Patent Publication Number: US-10763972-B2

Title: Timing recovery for optical coherent receivers in the presence of polarization mode dispersion

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 16/212,461 filed Dec. 6, 2018, which is a continuation of U.S. patent application Ser. No. 15/839,698 filed Dec. 12, 2017 (now U.S. Pat. No. 10,181,908 issued Jan. 15, 2019), which is a continuation of U.S. patent application Ser. No. 15/623,292 entitled “Timing Recovery for Optical Coherent Receivers in the Presence of Polarization Mode Dispersion” filed on Jun. 14, 2017 (now U.S. Pat. No. 9,876,583 issued Jan. 23, 2018), which is a continuation of U.S. patent application Ser. No. 14/869,676 entitled “Timing Recovery for Optical Coherent Receivers in the Presence of Polarization Mode Dispersion” filed on Sep. 29, 2015 (now U.S. Pat. No. 9,712,253 issued on Jul. 18, 2017), which is a continuation of U.S. patent application Ser. No. 14/095,789 entitled “Timing Recovery for Optical Coherent Receivers in the Presence of Polarization Mode Dispersion” filed on Dec. 3, 2013 (now U.S. Pat. No. 9,178,625, issued on Nov. 3, 2015), which claims the benefit of the follow U.S. Provisional Applications: U.S. Provisional Application No. 61/732,885 entitled “Timing Recovery for Optical Coherent Receivers In the Presence of Polarization Mode Dispersion” filed on Dec. 3, 2012 to Mario R. Hueda, et al.; U.S. Provisional Application No. 61/749,149 entitled “Timing Recovery for Optical Coherent Receivers In the Presence of Polarization Mode Dispersion” filed on Jan. 4, 2013 to Mario R. Hueda, et al.; U.S. Provisional Application No. 61/832,513 entitled “Timing Recovery for Optical Coherent Receivers In the Presence of Polarization Mode Dispersion” filed on Jun. 7, 2013 to Mario R. Hueda, et al.; and U.S. Provisional Application No. 61/893,128 entitled “Timing Recovery for Optical Coherent Receivers In the Presence of Polarization Mode Dispersion” filed on Oct. 18, 2013 to Mario R. Hueda, et al. The contents of each of the above referenced applications are incorporated by reference herein. 
    
