 3 . Those skilled in the pertinent art are capable of adding the intervening stages  1  through L−1. 
     In the embodiment of  FIG. 3 , a computational block  305  computes the complex conjugate of u 1 (k) and u 2 (k). Computational blocks  310  and  315  respectively compute 2·|v 1 (i)| 2 ×v 1 (i) and 2·|v 2 (i)| 2 ×v 2 (i). Computational blocks  320  and  325  respectively compute 2·(|v 1 (i)| 2 ×v 1 (i) and 2·(|v 2 (i)| 2 −1) 2 ×v 2 (i). Multipliers  330 ,  335 ,  340  and  345  respectively compute the product of the block  305  with each of the blocks  310 ,  315 ,  320  and  325 . Blocks  350  and  355  respectively compute the absolute values of v 1 (k) and v 2 (k). A selector  360  reorders the outputs from the multipliers  330 ,  335 ,  340  and  345  depending on the relative magnitudes of v 1 (k) and v 2 (k). Thus the selector  360  outputs to a multiplier  365  the left column of the matrix in Eq. 7 when |v 1 (k)|&gt;|v 2 (k)|, and the left column of the matrix in Eq. 8 when |v 2 (k)|&gt;|v 1 (k)|. Similarly, the selector  360  outputs to a multiplier  370  the right column of the matrix in Eq. 7 when |v 1 (k)|&gt;|v 2 (k)|, and the right column of the matrix in Eq. 8 when |v 2 (k)|&gt;|v 1 (k)|. 
     The multipliers  365  and  370  multiply their respective received vectors by a scalar −μ that represents a control gain. A larger value of μ results in a faster response time by the filter  200 , but may result in greater noise introduced in the filtered signal. An adder  375  sums the output of the multiplier  365  and the left column of the coefficient matrix W k ( 0 ), and an adder  380  sums the output of the multiplier  370  and the right column of the coefficient matrix W k ( 0 ). The adder  375  outputs coefficient matrix elements W 1,m   k+1 ( 0 ) (the left column) of an updated coefficient matrix W k+1 ( 0 ), and the adder  380  outputs coefficient matrix elements W 2,n   k+1 ( 0 ) (the right column) of the updated coefficient matrix. 
     Regarding the remaining L−1 tap update stages, a delay chain  385 - 1  . . .  385 (L−1) provides u n (k) . . . u n (k+L−1) to additional tap update stages. In this manner the l filter matrixes are updated for use in the i+1 st  filter clock period. 
     Frequency Estimation Module 
       FIG. 4  illustrates a computational block diagram of a method, e.g. a frequency compensator  400 , of frequency compensation of the received PS-QPSK signal. The illustrated embodiment presents a feedback architecture without limitation thereto. Those skilled in the pertinent art will appreciate that the method could be implemented using a feed-forward design with suitable modification. The frequency compensator  400  may be implemented in the frequency estimation module  160  of the optical receiver  100 . The frequency compensator  400  receives the [v 1 (k),v 2 (k)] symbol stream from, e.g. the polarization tracking and equalization module  150 . The frequency compensator  400  compensates for phase offsets between the two polarizations. A phase estimation module  500 , described below, relies on this compensation to provide a common phase estimation for both polarizations, as described further below. 
     The intermediate frequency compensation is adapted to perform this task. It uses the differential phase between two consecutive QPSK symbols as error signals for two integral controllers (one per polarization) to drive both the intermediate frequency and phase offset between the two polarizations to zero. In the case that the switch symbol is identical for two consecutive symbols, the differential phase is used to update the intermediate frequency estimate. In the case that the switch bit is different, the differential phase is used to update the phase offset. 
     In the following discussion the signal received by the frequency compensator  400  from the equalizing filter  200  is represented as a vector v k  as indicated by Eq. 9. The signal output by the frequency compensator  400  is represented as a vector x k  as indicated by Eq. 10. 
     
       
         
           
             
               
                 
                   
                     v 
                     k 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             v 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                       
                       
                         
                           
                             v 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     x 
                     k 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             x 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                       
                       
                         
                           
                             x 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The frequency compensator  400  operation is also data-dependent, via feedback from the x k  outputs to multipliers  405  and  410 . The multiplier  405  multiples v 1,k  by a first factor described below and outputs x 1,k . The multiplier  410  multiples v 2,k  by a second factor also described below and outputs x 2,k . A comparator  415  determines which of the output values x 1,k , x 2,k  is has a greater energy and outputs the larger of the two values. Eq. 11 describes this selection. The comparator  415  also outputs a tag i identifying the larger datum. This tag may have a value of 1 or 2 as described by Eq. 12. 
     
