Abstract:
A method reduces time-varying polarization crosstalk due to XPolM by transmitting multi-dimensional orthogonal constellations. Three variants of crosstalk-free constellations are provided: Grassmann constellations, unitary constellations, and rotation codes. The method uses the Grassmann constellations and the unitary constellations to deal with fiber nonlinearity by applying as a polarization-time coding. The rotation codes exploit a fiber channel characteristic to improve performance and to reduce computational complexity. The underlying orthogonality behind those constellations enables the receiver to decode it as if there is no polarization crosstalk. Moreover, the required computational complexity at the receiver is significantly reduced because neither crosstalk cancellers nor channel estimators are needed.

Description:
FIELD OF THE INVENTION 
     This invention relates generally to optical communication systems, and more particularly to avoiding depolarization effects due to fiber nonlinearity, specifically cross-polarization modulation. 
     BACKGROUND OF THE INVENTION 
     In optical communication systems that use polarization multiplexing (POLMUX), two optical signals are polarized orthogonally with respect to each other before being transmitted in an optical channel at a certain wavelength. This doubles the data rate of the channel. To realize a high-speed transmission over 100 Gbps, both POLMUX and dense-wavelength-division multiplexing (DWDM) are needed. 
     However, the constantly alternating bit sequences of the POLMUX signal in DWDM systems cause the resulting polarization to change over time. This depolarization effect results in cross-polarization modulation (XPolM), which is essentially noise. The XPolM impairment is induced by inter-channel fiber nonlinearity particularly for DWDM optical communication systems. Due to the XPolM, the dual-polarized (DP) signals, which originally were polarized orthogonally, are no longer so, and cannot be separated at a receiver by a polarization splitter. 
     XPolM causes significant performance degradation in bit error rate or in quality factor for DP coherent optical systems. The induced polarization crosstalk has a low-pass response because of chromatic dispersion, resulting in a short coherence time of tens of symbols. Such a fast time-varying polarization crosstalk is hard to be compensated at receivers for current optical communication systems. 
     Most conventional techniques focus on compensating the time-varying XPolM effect at the receiver, through the use of a crosstalk cancellation with adaptive channel estimation. However, the performance improvement is limited by the channel estimation accuracy, which is severely degraded by rapid time-varying channels, particularly due to the XPolM. In addition, nonlinear compensation methods at the receiver have a high-computational complexity to implement in circuits. 
     Some nonlinear rotations such as self-phase modulation (SPM) are well treated by differential coding schemes like dual-polarized differential quadrature phase-shift keying (DP-DQPSK) even without accurate channel estimations. However, DP-DQPSK suffers from severe performance degradations in the presence of polarization crosstalk due to XPolM and polarization mode dispersion (PMD). 
     SUMMARY OF THE INVENTION 
     The embodiments of the invention provide a method for tolerating nonlinear effects of XPolM, cross-phase modulation (XPM), self-phase modulation (SPM), and linear effects of PMD in coherent fiber-optic communication networks, wherein multiple wavelengths with POLMUX signals are densely multiplexed as DWDM systems. 
     The method uses dimension-extended constellations, which automatically resolve any polarization crosstalk at a transmitter. Such a self-polarization-isolation property comes from a specific constellation design, which mutually orthogonalizes x-polarization and y-polarization signals. The joint design of x-/y-polarization signals reduces a significant amount of computational complexity at the receiver side because neither crosstalk cancellers nor channel estimations are required. 
     The embodiments of the invention provide three variants of crosstalk-free constellation designs, more specifically, Grassmann constellations, unitary constellations, and rotation codes. 
     The Grassmann constellations and the unitary constellations have been used as space-time coding for wireless communications. The invention adapts those constellations as polarization-time coding for optical communications to deal with XPolM, XPM, SPM and PMD. The rotation codes leverage a unique property of optical channels, i.e., near-unitary characteristics, to improve performance and to reduce computational complexity. 