    
     BACKGROUND 
     1. Field of the Art 
     The disclosure relates generally to communication systems, and more specifically, to timing recovery in an optical receiver. 
     2. Description of the Related Art 
     The most recent generation of high-speed optical transport network systems has widely adopted receiver technologies with electronic dispersion compensation (EDC). In coherent as well as in intensity modulation direct detection (IM-DD) receivers, EDC mitigates fiber impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD). Timing recovery (TR) in the presence of differential group delay (DGD) caused by PMD has been identified as one of the most critical challenges for intradyne coherent receivers. This can result in the receiver failing to recover data received over the fiber channel, thereby decreasing performance of the optical network system. 
     SUMMARY 
     A receiver performs a timing recovery method to recover timing of an input signal. The input signal is received over an optical communication channel. In an embodiment, the optical channel may introduce an impairment into the input signal including at least one: a half symbol period differential group delay, a cascaded differential group delay, a dynamic polarization mode dispersion, and a residual chromatic dispersion. The receiver samples the input signal based on a sampling clock. The receiver generates a timing matrix representing coefficients of a timing tone detected in the input signal. The timing tone representing frequency and phase of a symbol clock of the input signal and has a non-zero timing tone energy. The receiver computes a rotation control signal based on the timing matrix that represents an accumulated phase shift of the input signal relative to the sampling clock. A numerically controlled oscillator is controlled to generate the sampling clock based on the rotation control signal. 
     In one embodiment, the receiver includes a resonator filter to filter the input signal to generate a band pass filtered signal. An in-phase and quadrature error signal is computed based on the band pass filter signal. The in-phase and quadrature error signal represents an amount of phase error in each of an in-phase component and a quadrature component of the input signal. 
     The rotation control signal is computed based on the in-phase and quadrature error signal. In various embodiments, a determinant method or a modified wave difference method can be used to determine the rotation control signal based on the timing matrix. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention has other advantages and features which will be more readily apparent from the following detailed description of the invention and the appended claims, when taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a system diagram of an embodiment of an optical communication system. 
         FIG. 2  is a plot illustrating an effect of half baud DGD in an uncompensated optical receiver where a single polarization is used for timing recovery. 
         FIG. 3  is a plot illustrating an effect of half baud DGD in an uncompensated optical receiver where two polarizations are used for timing recovery. 
         FIG. 4  is a diagram modeling PMD effects in an optical channel. 
         FIG. 5  is a block diagram illustrating an embodiment of a timing recovery system. 
         FIG. 6  is a block diagram illustrating an embodiment of a resonator filter for a timing recovery system. 
         FIG. 7  is a block diagram illustrating an embodiment of an in-phase and quadrature error computation block filter for a timing recovery system. 
         FIG. 8A  is a block diagram illustrating an embodiment of a phase error computation block for a timing recovery system that applies a determinant method. 
         FIG. 8B  is a block diagram illustrating an embodiment of a phase error computation block for a timing recovery system that applies a modified wave difference method. 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     A receiver architecture and method for timing recovery is described for transmissions received over an optical fiber channel in the presence of differential group delay (DGD) (caused, for example, by polarization mode dispersion) that affects the detectability of a reliable timing tone. Timing recovery can then be performed on the transformed signal to recover a clock signal. 
     High Level System Architecture 
       FIG. 1  is a block diagram of a communication system  100 . The communication system  100  comprises a transmitter  110  for encoding data as an electrical signal, an optical transmitter  120  for converting the electrical signal produced by the transmitter  110  to an optical signal suitable for transmission over a communication channel  130 , an optical front end  150  for converting the received optical signal to an electrical signal, and a receiver for receiving and processing the electrical signal encoding the data from the optical front end  150 . In one embodiment, the communication system  100  comprises an ultra-high speed (e.g., 40 Gb/s or faster) optical fiber communication system, although the described techniques may also be applicable to lower speed optical communication systems. 
     The transmitter  110  comprises an encoder  112 , a modulator  114 , a transmitter (Tx) digital signal processor (DSP)  116 , and Tx analog front end (AFE)  118 . The encoder  112  receives input data  105  and encodes the data for transmission over the optical network. For example, in one embodiment, the encoder  112  encodes the input data  105  using forward error correction (FEC) codes that will enable the receiver  160  to detect, and in many cases, correct errors in the data received over the channel  130 . The modulator  114  modulates the encoded data via one or more carrier signals for transmission over the channel  130 . For example, in one embodiment, the modulator  114  applies phase-shift keying (PSK) or differential phase-shift keying (DPSK) to the encoded data. The Tx DSP  116  adapts (by filtering, etc.) the modulator&#39;s output signal according to the channel characteristics in order to improve the overall performance of the transmitter  110 . The Tx AFE  118  further processes and converts the Tx DSP&#39;s digital output signal to the analog domain before it is passed to the optical transmitter (Optical Tx)  120  where it is converted to an optical signal and transmitted via the channel  130 . One example of the optical transmitter  120  transmits independent modulations on both polarizations of the optical carrier. An example modulation is QPSK, though other modulations can be used, and the choice can be made to transmit on either one or both polarizations. 
     In addition to the illustrated components, the transmitter  110  may comprise other conventional features of a transmitter  110  which are omitted from  FIG. 1  for clarity of description. Furthermore, in one embodiment, the transmitter  110  is embodied as a portion of a transceiver device that can both transmit and receive over the channel  130 . 
     The channel  130  may have a limited frequency bandwidth and may act as a filter on the transmitted data. Transmission over the channel  130  may add noise to the transmitted signal including various types of random disturbances arising from outside or within the communication system  100 . Furthermore, the channel  130  may introduce fading and/or attenuation effects to the transmitted data. Additionally, the channel  130  may introduce chromatic dispersion (CD) and polarization mode dispersion (PMD) effects that cause a spreading of pulses in the channel  130 . Based on these imperfections in the channel  130 , the receiver  160  is designed to process the received data and recover the input data  105 . 
     In general, the optical front end  150  receives the optical signal, converts the optical signal to an electrical signal, and passes the electrical signal to the receiver  160 . The receiver  160  receives the encoded and modulated data from the transmitter  110  via the optical transmitter  120 , communication channel  130 , and optical front end  150 , and produces recovered data  175  representative of the input data  105 . The receiver  160  includes a receiver (Rx) analog front end (AFE)  168 , an RX DSP  166 , a demodulator  164 , and a decoder  162 . The Rx AFE  168  samples the analog signal from the optical front end  150  based on a clock signal  181  to convert the signal to the digital domain. The Rx DSP  166  further processes the digital signal by applying one or more filters to improve signal quality. As will be discussed in further detail below, the Rx DSP  166  includes a timing recovery block  179  that operates to generate the sampling clock  181  and to adjust the sampling frequency and phase of the sampling clock signal  181  to ensure that the sampling clock remains synchronized with the symbol rate and phase of the incoming optical signal. This timing recovery problem becomes challenging due to the imperfections in the channel  130  that may alter the received optical signal. For example, chromatic dispersion (CD) and polarization mode dispersion (PMD) effects may cause a spreading of pulses in the channel  130 , thereby increasing the difficulty of timing recovery, as will be explained below. 
     The demodulator  164  receives the modulated signal from the Rx DSP  166  and demodulates the signal. The decoder  162  decodes the demodulated signal (e.g., using error correction codes) to recover the original input data  105 . 
     In addition to the illustrated components, the receiver  160  may comprise other conventional features of a receiver  160  which are omitted from  FIG. 1  for clarity of description. Furthermore, in one embodiment, the receiver  160  is embodied as a portion of a transceiver device that can both transmit and receive over the channel  130 . 
     Components of the transmitter  110  and the receiver  160  described herein may be implemented, for example, as an integrated circuit (e.g., an Application-Specific Integrated Circuit (ASIC) or using a field-programmable gate array (FPGA), in software (e.g., loading program instructions to a processor (e.g., a digital signal processor (DSP)) from a computer-readable storage medium and executing the instructions by the processor), or by a combination of hardware and software. 
     Impact of Channel Impairments on Timing Recovery 
     In order for the timing recovery block  179  to properly generate the sampling clock  181 , a timing tone is detected in the digital input signal that will ideally appear at a frequency representative of the symbol rate. A common technique for timing recovery is the nonlinear spectral line method as described in J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication. KAP, third ed. 2004. In this method, a timing tone is detected in the digital input signal and the frequency of the timing tone is used to synchronize the sampling clock  181  to the incoming signal. However, under certain conditions, the timing tone can be lost or offset when using a conventional spectral line method timing recovery technique. For example, the timing tone may disappear in the presence of half-baud DGD. Furthermore, when a cascaded DGD is present, the nonlinear spectral line method may generate a timing tone with a frequency offset which causes a loss of synchronization. Other channel conditions such as general dynamic PMD and/or residual chromatic dispersion can furthermore cause the traditional spectral line method to fail due to frequency or phase offset between the symbol clock and the sampling clock. 
     The following description explains how the PMD is expressed mathematically, which provides a basis for an explanation of the effects of particular PMD on the timing information in the received signal. The signal from the optical front end  150  presented to the receiver  160  consists of electrical signals from both polarizations of the optical signal received by the optical front end  150 . These two polarizations can be treated mathematically as a two-dimensional complex vector, where each component corresponds to one of the polarizations of the received optical signal. Alternatively, the two polarizations can be treated mathematically as a four-dimensional real vector, where two of the dimensions correspond to the in-phase and quadrature components of one polarization, and the other two components correspond to the in-phase and quadrature components of the other polarization. 
     This example demonstrates the embodiment where the two polarizations at the transmitter are each modulated independently. Let {a k } and {b k } respectively represent the symbol sequences transmitted on the horizontal and vertical polarizations of the optical signal, where the symbols are in general complex. For this example, it is assumed that a k  and b k  are independent and identically distributed complex data symbols with E{a k a m *}=E{b k b m *)}=δ m-k  where δ k  is the discrete time impulse function and E{.} denotes expected value. Let X(ω) be the Fourier transform of the channel input 
                 ∑   k     ⁢       [           a   k               b   k           ]     ⁢     δ   ⁡     (     t   -   kT     )           ,         
where δ(t) is the continuous time impulse function (or delta function), and T is the symbol period (also called one baud). Let S(ω) be the Fourier transform of the transmit pulse, s(t). In the presence of CD and PMD, the channel output can be written as H(ω)X(ω), with the channel transfer matrix expressed as
 
 H (ω)= e   −jβ(ω)L   J (ω) S (ω),  (1)
 
where ω is the angular frequency, L is the fiber length of the optical channel  130 , β(ω) is the CD parameter,
 