       
         
           
             
               
                 
                   
                     x 
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             x 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     1 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                       
                         
                           
                             x 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     2 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     i 
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     1 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     2 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     A computational block  420  computes an instantaneous differential phase Δφ k  described by Eq. 13, in which x k  and x k−1  are each raised to the fourth power. 
     
       
         
           
             
               
                 
                   
                     Δφ 
                     k 
                   
                   = 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     arg 
                     ⁢ 
                     
                       { 
                       
                         
                           ( 
                           
                             x 
                             k 
                             4 
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               x 
                               
                                 k 
                                 - 
                                 1 
                               
                               4 
                             
                             ) 
                           
                           * 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The intermediate frequency estimation is updated as follows. A comparator  425  determines if the tag i (1 or 2) of a datum at time index k−1 is equal to the tag of the next datum at time index k. If the tags are equal, then the comparator  425  selects a switch  430   a . If the tags are not equal then the comparator  425  selects a switch  430   b.    
     A multiplier  435  computes the product of μ f  and Δφ k , where μ f  is a first adjustable control gain. A multiplier  440  computes the product of μ φ  and Δφ k , where μ φ  is a second adjustable control gain. A multiplier  445  multiplies μ φ Δφ k  by a sign computed by a signum function  450  that outputs +1 if i k =1 and i k−1 =2, and outputs −1 if i k−1 =2 and i k−1 =1. Thus, the multiplier  445  outputs ±μ φ Δφ k . 
     The operation of the switch  430   a , multiplier  435  and a delay loop  460  implements Eq. 14: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               f 
                             
                             + 
                             
                               
                                 μ 
                                 f 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ϕ 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 i 
                                 k 
                               
                             
                             = 
                             
                               i 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 i 
                                 k 
                               
                             
                             ≠ 
                             
                               i 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The operation of the switch  430   b , the multiplier  445  and a delay loop  462  implements Eq. 15: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       φ 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 k 
                               
                             
                             + 
                             
                               
                                 μ 
                                 f 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 i 
                                 k 
                               
                             
                             = 
                             
                               
                                 1 
                                 ⋂ 
                                 
                                   i 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               = 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 k 
                               
                             
                             - 
                             
                               
                                 μ 
                                 f 
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 i 
                                 k 
                               
                             
                             = 
                             
                               
                                 2 
                                 ⋂ 
                                 
                                   i 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               = 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               φ 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 i 
                                 k 
                               
                             
                             = 
                             
                               i 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Sigma block  455  computes a running total Σ(Δf k ). A multiplier  465  scales Δφ k  by ½. Adders  470  and  475  respectively compute the sum and difference of Σ(Δf n ) and ½Δφ k . An exponentiation block  480  computes the exponential of the adder  470  output scaled by −j, while an exponentiation block  485  computes the exponential of the adder  475  output scaled by −j. The multiplier  405  computes the product of v 1,k  and the result from the block  480  to produce x 1,k . The multiplier  410  computes the product of v 2,k  and the result from the block  485  to produce x 2,k . The described operation of the elements  405 ,  410 ,  465 ,  470 ,  475 ,  480  and  485  implement Eq. 16. 
     
       
         
           
             
               
                 
                   
                     x 
                     
                       1 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       v 
                       
                         1 
                         , 
                         k 
                       
                     
                     ⁢ 
                     exp 
                     ⁢ 
                     
                       { 
                       
                         - 
                         
                           j 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ± 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ϕ 
                                       n 
                                     
                                   
                                   2 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   k 
                                 
                                 ⁢ 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     f 
                                     n 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Phase Estimation Module 
     Referring back to  FIG. 1 , embodiments of the phase estimation module  170  are now described.  FIG. 5  illustrates a computational block diagram of a method, e.g. a phase compensator  500 , of performing phase compensation on the received PS-QPSK signal. The phase compensator  500  takes into account the switch symbol in PS-QPSK symbol pair. Because the switch symbol essentially has no phase (e.g. a null signal), the switch symbol cannot typically be used to estimate frequency and phase offsets. In the case of phase recovery, the received QPSK point from each symbol time is extracted into a single data stream, and recovery is performed using this data stream. This typically requires that phase offsets between the two polarizations (introduced, e.g. by the equalization module  150 ) are compensated beforehand as previously described. 
     The phase compensator  500  receives the output 
               x   k     =     [           x     1   ,   k                 x     2   ,   k             ]           
from the frequency estimation module  160 , e.g. executing the frequency compensator  400 , outputs a phase-compensated data stream
 
     
       
         
           
             
               y 
               k 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         y 
                         
                           1 
                           , 
                           k 
                         
                       
                     
                   
                   
                     
                       
                         y 
                         
                           2 
                           , 
                           k 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     A comparator  505  selects the QPSK symbol, x 1,k  or x 2,k , that has the higher energy (Eq. 17). 
     