     At the receiver, the crosstalk-free multi-dimensional constellations are decoded by generalized-likelihood ratio test (GLRT) process without estimating crosstalk channels. The GLRT decoding is further improved by multi-symbol decoding, unitary constraint, and a high-order approach to deal with rapid time-varying channels. The joint use of trellis-coded modulation (TCM), and the crosstalk-free multi-dimensional constellations offers further gains. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic of a coherent fiber-optic communications with DWDM and POLMUX, according to embodiments of the invention; 
         FIG. 2  is a schematic of a multi-dimensional orthogonal constellation based on Grassmann polarization-time codes, according to embodiments of the invention; 
         FIG. 3  is a block diagram of a differential unitary constellation based on polarization-time coding, according to embodiments of the invention; and 
         FIG. 4  is a schematic of a multi-dimensional orthogonal constellation based on double-differential rotation codes, according to embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       FIG. 1  shows the preferred embodiments of coherent fiber-optic communication systems  100  which use POLMUX and WDM. At a channel of interest, x-polarization signal S x (k) and y-polarization signal S y (k), at the k-th symbol instance, are generated by electrical modulators  110  for multi-dimensional constellations. After passing through an electrical filter, x/y-polarization signals S x (k) and S y (k) are up-converted to optical signal by an electrical-to-optical (E/O) modulator with POLMUX  120 , which orthogonally polarizes the signals S x (k) and S y (k). 
     The up-converted optical signal is then multiplexed with a number of different wavelengths  131 , using a transmitter circuit (Tx), by a WDM multiplexer, such as an arrayed waveguide grating device  130 . The WDM optical signals propagate through an optical fiber channel  140 , which consists of multiple-span sets of single-mode fibers (SMF)  141 , optical amplifiers (OA) such as erbium doped fiber amplifier  142  and  144 , and dispersion compensation fibers (DCF)  143 . 
     At a receiver, WDM signals are first de-multiplexed by a de-WDM device  150  to separate different wavelengths to the corresponding receiver (Rx) circuits  151 . For the channel of interest, the optical signal is down-converted by optical-to-electrical (O/E) converter with de-POLMUX  160 , which obtains x-polarization received signal R x (k) and y-polarization received signal R y (k) at the k-th symbol instance. To decode the multi-dimensional orthogonal constellations transmitted from the modulator  110 , a blind process with generalized-likelihood ratio test (GLRT) decoding  170  is used at the receiver. 
     For the dispersion-managed fibers, the received signals R x (k) and R y (k) are well modeled by a linear crosstalk representation as follows: 
                 [             R   x     ⁡     (   k   )                   R   y     ⁡     (   k   )             ]     =         A   ⁢           ⁢         ⅇ     j   ⁢           ⁢   B       ⁡     [             ⅇ     j   ⁢           ⁢   C       ⁢       1   -            W   ⁡     (   k   )            2                   -   W     *     (   k   )                 W   ⁡     (   k   )               ⅇ       -   j     ⁢           ⁢   C       ⁢       1   -            W   ⁡     (   k   )            2                 ]       ⁡     [             S   x     ⁡     (   k   )                   S   y     ⁡     (   k   )             ]         +       [             N   x     ⁡     (   k   )                   N   y     ⁡     (   k   )             ]     ⁢     
     ⟹     R   ⁡     (   k   )           =         H   ⁡     (   k   )       ⁢     S   ⁡     (   k   )         +     N   ⁡     (   k   )             ,         
where A corresponds to a received signal magnitude, B is a common phase rotation caused by self-phase modulation (SPM) and cross-phase modulation (XPM), C corresponds to the phase difference between x-polarization and y-polarization, W(k) denotes the crosstalk factor from x-polarization to y-polarization, and [N x (k), N y (k)] is additive noise for dual polarizations at the k-th symbol.
 
     Here, denotes an imaginary unit and the superscript [.]* is the complex conjugate. Accordingly, the received signal vector R(k)=[R x (k), R y (k)] is given by a channel matrix M(k) and the transmitted signal vector S(k)=[S x (k), S y (k)] plus the noise vector N(k)=[N x (k), N y (k)]. The crosstalk W(k) is caused by cross-polarization modulation (XPolM) and polarization-mode dispersion (PMD). The covariance of the crosstalk W(k) has a short coherence time in particular due to XPolM, which is caused by WDM interfering channels and POLMUX signals. Because of the crosstalk between x-polarization signal and y-polarization signal, the DP received signals are no longer orthogonal to each other. 
     One component of the invention is the multi-dimensional orthogonal modulator  110 , which is either based on Grassmann constellations  200 , unitary constellations  300  or rotation codes  400 , see  FIGS. 2-4 . 
     The underlying orthogonality between x-polarization signal and y-polarization signal of the constellations automatically resolves any crosstalk even without estimating the channels and canceling the crosstalk. Therefore, the method significantly reduces the computational complexity of a digital signal processing to compensate polarization crosstalk at the receiver. 