                       S   ⁡     (   ω   )       =     [           S   ⁡     (   ω   )           0           0         S   ⁡     (   ω   )             ]       ,           (   2   )               
and J(ω) is the Jones matrix. The components of J(ω) are defined by
 
                       J   ⁡     (   ω   )       =     [           U   ⁡     (   ω   )             V   ⁡     (   ω   )                 -       V   *     ⁡     (   ω   )                 U   *     ⁡     (   ω   )             ]       ,           (   3   )               
where * denotes complex conjugate. Matrix J(ω) is special unitary (i.e., J(ω) H  J(ω)=I, det(J(ω))=1, where I is the 2×2 identity matrix and H denotes conjugate transpose) and models the effects of the PMD. For example, the Jones matrix for first-order PMD reduces to
 
                     J   ⁡     (   ω   )       =       R   ⁡     (       θ   0     ⁢     ϕ   0       )       ⁡     [           e       j   ⁢           ⁢   ω   ⁢     τ   2       +     j   ⁢       ψ   0     2               0           0         e         -   j     ⁢           ⁢   ω   ⁢     τ   2       -     j   ⁢       ψ   0     2                 ]               (   4   )               
where τ is the differential group delay (DGD), ψ 0  is the polarization phase, and R(.,.) is the rotation matrix given by
 
                     R   ⁡     (     θ   ,   ϕ     )       =       [           e     j   ⁢           ⁢     ϕ   2             0           0         e       -   j     ⁢           ⁢     ϕ   2               ]     ⁡     [           cos   ⁡     (   θ   )             sin   ⁡     (   θ   )                 -     sin   ⁡     (   θ   )               cos   ⁡     (   θ   )             ]               (     5   ⁢   a     )               
where θ is the polarization angle and ϕ is a random phase angle. Note that first order DGD may be more generally expressed as:
 
                       R   ⁡     (       θ   0     ⁢     ϕ   0       )       ⁡     [           e       j   ⁢           ⁢   ω   ⁢     τ   2       +     j   ⁢       ψ   0     2               0           0         e         -   j     ⁢     τ   2       -     j   ⁢       ψ   0     2                 ]       ⁢       R     -   1       ⁡     (       θ   1     ⁢     ϕ   1       )               (     5   ⁢   b     )               
but the last rotation matrix does not affect the strength of the timing tone, so the simpler form given in (4) is used herein.
 
     Assuming that CD is completely compensated in the receiver Rx DSP  166  of the receiver  160  or prior to the signal reaching the receiver Rx DSP  166 , (i.e., H(ω)=J(ω)S(ω)), the noiseless signal entering the timing recovery block  179  can be expressed as: 
                       r   ⁡     (   t   )       =       [             r   x     ⁡     (   t   )                   r   y     ⁡     (   t   )             ]     =     [               ∑     k   =     -   ∞       ∞     ⁢         h   11     ⁡     (     t   -   kT     )       ⁢     a   k         +         h   12     ⁡     (     t   -   kT     )       ⁢     b   k                       ∑     k   =     -   ∞       ∞     ⁢         h   21     ⁡     (     t   -   kT     )       ⁢     a   k         +         h   22     ⁡     (     t   -   kT     )       ⁢     b   k               ]         ,           (   6   )               
where 1/T is the symbol rate, while
 
 h   11 ( t )=   −1   {U (ω) S (ω)}, h   12 ( t )=   −1   {V (ω) S (ω)},
 
 h   21 ( t )=   −1   {−V *(ω) S (ω)}, h   22 ( t )=   −1   {U *(ω) S (ω))},  (7)
 
where    −1 {.} denotes the inverse Fourier transform.
 
     In the nonlinear spectral line method for timing recovery, the timing recovery block  179  processes the received signal by a memoryless nonlinearity in order to generate a timing tone with frequency 1/T. A magnitude squared nonlinearity is used as the memoryless nonlinearity applied to the received signal. Then, the mean value of the magnitude squared of the received signal is periodic with period T and can be expressed through a Fourier series as 
     
       
         
           
             
               
                 
                   
                       
                   
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     The timing information can be extracted either from the periodic signal derived from the received x polarization (E {|r x (t)| 2 }), or from the periodic signal derived from the received y polarization (E {|r y (t)| 2 }), or from some combination of these two periodic signals. Additional details regarding the nonlinear spectral line method for timing recovery is described in J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication. KAP, third ed. 2004. 
     Timing Information Using One Polarization 
     The following description explains the effects of PMD when one polarization is used for timing recovery and gives example conditions where the timing recovery information can disappear. From (8)-(9), it can be seen that the clock signal  181  will be different from zero in |r x (t)| 2  (or |r y (t)| 2 ) if the magnitude of the Fourier coefficient |z x,1 |&gt;0 (or |z y,1 |&gt;0). On the other hand, from (10) and (11) it can be verified that the timing tone in each polarization component will be zero if 
                       V   ⁡     (   ω   )       =       e     j   ⁢           ⁢   ψ       ⁢     e       ±           ⁢   j     ⁢           ⁢   ω   ⁢           ⁢     T   2         ⁢     U   ⁡     (   ω   )           ,           (   12   )               
where ψ is an arbitrary phase. Since the Jones matrix is special unitary (so |U(ω)| 2 +|V(ω)| 2 =1), the following expression can be derived from (12)
 
     
       
         
           
             
               
                 
                   
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     Let γ(ω)/2 be the phase response of U(ω). Then, from (12) and (13), the clock signal disappears when the Jones matrix representing the PMD at the input of the timing recovery block  179  can be expressed as 
     
       
         
           
             
               
                 
                   
                     
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                                           ( 
                                           ω 
                                           ) 
                                         
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     
                                       
                                         γ 
                                         ⁡ 
                                         
                                           ( 
                                           ω 
                                           ) 
                                         
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     1 
                                     
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                       
                                         γ 
                                         ⁡ 
                                         
                                           ( 
                                           ω 
                                           ) 
                                         
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                       
                                         γ 
                                         ⁡ 
                                         
                                           ( 
                                           ω 
                                           ) 
                                         
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 
                                   
                                     
                                       ± 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                       T 
                                       4 
                                     
                                   
                                   + 
                                   
                                     j 
                                     ⁢ 
                                     
                                       ψ 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 e 
                                 
                                   
                                     
                                       ∓ 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                       T 
                                       4 
                                     
                                   
                                   - 
                                   
                                     j 
                                     ⁢ 
                                     
                                       ψ 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           R 
                           ⁡ 
                           