       
         
           
             
               
                 
                   
                     x 
                     k 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             x 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     1 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                       
                         
                           
                             x 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     2 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             ≥ 
                             
                                
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     A block  510  computes an estimated phase φ k  with a filter half-width of M. The computational algorithm is illustrated by, e.g. Eq. 18. A module  515  computes x k   4  and provides this value to a first delay module  520 - 1 . The module  515  then computes x k+1   4  and provides this value to the first delay module  520 - 1  as the first delay module  520 - 1  passes x k   4  to a second delay module (not shown). This operation is repeated for an additional 2M-2 symbols. When x k   4  reaches the delay module  520 - 2 M, an adder  525  computes a summation of the values output by each delay module  520 . A module  530  computes the arg{ } function of the negated summation and scales the result by ¼. 
     
       
         
           
             
               
                 
                   
                     φ 
                     k 
                   
                   = 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     arg 
                     ⁢ 
                     
                       { 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               M 
                             
                           
                           M 
                         
                         ⁢ 
                         
                           x 
                           
                             k 
                             - 
                             m 
                           
                           4 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     A module  535  then computes the exponential of the estimated phase φ k  scaled by −j. Multipliers  540  and  545  respectively compute the product of x 1,k  and x 2,k  and the output of the module  535 . Delay modules  550 ,  555  respectively align x 1,k  and x 2,k  with the computation by the block  510 . The output of the method phase compensator  500  is shown by Eq. 19. 
     
       
         
           
             
               
                 
                   
                     y 
                     k 
                   
                   = 
                   
                     
                       ⌊ 
                       
                         
                           
                             
                               x 
                               
                                 1 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               x 
                               
                                 2 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       ⌋ 
                     
                     ⁢ 
                     exp 
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           φ 
                           k 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     The described method of carrier phase recovery may be extended to cases of nonlinear noise.  FIG. 6  illustrates a method, e.g. a phase compensator  600 , in an illustrative embodiment. In  FIG. 6  the input and output data streams are again designated 
     
       
         
           
             
               x 
               k 
             
             = 
             
               
                 
                   x 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             v 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                       
                       
                         
                           
                             x 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   y 
                   k 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           y 
                           
                             1 
                             , 
                             k 
                           
                         
                       
                     
                     
                       
                         
                           y 
                           
                             2 
                             , 
                             k 
                           
                         
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     The QPSK symbol with the greater energy is selected, as described by Eqs. 20 and 21, with the lower-energy symbol being set to zero. The resulting data vector is designated 
     
       
         
           
             
               
                 x 
                 ~ 
               
               k 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           x 
                           ~ 
                         
                         
                           1 
                           , 
                           k 
                         
                       
                     
                   
                   
                     
                       
                         
                           x 
                           ~ 
                         
                         
                           2 
                           , 
                           k 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       x 
                       ~ 
                     
                     
                       1 
                       , 
                       k 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             x 
                             
                               1 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     1 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     1 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             ≤ 
                             
                                
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       ~ 
                     
                     
                       2 
                       , 
                       k 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             x 
                             
                               2 
                               , 
                               k 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     2 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             &gt; 
                             
                                
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                  
                                 
                                   x 
                                   
                                     2 
                                     , 
                                     k 
                                   
                                 
                                  
                               
                             
                             ≤ 
                             
                                
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                                
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Thus, a comparator module  604  provides {tilde over (x)} 1,k  when |x 1,k |&gt;|x 2,k |, and a comparator module  608  provides {tilde over (x)} 2,k  when |x 2,k |&gt;| 1,k |. 
     Equations 22 and 23 describe a phase estimate determined as a function of {tilde over (x)} 1,k  and {tilde over (x)} 2,k . The equations apply a filter half width of M, and include a correlation factor c that determines a contribution from x 2,k  to φ 1,k  and a contribution of x 1,k  to φ 2,k . 
     