                             ( 
                             
                               
                                 π 
                                 / 
                                 4 
                               
                               , 
                               
                                 γ 
                                 ⁡ 
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   e 
                                   
                                     
                                       
                                         ± 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ω 
                                       ⁢ 
                                       
                                         T 
                                         4 
                                       
                                     
                                     + 
                                     
                                       j 
                                       ⁢ 
                                       
                                         ψ 
                                         2 
                                       
                                     
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   e 
                                   
                                     
                                       
                                         ∓ 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ω 
                                       ⁢ 
                                       
                                         T 
                                         4 
                                       
                                     
                                     - 
                                     
                                       j 
                                       ⁢ 
                                       
                                         ψ 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     For example, the impact of the first-order PMD defined by (4) is analyzed in  FIG. 2 . Here, the plot illustrates the normalized magnitude of the timing tone coefficient z x,1  derived from (10), versus DGD (τ) and the rotation angle (θ) assuming an ideal lowpass pulse s(t) with bandwidth excess &lt;100%. In particular, for τ=T/2, and θ 0 =π/4, the following expression can be derived from (4): 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           
                             
                               π 
                               / 
                               4 
                             
                             , 
                             
                               ϕ 
                               0 
                             
                           
                           ) 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                       T 
                                       4 
                                     
                                   
                                   + 
                                   
                                     j 
                                     ⁢ 
                                     
                                       ψ0 
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 e 
                                 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                       T 
                                       4 
                                     
                                   
                                   - 
                                   
                                     j 
                                     ⁢ 
                                     
                                       ψ0 
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The matrix in (15) can be written as (14) (with γ(ω)=ϕ 0  and ψ=ψ 0 ). It can be inferred from the equations above that the clock signal may disappear with half-baud DGD, as illustrated in  FIG. 2 . 
     Timing Information Using Two Polarizations 
     The effects of PMD are now described when both polarizations are used for timing recovery and conditions where the timing recovery information can disappear are presented. In particular, the sum of the squared signals of both polarizations may be used for timing recovery (i.e., |r x (t)| 2 +|r y (t)| 2 ). Then, the total timing tone coefficient z x+y,1  can be expressed as: 
                       z       x   +   y     ,   1       =         z     x   ,   1       +     z     y   ,   1         =         ∫     -   ∞     ∞     ⁢     2   ⁢   ⁢     {       U   ⁡     (   ω   )       ⁢       U   *     ⁡     (     ω   -       2   ⁢   π     T       )         }     ⁢     S   ⁡     (   ω   )       ⁢       S   *     ⁡     (     ω   -       2   ⁢           ⁢   π     T       )       ⁢   d   ⁢           ⁢   ω       +       ∫     -   ∞     ∞     ⁢     2   ⁢   ⁢     {       V   ⁡     (   ω   )       ⁢       V   *     ⁡     (     ω   -       2   ⁢   π     T       )         }     ⁢     S   ⁡     (   ω   )       ⁢       S   *     ⁡     (     ω   -       2   ⁢           ⁢   π     T       )       ⁢   d   ⁢           ⁢   ω             ,           (   16   )               
where  {.} denotes the real part of the expression.
 
     For example, from (16) it is observed that the clock signal will be zero if 
     
       
         
           
             
               
                 
                   
                     
                       U 
                       ⁡ 
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     = 
                     
                       
                          
                         
                           U 
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                          
                       
                       ⁢ 
                       
                         e 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                             T 
                             4 
                           
                         
                       
                       ⁢ 
                       
                         e 
                         
                           j 
                           ⁢ 
                           
                             
                               ψ 
                               U 
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         V 
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                       = 
                       
                         
                            
                           
                             V 
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                            
                         
                         ⁢ 
                         
                           e 
                           
                             
                               ± 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                               T 
                               4 
                             
                           
                         
                         ⁢ 
                         
                           e 
                           
                             j 
                             ⁢ 
                             
                               
                                 ψ 
                                 V 
                               
                               2 
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     where ψ U  and ψ V  are arbitrary angles. As can be seen, the first-order PMD defined by (4) with τ=T/2 satisfies condition (17) for any combination of θ 0 , ϕ 0  and ψ 0 . Therefore, the clock signal contained in |r x (t)| 2 +|r y (t)| 2  is lost in the presence of half-baud DGD. This is illustrated in  FIG. 3 , which is a plot of the normalized timing tone magnitude in |r x (t)| 2 +|r y (t)| 2  versus DGD (τ) and the rotation angle (θ). 
     Timing Information in Cascaded DGD with Two Segments 
     Under certain conditions, the timing tone generated using the nonlinear spectral line method may generate a timing tone with a frequency offset which causes a loss of synchronization. An example channel  430  (which may be used as channel  130 ) having these conditions is illustrated in  FIG. 4 . Here, the channel is modeled as a first DGD block  402 , a first matrix rotation block  404 , a second DGD block  406 , and a second matrix rotation block  408 , which each represent distortions in the channel. The signal sent over the channel  430  comprises an optical signal having two polarizations (e.g., a horizontal and vertical polarization, or more generally, an “X” and “Y” polarization) of the optical signal received by the optical front end  150 . These two polarizations can be treated mathematically as a two-dimensional complex vector, where each component corresponds to one of the polarizations of the received optical signal and is expressed as a complex representation of in-phase and quadrature signals. The first DGD block  402  and second DGD block  406  each apply a transform 
               [           e     j   ⁢           ⁢   ω   ⁢           ⁢     T   /   4             0           0         e       -   j     ⁢           ⁢   ω   ⁢           ⁢     T   /   4               ]               
to the incoming signal. The first matrix rotation block  404  and the second matrix rotation block  408  each apply a transform
 
             MR   =     [           cos   ⁡     (   θ   )             sin   ⁡     (   θ   )                 -     sin   ⁡     (   θ   )               cos   ⁡     (   θ   )             ]           
where in this example, θ=ω R t for the first matrix rotation block  404  and, θ=π/4 for the second matrix rotation block  408 .
 
     The timing tone coefficients for each polarization (z x,1  and z y,1 ) are given by the diagonal elements of the 2×2 matrix 
                     M   t     =       [           z     x   ,   1             z   xy               z   yx           z     y   ,   1             ]     =       ∫     -   ∞     ∞     ⁢       S   ⁡     (   ω   )       ⁢     S   H     ⁢     {     ω   -       2   ⁢   π     T       }     ⁢     J   ⁡     (   ω   )       ⁢     J   H     ⁢     {     ω   -       2   ⁢   π     T       }     ⁢   d   ⁢           ⁢   ω                 (   18   )               
where S(ω) is the transfer function of the transmit and receive filters and J(ω) is the Jones matrix. The components of J(ω) are defined by
 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             U 
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                         
                           
                             V 
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               
                                 V 
                                 * 
                               
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               U 
                               * 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     where * denotes complex conjugate. Matrix J(ω) is special unitary (i.e., J(ω) H  J(ω)=I, det(J(ω))=1, where I is the 2×2 identity matrix and H denotes conjugate transpose) and models the effects of the PMD. 
     Since ω R &lt;&lt;2π/T, the PMD matrix of the channel  430  described previously is given by: 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   ≈ 
                   
                     
                       1 
                       
                         2 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         R 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                       2 
                                     
                                   
                                 
                               
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         R 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           R 
                                         
                                         ⁢ 
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                       2 
                                     
                                   
                                 
                               
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         R 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         T 
                                       
                                       2 
                                     
                                   
                                 
                               
                               - 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Assuming that the bandwidth excess is small-moderate (i.e., S(ω)S H (ω−2π/T) is concentrated around ω=π/T), then the timing matrix M t  can be approximated as: 
                     M   t     ≈       K   s     ⁢     J   ⁡     (     π   T     )       ⁢       J   H     ⁡     (     -     π   T       )                 (   21   )               
where K s  is a given complex constant and
 
     
       
         
           
             
               
                 
                   
                     J 
                     ⁡ 
                     
                       ( 
                       
                         π 
                         T 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 je 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     R 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   je 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   je 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   je 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                       
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       J 
                       H 
                     
                     ⁡ 
                     
                       ( 
                       
                         - 
                         
                           π 
                           T 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               je 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   R 
                                 
                                 ⁢ 
                                 t 
                               
                             
                           
                           
                             
                               - 
                               
                                 je 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     R 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 je 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     R 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 je 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     R 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Based on the above equations, 
     
       
         
           
             
               
                 
                   
                     M 
                     t 
                   
                   ≈ 
                   
                     
                       - 
                       
                         
                           K 
                           s 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ω 
                                       R 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                       
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Note that the timing tone energy in a given polarization is maximized. The resulting clock tone in a given polarization results in 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             c 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           2 
                           ⁢ 
                           ⁢ 
                           
                             { 
                             
                               
                                 z 
                                 
                                   x 
                                   , 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     / 
                                     T 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           ⁢ 
                           
                             { 
                             
                               
                                 K 
                                 s 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     R 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     / 
                                     T 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             2 
                           
                           ⁢ 
                           ⁢ 
                           
                             { 
                             
                               
                                 K 
                                 s 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           / 
                                           
                                             T 
                                             R 
                                           
                                         
                                         + 
                                         
                                           1 
                                           / 
                                           T 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   t 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     R 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         R 
                       
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         π 
                         / 
                         
                           T 
                           R 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     From the above, it can be observed that the frequency of the clock signal provided by the spectral line timing recovery algorithm is shifted by 2/T R . In other words, the timing tone is modulated by the time variations of the DGD which is affected by the frequency of the rotation matrices between the DGD segments. Therefore, under these conditions, a proper clock signal with frequency 1/T will not be properly detected using a conventional spectral line method technique. 
     The channel  430  having two DGD elements with a rotating element between them is only one example of a channel having dynamic PMD. A more general channel with dynamic PMD is modeled as an arbitrarily large number of DGD elements, with randomly varying rotation angles between the DGD elements. These channels can cause timing recovery methods based on a traditional nonlinear spectral line method to fail by causing a frequency offset (as described above) or by causing randomly varying phase between the received symbol clock and the sampling clock. The randomly varying phase can accumulate to cause an offset of multiple symbol periods between the received signal and the sampling clock, resulting in eventual failure of the receiver once the receiver can no longer compensate for the shift in timing. This situation is compounded by the presence of residual chromatic dispersion that reaches the timing recovery circuit, which can cause the amplitude of the timing tone to fade. The disclosed embodiments solve for these impairments and perform robustly in the presence of general dynamic PMD and in the presence of residual chromatic dispersion. 
     Timing Recovery Architecture and Method 
       FIG. 5  illustrates an example embodiment of a timing recovery system that generates a timing recovery signal without the problems of the spectral line method discussed above. In one embodiment, the timing recovery block  179  comprises a resonator filter  504 , an in-phase and quadrature error computation block  506 , a rotation computation block  508 , and a numerically controlled oscillator (NCO) control block  510 . 
     The resonator filter  504  receives oversampled input samples of the two polarizations, which may be received directly from an oversampling analog-to-digital converter (ADC) or from an interpolator filter between the ADC and the timing recovery block  179 . The resonator filter  504  filters the input signal by applying, for example, a band pass filter having a center frequency of approximately 1/(2T). This center frequency is used because the timing information in a spectral line timing recovery scheme is generally within the vicinity of this frequency band. The phase and quadrature error computation block  506  interpolates the filtered signal (e.g., from two samples per baud to four samples per baud per polarization) from the resonator filter  504  and determines in-phase and quadrature error signals from the interpolated signal. The rotation computation block  508  generates a timing matrix representing detected timing tones and estimates a rotation error based on the timing matrix. The rotation error signal controls the NCO control block  510  which adjusts the sampling phase or frequency, or both, of the sampling clock  181  based on the rotation error. In alternative embodiments, the NCO control block  510  may instead control sampling phase of an interpolator filter between the ADC and the timing recovery block, rather than controlling sampling phase and frequency of the sampling clock  181  directly. 
       FIG. 6  illustrates the input and output signals from the resonator filter  504  in more detail. The resonator filter  504  receives an oversampled input signal {tilde over (R)} which may be generated by, for example, an interpolator filter or directly from an analog-to-digital converter in the Rx AFE  168 . For example, in one embodiment {tilde over (R)} has a sampling rate of 2/T per polarization where T is the period of the timing tone. In this example, {tilde over (R)} comprises four vector components:
 
 {tilde over (R)}   x ( k, 0)=[ r   x ( kN   TR ,0), r   x ( kN   TR +1,0), . . .  r   x ( kN   TR   +N   TR −1,0)]
 
 {tilde over (R)}   x ( k, 2)=[ r   x ( kN   TR ,2), r   x ( kN   TR +1,2), . . .  r   x ( kN   TR   +N   TR −1,2)]
 