       
         
           
             
               
                 
                   
                     φ 
                     
                       1 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     arg 
                     ⁢ 
                     
                       { 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               M 
                             
                           
                           M 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 1 
                                 , 
                                 
                                   k 
                                   - 
                                   m 
                                 
                               
                               4 
                             
                             + 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   x 
                                   ~ 
                                 
                                 
                                   2 
                                   , 
                                   
                                     k 
                                     - 
                                     m 
                                   
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     φ 
                     
                       2 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     arg 
                     ⁢ 
                     
                       { 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               M 
                             
                           
                           M 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 x 
                                 ~ 
                               
                               
                                 2 
                                 , 
                                 
                                   k 
                                   - 
                                   m 
                                 
                               
                               4 
                             
                             + 
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   v 
                                   
                                     ~ 
                                     ~ 
                                   
                                 
                                 
                                   1 
                                   , 
                                   
                                     k 
                                     - 
                                     m 
                                   
                                 
                                 4 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The phase-compensated symbol stream 
               y   k     =     [           y     1   ,   k                 y     2   ,   k             ]           
is then computed as shown in Eq. 24.
 
     
       
         
           
             
               
                 
                   
                     y 
                     k 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               
                                 1 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               
                                 2 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 
                                   1 
                                   , 
                                   k 
                                 
                               
                               ⁢ 
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     φ 
                                     
                                       1 
                                       , 
                                       k 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 
                                   2 
                                   , 
                                   k 
                                 
                               
                               ⁢ 
                               exp 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     φ 
                                     
                                       2 
                                       , 
                                       k 
                                     
                                   
                                 
                                 } 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Referring to  FIG. 6 , modules  612  and  616  respectively compute the fourth power of {tilde over (x)} 1,k  and {tilde over (x)} 2,k . Delay modules  620 - 1  . . .  620 - 2 M sequentially delay {tilde over (x)} 1,k , with outputs of each delay module being summed by an adder  624 . Delay modules  628 - 1  . . .  628 - 2 M and an adder  632  operate analogously with respect to {tilde over (x)} 2,k . A multiplier  636  scales the output of the adder  624  by the correlation factor c, with the scaled output being combined with the output of the adder  632  by an adder  640 . Similarly, a multiplier  644  scales the output of the adder  632  by the correlation factor c, with the scaled output being combined with the output of the adder  624  by an adder  648 . 
     A module  652  computes the arg{ } function of the output of the adder  648  and scales this value by ¼. A module  656  scales the output of the module  652  by −j and computes the exponential of the scaled value. A multiplier  660  computes the product of the exponential value from the module  656  and {tilde over (x)} 1,k  delayed by M clock cycles by a delay module  664  to produce y 1,k . 
     Similarly, a module  668  computes the arg{ } function of the output of the adder  640  and scales this value by ¼. A module  672  scales the output of the module  668  by −j and computes the exponential of the scaled value. A multiplier  676  computes the product of the exponential value from the module  672  and {tilde over (x)} 2,k  delayed by M clock cycles by a delay module  680  to produce y 2,k . 
     Decision Module 
     The decision module  180  receives the vector 
               y   k     =     ⌊           y     1   ,   k                 y     2   ,   k             ⌋           
from the phase estimation module  170 , where k is the time index. The vector components can each be viewed in terms of in-phase and quadrature components, e.g.
 
 y   2   =y   1i   +jy   1q   (25)
 
 y   2   =y   2i   +jy   2q   (26)
 
     where the subscripts i and q respectively denote in-phase and quadrature components, and the time index k is assumed. The vector y can be viewed as a four-dimensional vector in a 4-D space. 
     The ML detection scheme determines the symbol ŷ represented by the vector y. There are eight possible symbols in PS-QPSK represented by a constellation Y, where Y={(0,1+j), (0,1−j), (0,−1+j), (0,−1−j) (1+j,0), (1−j,0) (−1+j,0), (−1−j,0)}, where the customary normalization factor √2/2 is omitted. 
     The ML decision is formulated as 
     
       
         
           
             
               
                 
                   
                     y 
                     ^ 
                   
                   = 
                   
                     arg 
                     ⁡ 
                     
                       ( 
                       
                         
                           max 
                           
                             z 
                             ∈ 
                             Y 
                           
                         
                         ⁢ 
                         
                           p 
                           ⁡ 
                           
                             ( 
                             
                               y 
                               | 
                               z 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     where p(y|z) is the conditional probability density function of y given z. It can be shown that in presence of independent additive white Gaussian noise (AWGN) in each of the four dimensions of the 4-D vector space, the ML detection problem reduces to 
     
       
         
           
             
               
                 
                   
                     y 
                     ^ 
                   
                   = 
                   
                     arg 
                     ⁡ 
                     
                       ( 
                       
                         
                           min 
                           
                             z 
                             ∈ 
                             Y 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               y 
                               - 
                               z 
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     or restated, 
     