 {tilde over (R)}   y ( k, 0)=[ r   y ( kN   TR ,0), r   y ( kN   TR +1,0), . . .  r   y ( kN   TR   +N   TR −1,0)]
 
 {tilde over (R)}   y ( k, 2)=[ r   y ( kN   TR ,2), r   y ( kN   TR +1,2), . . .  r   y ( kN   TR   +N   TR −1,2)]  (25)
 
where {tilde over (R)} x (k,0) represents an N TR  dimensional vector with even samples of a first polarization “X”; {tilde over (R)} x (k,2) represents an N TR  dimensional vector with odd samples of the first polarization “X”; {tilde over (R)} y (k,0) represents an N TR  dimensional vector with odd samples of a second polarization “Y”; and R y (k,2) represents and N TR  dimensional vector with even samples of the second polarization “Y,” and k is an index number associated with each set of 4N TR  samples. In one embodiment, N TR =64, although different dimensionalities may be used in alternative embodiments. The polarizations “X” and “Y” are used generically to refer to signals derived from two different polarizations of an optical signal and in one embodiment correspond to horizontal and vertical polarizations respectively, or vice versa.
 
     Each element of the N TR  dimensional vectors are in the form r a (n,l) a∈(x,y) where n represents the number of the symbol and l∈[0, 1, 2, 3] represents the sampling phase with four sample phases per symbol per polarization at a sampling rate of 4/T, two of which (0 and 2) are used at a sampling rate of 2/T. For example, the two samples associated with the first polarization “X” in the symbol n are represented by r x (n,0) r x (n,2) when the signal derived from polarization X is sampled at a sampling rate of 2/T. The two samples associated with the second polarization “Y” in the symbol n are represented by r y (n,0) r y (n,2) when the signal derived from polarization Y is sampled at a sampling rate of 2/T. 
     The resonator filter  504  applies a unit pulse response h rf (n) to the incoming signal R where 
                       h   rf     ⁡     (   n   )       =       2     N   rf       ⁢     sin   ⁡     (     n   ⁢           ⁢     π   2       )       ×       rect     N   rf       ⁡     (   n   )                 (   28   )                   rect     N   rf       ⁡     (   n   )       =       ∑     i   =   0         N   rf     -   1       ⁢     δ     n   -   i                                 
where N rf  represents the number of taps in the resonator filter  504 . In this notation, the unit pulse response h rf (n) is based on a sampling rate of 2/T. For example, if the input signal is sampled at 2/T, then in one embodiment,
 
     
       
         
           
             
               
                 
                   
                     
                       N 
                       rf 
                     
                     = 
                     
                       
                         16 
                         → 
                         
                           2 
                           
                             N 
                             rf 
                           
                         
                       
                       = 
                       
                         2 
                         
                           - 
                           3 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   or 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       N 
                       rf 
                     
                     = 
                     
                       
                         32 
                         → 
                         
                           2 
                           
                             N 
                             rf 
                           
                         
                       
                       = 
                       
                         2 
                         
                           - 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     The unit pulse response h rf (n) can also be written as: 
     
       
         
           
             
               
                 
                   
                     
                       h 
                       rf 
                     
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         N 
                         rf 
                       
                     
                     [ 
                     
                       0 
                       , 
                       1 
                       , 
                       0 
                       , 
                       
                         - 
                         1 
                       
                       , 
                       0 
                       , 
                       1 
                       , 
                       … 
                     
                     ⁢ 
                     
                         
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     By filtering the signal derived from polarization a sampled at a rate of 2/T, where a∈(x,y), with the unit pulse response h rf (n) above, the resonator filter  504  generates a signal {tilde over (R)} comprising four vector components: {tilde over (R)} x (k,0), {tilde over (R)} x (k,2), {tilde over (R)} y (k,0), {tilde over (R)} y (k,2) having the general form: 
     
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         ^ 
                       
                       a 
                     
                     ⁡ 
                     
                       ( 
                       
                         k 
                         , 
                         l 
                       
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             r 
                             _ 
                           
                           a 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               kN 
                               TR 
                             
                             , 
                             l 
                           
                           ) 
                         
                       
                       , 
                       
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               r 
                               _ 
                             
                             a 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   kN 
                                   TR 
                                 
                                 + 
                                 
                                   N 
                                   TR 
                                 
                                 - 
                                 1 
                               
                               , 
                               l 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   a 
                   ∈ 
                   
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     l 
                   
                   ∈ 
                   
                     ( 
                     
                       0 
                       , 
                       2 
                     
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 where 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         r 
                         _ 
                       
                       a 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             kN 
                             TR 
                           
                           + 
                           m 
                         
                         , 
                         l 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       
                         N 
                         rf 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           
                             
                               N 
                               rf 
                             
                             / 
                             2 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           i 
                         
                         ⁢ 
                         
                           
                             r 
                             a 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   kN 
                                   TR 
                                 
                                 + 
                                 m 
                                 - 
                                 i 
                               
                               , 
                               l 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   m 
                   ∈ 
                   
                     [ 
                     
                       0 
                       , 
                       1 
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         
                           N 
                           TR 
                         
                         - 
                         1 
                       
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   a 
                   ∈ 
                   
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     l 
                   
                   ∈ 
                   
                     ( 
                     
                       0 
                       , 
                       2 
                     
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     An effect of the resonator filter  504  is to band pass filter the signal around the frequency 1/(2T) where the timing information is contained. Although chromatic dispersion may be compensated for in the Rx DSP  166  or by other means prior to the timing recovery block  179 , residual chromatic dispersion (that is, chromatic dispersion that has not been compensated for prior to the timing recovery circuit  179 ) has the effect of varying the timing tone level particularly in the presence of a high roll-off factor of the transmit pulse. The resonator filter  504  limits the bandwidth excess and reduces high fluctuations of the timing tone level caused by residual chromatic dispersion, and thereby reduces the dependence of the timing tone level on the chromatic dispersion. 
       FIG. 7  is block diagram illustrating an example embodiment of the in-phase and quadrature error computation block  506 . In the illustrated embodiment, the block  506  comprises an interpolator filter  702 , a correlation computation block  704 , and a difference computation block  706 . 
     The interpolator filter  702  interpolates the signal {circumflex over (R)} which has two samples per polarization per symbol, to generate an interpolated signal R having four samples per polarization per symbol. Since the frequency of the timing tone is 1/T, the sampling rate of the input signal should be higher than 2/T to recover the timing tone. The interpolator filter  702  increases the sampling rate (e.g., to 4/T per polarization) in order to have a sampling rate high enough to accurately estimate the timing tone. The signal R generated by the interpolator filter  702  has eight components (four for each polarization) having form:
 
 R   a ( k,l )=[ r   a ( kN   TR   ,l ), . . .  r   a ( kN   TR   +N   TR −1, l )]
 
 a ∈( x,y ) l ∈(0,1,2,3)  (33)
 
where:
 
 r   a ( kN   TR   +m, 0)= {tilde over (r)}   a ( kN   TR   +m−d   if ,0)
 
 r   a ( kN   TR   +m, 1)=Σ i=0   N     if     −1   c   0 ( i ) {tilde over (r)}   a ( kN   TR   +m− 1− i, 2)+Σ i=0   N     if     −1   c   1 ( i ) {tilde over (r)}   a ( kN   TR   +m−i, 0)
 
 r   a ( kN   TR   +m, 2)=   r     a ( kN   TR   +m−d   if ,2)
 
 r   a ( kN   TR   +m, 3)=Σ i=0   N     if     −1   c   0 ( i ) {tilde over (r)}   a ( kN   TR   +m−i, 0)+Σ i=0   N     if     −1   c   1 ( i ) {tilde over (r)}   a ( kN   TR   +m−i, 2)  (34)
 
for a∈(x,y) and m∈[0, 1 . . . , N TR −1] and where 2N if  is the tap number of the interpolation filter  702  and d if  is a delay where d if &lt;N if .
 
     Information of a given transmit polarization may be contained in both received polarizations due to fiber channel impairments such as PMD. Thus, in order to recover the timing tones, a correlation of the received signals is performed by the correlation computation block  704 . The correlation computation block  704  receives the interpolated signal R (e.g., having 8 N TR  dimensional vector components) and generates correlated signals r having 16 components which are each a scalar complex value. For example, in one embodiment, the correlated signal includes components r xx (k,0), r xy (k,0), r yx (k,0), r yy (k,0), r (k,1), . . . r xx (k, 3), r xy (k,3), r yx (k,3), r yy (k,3) where each component is given by:
 
 r   ab ( k,i )= R   a ( k,i )× R   b   H ( k,i )
 
 a,b ∈( x,y )  (35)
 
where (.) H  indicates a transpose and conjugate. All of the timing information is contained in the correlated signals r at the output of the correlation computation block  704 .
 
     To extract the amplitude and phases of the four timing tones of frequency 1/T, the difference computation block  706  computes in-phase error p and quadrature error q, which are each defined as the difference of two samples separated by T/2. In one embodiment, the difference computation block  706  receives the correlated signal and generates a signal representative of the in-phase and quadrature error in the block of samples. In one embodiment, the in-phase and quadrature error signals comprise four in-phase error components: p xx (k), p xy (k), p yx (k), p yy (k) and four quadrature error components: q, (k), qx y (k), q yx (k), q yy (k). 
     In one embodiment, the in-phase error components are given by:
 
 p   ab ( k )= r   ab ( k, 0)− r   ab ( k, 2) a,b ∈( x,y )  (36)
 
and the quadrature error components are given by:
 
 q   ab ( k )= r   ab ( k, 1)− r   ab ( k, 3) a,b ∈( x,y )  (37)
 
       FIGS. 8A and 8B  illustrate alternative embodiments of the rotation computation block  508 , where the alternative embodiments are designated  508 -A and  508 -B in  FIGS. 8A and 8B  respectively. In the embodiment of  FIG. 8A , the rotation computation block  508 -A estimates the phase error using a determinant method. In the embodiment of  FIG. 8B , the rotation computation block  508 -B estimates the phase error using a modified wave difference method. Referring first to  FIG. 8A , in one embodiment, the rotation computation block  508 -A comprises a matrix estimator  802 , a cycle slip computation block  804 , a DTM block  806 , and a loop filter  808 . The timing matrix estimation block  802  estimates a timing matrix from the in-phase and quadrature error signals where the timing tone coefficient for each element of the matrix is the complex number with real and imaginary parts given by p and q respectively. In one embodiment, the timing matrix has the following form: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       t 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               M 
                               t 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               M 
                               t 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               M 
                               t 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               M 
                               t 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     The timing matrix M t (k) is calculated as follows:
 
 M   t   (1,1) ( k )=β M   t   (1,1) ( k− 1)+(1−β)[ p   xx ( k )− jq   xx ( k )]
 
 M   t   (1,2) ( k )=β M   t   (1,2) ( k− 1)+(1−β)[ p   xy ( k )− jq   xy ( k )]
 
 M   t   (2,1) ( k )=β M   t   (2,1) ( k− 1)+(1−β)[ p   yx ( k )− jq   yx ( k )]
 
 M   t   (2,2) ( k )=β M   t   (2,2) ( k− 1)+(1−β)[ p   yy ( k )− jq   yy ( k )]  (39)
 
where β is a constant and M t (k) can assume an initial value of all zeros. For example, in one embodiment, β=1−2 −m  (e.g., m=6), although other values of β may be used in alternative embodiments.
 
     The cycle slip computation block  804  receives the timing matrix M t (k), and generates a cycle slip number signal ϵ cs (k). The cycle slip number signal ϵ cs (k) represents a number of cycle slips of the sampling phase relative to the received signal. A cycle slip occurs when the sampling phase shifts a symbol period T. This effect generally occurs during the start-up period and its frequency depends on clock error between the transmit and receive signals. 
     To determine the number of cycle slips, the cycle slip computation block  804  calculates a determinant of the timing matrix M t (k) as follows:
 
ρ( k )= M   t   (1,1) ( k )* M   t   (2,2) ( k )* M   t   (1,2) ( k )* M   t   (2,1) ( k )  (40)
 
In-phase and quadrature phase error signals and then computed as:
 
                       p   ~     ⁡     (   k   )       =     sign   ⁡     [     real   ⁢     {     ρ   ⁡     (   k   )       }       ]               (   41   )                   q   ~     ⁡     (   k   )       =     sign   ⁡     [       i   ⁢   mag     ⁢     {     ρ   ⁡     (   k   )       }       ]                             where                           sign   ⁡     [   x   ]       =     {         1         x   ≥   0               -   1           x   &lt;   0                     (   42   )               
real[x] is the real part of complex x and imag[x] is the imaginary part of complex x. The accumulated number of cycle slips is determined as:
 