       
         
           
             
               
                 
                   
                     y 
                     ^ 
                   
                   = 
                   
                     arg 
                     ⁢ 
                     
                       ⌊ 
                       
                         
                           min 
                           
                             z 
                             ∈ 
                             Y 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     
                                       1 
                                       ⁢ 
                                       i 
                                     
                                   
                                   - 
                                   
                                     z 
                                     
                                       1 
                                       ⁢ 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     
                                       1 
                                       ⁢ 
                                       q 
                                     
                                   
                                   - 
                                   
                                     z 
                                     
                                       1 
                                       ⁢ 
                                       q 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                   
                                   - 
                                   
                                     z 
                                     
                                       2 
                                       ⁢ 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     
                                       2 
                                       ⁢ 
                                       q 
                                     
                                   
                                   - 
                                   
                                     z 
                                     
                                       2 
                                       ⁢ 
                                       q 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       ⌋ 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     In another embodiment, the symbol detection is performed using a data-dependent technique, in which the symbol decision depends on the relative energy of the received symbols y 1  and y 2 . An value associated with the energy may be calculated for each symbol as follows:
 
 e   1   =y   i1   2   +y   q1   2   (30)
 
 e   2   =y   i2   2   +y   q2   2   (31)
 
     If e 1 &gt;e 2 , then the method assumes that y 2 =0 (switch symbol), and determines y 1  as for a normal QPSK symbol decision. If e 2 &gt;e 1  then the method analogously determines the value of y 2 . 
     In some cases the ML detection method is preferred over the data-dependent method, as the ML method provides a more optimized detection. For example, at lower Signal to Noise Ratios (SNRs), the ML method provides more accurate data detection. This is because it is the optimal decision method, e.g. in an additive white Gaussian noise channel. At high SNRs, the performance difference between the ML and data-dependent methods is expected to be very close. With certain low-probability noise realizations, the data-dependent method can be incorrect while the ML-detection chooses the right symbols. This requires a somewhat larger magnitude of noise, which is more likely to occur at lower SNRs. 
     Turning to  FIG. 7 , illustrated is a simulated bit error rate (BER) characteristic as a function of electrical signal-to-noise ratio (SNR) for two cases. The characteristic traced with a dashed line corresponds to a representative conventional PDM-QPSK implementation. The characteristic traced by the solid line corresponds to PDM-QPSK implemented as by various embodiments of PS-QPSK described herein. It can be seen that the BER of the PS-QPSK method is lower for all values of SNR within the simulation space, indicating greater noise tolerance of embodiments of the invention relative to PDM-QPSK modulation. It is noted that the overall data rate is lower for PS-QPSK. This illustrated simulation assumes that both signals have the same bandwidth, but in this case, PS-QPSK has 75% of the data rate of PDM-QPSK. However, as described previously, the sacrifice in data rate is balanced by having greater noise immunity (e.g., higher sensitivity), which will allow transmission of longer distances. 
       FIG. 8  presents method  800 , e.g. for forming an optical receiver system such as the system  100 . The steps of the method  800  are described without limitation by reference to elements previously described herein, e.g. in  FIGS. 1-6 . The steps of the method  800  may be performed in another order than the illustrated order, and in some embodiments may be omitted altogether. 
     In a step  810  an optical front end, e.g. the receiver  100 , is configured to receive an optical signal bearing first and second bits on respective first and second polarization channels. In a step  820  an equalizer, e.g. the polarization tracking and equalization module  150 , is configured to 1) select a first cost function if said first bit has greater energy than said second bit, 2) elect a second different cost function if said second bit has a greater energy than said first bit, and 3) update coefficients of an adaptive filter configured to demultiplex and equalize said first and second polarization channels. 
     In a step  830  a frequency compensator, e.g. the frequency estimation module  160 , is configured to update a frequency estimate of said first and second polarization channels if said first bit has greater energy than said second bit for two adjacent bit periods. 
     In a step  840  a phase compensator, e.g. the phase estimation module  170 , is configured to compute an estimated phase of said optical signal based on said first bit if said first bit has a greater energy than said second bit, and to compute said estimated phase based on said second bit if said second bit has a greater energy than said first bit. 
     In a step  850  the phase compensator is configured to estimate a phase of said first bit based on a first sequence of bits received on said first polarization channel and a second sequence of bits received on said second polarization channel. 
     In a step  860  a programmable gate array is configured to implement said equalizer module. 
     Those skilled in the art to which this application relates will appreciate that other and further additions, deletions, substitutions and modifications may be made to the described embodiments.