ϵ cs ( k )=Σ i=0   k   cs ( i )  (43)
 
where:
 
 cs ( k )=−1 when( p ( k− 1), q ( k− 1))=(1,1) and ( {tilde over (p)} ( k ), {tilde over (q)} ( k ))=(1,−1)
 
 cs ( k )=−1 when( p ( k− 1), q ( k− 1))=(−1,−1) and ( p ( k ), q ( k ))=(−1,1)
 
 cs ( k )=+1 when( p ( k− 1), q ( k− 1))=(1,−1) and ( p ( k ), q ( k ))=(1,1)
 
 cs ( k )=+1 when( p ( k− 1), q ( k− 1))=(−1,1) and ( p ( k ), q ( k ))=(−1,−1)
 
 cs ( k )=0 otherwise.  (44)
 
     The DTM block  806  computes a phase error signal ϵ phase (k) representing the estimated phase error of a local clock relative to the received signal. In the embodiment of  FIG. 8A , the phase error signal ϵ phase (k) is calculated using a determinant method. 
     In the determinant method, the phase error signal is given by
 
ϵ phase ( k )=ϵ DTM ( k )=γ( k )=arg{ρ( k )}  (45)
 
     In one embodiment, the argument of the determinant is determined using a lookup table. In one embodiment, the DTM block  806  and cycle slip computation block  804  are integrated such that the determinant of the timing matrix M t (k) is determined only once, and may then be used to compute both ϵ cs (k) and ϵ phase (k). 
     The loop filter  808  filters and combines the phase error signal ϵ phase (k) and the cycle slip number signal ϵ cs (k) to generate a control signal to the NCO  510  that controls a total accumulated phase shift in the received signal to be compensated. For example, in one embodiment, the loop filter  808  applies a proportional-integral (PI) filter to the phase error signal ϵ phase (k) to generate a filtered phase error signal ϵ′ phase (k). For example, in one embodiment, the PI filter has an output L(z)=K p +K i /1−z −1  where K p  and K i  are gain constants. The loop filter  808  then combines the number of cycle slips ϵ cs (k) with the filtered phase error ϵ′ phase (k). to detect the total rotation and provides this information as a control signal to the NCO  510 . For example, in one embodiment, the loop filter  808  implements the function:
 
NCO control ( k )=ϵ′ phase ( k )− k   cs ϵ cs ( k )  (46)
 
where k cs  is a constant that represents a gain of the loop filter  808 . For example, in one embodiment, k cs =2-3 although other gain constants may be used.
 
       FIG. 8B  illustrates an alternative embodiment of the rotation computation block  508 -B which instead uses a modified wave difference method to determine the phase error signal ϵ phase (k). In the embodiment of  FIG. 8B , the rotation computation block  508 -B comprises a matrix estimator  852 , a cycle slip computation block  854 , a WDM block  856 , and a loop filter  858 . The matrix estimator  852  operates similarly to the matrix estimator  802  described above but generates both a timing matrix M t (k) and a noisy timing matrix {circumflex over (M)} t (k). The timing matrix M t (k) is computed in the same manner described above. The noisy timing matrix {circumflex over (M)} t (k) is defined as the timing matrix M t (k) where β=0. Thus, the components of {circumflex over (M)} t (k) are given as:
 
 {circumflex over (M)}   t   (1,1) ( k )= p   xx ( k )− jq   xx ( k )
 
 {circumflex over (M)}   t   (1,2) ( k )= p   xy ( k )− jq   xy ( k )
 
 {circumflex over (M)}   t   (2,1) ( k )= p   yx ( k )− jq   yx ( k )
 
 {circumflex over (M)}   t   (2,2) ( k )= p   yy ( k )− jq   yy ( k )
 
     The cycle slip computation block  854  generates a cycle slip number signal ϵ cs (k) representing the detected number of cycle slips based on the timing matrix M t (k) in the same manner described above. 
     The WDM block  856  determines the phase error signal based on the in-phase and quadrature error signals from the in-phase and quadrature error computation block  506 , the timing matrix M t (k), and the noisy timing matrix {circumflex over (M)} t (k) using a modified wave difference method. In this technique, the phase error signal is computed as: 
                       ϵ   phase     ⁢           ⁢       ϵ   MWDM     ⁡     (   k   )         =       -     ⁢           ⁢     {               e       j   ⁢           ⁢     γ   ⁡     (   n   )         2       ⁡     [           M   ^     t     (     1   ,   1     )       ⁡     (   k   )       ⁢           ⁢         M   ^     t     (     1   ,   2     )       ⁡     (   k   )       ⁢           ⁢         M   ^     t     (     2   ,   1     )       ⁡     (   k   )       ⁢           ⁢         M   ^     t     (     2   ,   2     )       ⁡     (   k   )         ]       ×                 [         M   t     (     1   ,   1     )       ⁡     (   k   )       ⁢           ⁢       M   t     (     1   ,   2     )       ⁡     (   k   )       ⁢           ⁢       M   t     (     2   ,   1     )       ⁡     (   k   )       ⁢           ⁢       M   t     (     2   ,   2     )       ⁡     (   k   )         ]     H           }               (   48   )               
where  {.} denotes the imaginary part, [.] H  denotes the complex conjugate and transpose, and
 
γ( k )=unwrap{γ( k− 1),arg(det{ M   t ( k )})}  (49)
 
where γ(k)∈[−2π, 2π) is the new unwrapped angle based on the old unwrapped angle and the new argument of the determinant. In more detail, z=unwrap{y,x} computes the unwrapped angle z∈[−2π, 2π) based on the inputs y∈[−2π, 2π), x∈[−π,π) by first adding a multiple of 2π to x to give an intermediate result in the range [y−π, y+π), then adding a multiple of 4π to give a final result z in the range [−2π, 2π). Note that γ(k) is defined for the range [−2π, 2π) because it is halved when computing the phase error. The initial value of γ(k) can be set to 0.
 
     The loop filter  858  operates similarly to the loop filter  808  described above. 
     Although the detailed description contains many specifics, these should not be construed as limiting the scope but merely as illustrating different examples and aspects of the described embodiments. It should be appreciated that the scope of the described embodiments includes other embodiments not discussed in detail above. For example, the functionality of the various components and the processes described above can be performed by hardware, firmware, software, and/or combinations thereof. 
     Various other modifications, changes and variations which will be apparent to those skilled in the art may be made in the arrangement, operation and details of the method and apparatus of the described embodiments disclosed herein without departing from the spirit and scope of the invention as defined in the appended claims. Therefore, the scope of the invention should be determined by the appended claims and their legal equivalents.