       FIG. 2  shows a signal constellation design over a Grassmannian manifold, which maps any arbitrary multi-dimensional modulation into a hyper-sphere surface by exponential projections. The signal modulation includes five steps: digital data buffering  210 , non-orthogonal multi-dimensional modulating  220 , exponential mapping  230 , Grassmann orthogonal multi-dimensional modulating  240 , and DP signal transmitting  250 . 
     Let q be the number of bits to be transmitted per symbol per polarization, e.g., q=2. The Grassmann constellations are of a dimension 2×L, where 2 corresponds to dual polarization and L is the block length in time, e.g., L=4. Thus, the total number of bits per block becomes 2qL, and the cardinality of the Grassmann constellations is 2 2qL . A buffer  210  of 2qL bits for x-/y-polarization over L symbols  212  produces a constellation index m (m&lt;2 2qL )  211 . 
     For the index m, a non-orthogonal multi-dimensional constellation D m    221  is generated e.g. by a lattice modulation or simply by individual quadrature amplitude modulations (QAM). The dimension of the constellation D m    222  is set to be 2×(L−2), which is smaller than the desired dimension of 2×L. The constellation D m  is then mapped into a Grassmannian manifold by exponential projection  231  as follows: 
                 G   m     =       [           I   2         0         ]     ⁢     exp   ⁡     [         0         gD   m               -     gD   m   H           0         ]           ,         
where I 2  is a (2×2)-dimensional identity matrix, g is a constant value which controls the codeword distance, and [.] H  denotes the Hermitian transpose. The exponential mapping projects a Cartesian coordinated constellation point  232  to a surface point of a higher-dimensional hyper sphere  233 . The projected Grassmann constellation G m    241  is of a desired dimension of (2×L)  242 , and every codeword has an orthogonal property, more specifically, G m G m   H =I 2 . It implies that the x-polarization signal is perpendicular to the y-polarization signal and vice versa. The m-th constellation G m  is sequentially transmitted for x-/y-polarization signals over L symbols, [S(kL+1), S(kL+2), . . . , S(kL+L)]  251  for the k-th block. Note that the m-th codeword constellations are pre-determined given a fixed value for g and original modulations D m .
 
     To decode the Grassmann constellations, the GLRT process provides the maximum-likelihood (ML) performance because of the codeword orthogonality by searching for the maximum squared norm over the sub-space codeword projection of the L-symbol received signal block as follows: 
                 max   m     ⁢            [           R   ⁡     (     kL   +   1     )           …           R   ⁡     (     kL   +   L     )       ⁢     G   m   H             ]          2       ,         
where ∥.∥ denotes the Euclidean norm. Because x-/y-polarization signals are mutually orthogonal, the GLRT process obtains the ML estimate of the transmitted data regardless of any polarization crosstalk. Hence, crosstalk cancellations and channel estimations are not necessary at the receiver for any depolarized fiber channels. The performance is improved with multi-block decoding with high-order GLRT.
 
     Although the Grassmann constellation achieves near-capacity performance when the block length goes to infinity, it becomes hard to optimize all the constellation points for a large cardinality. The minimum allowable block length for Grassmann constellations is L=4, for which the cardinality becomes more than 65 thousand to achieve 2 bits per second per polarization. 
     As shown in  FIG. 3 , as an alternative approach which reduces the cardinality, the embodiments of invention provide a method  300  to make a full use of unitary constellations which allow differential encoding for (2×2)-dimensional modulations. As an example of unitary constellations, the m-th codeword U m    312  of the unitary code set U  311  is given by the following form: 
               U   m     =         [           ⅇ     j   ⁢       2   ⁢   π   ⁢           ⁢   m     M             0           0         ⅇ     j   ⁢           ⁢       2   ⁢   π   ⁢           ⁢   m   ⁢           ⁢     k   1       M               ]     ⁡     [           cos   ⁡     (       2   ⁢   π   ⁢           ⁢   m   ⁢           ⁢     k   2       M     )             sin   ⁡     (       2   ⁢   π   ⁢           ⁢   m   ⁢           ⁢     k   2       M     )                 -     sin   ⁡     (       2   ⁢   π   ⁢           ⁢     mk   2       M     )               cos   ⁡     (       2   ⁢   π   ⁢           ⁢   m   ⁢           ⁢     k   2       M     )             ]       ⁢             [           ⅇ     j   ⁢           ⁢       2   ⁢   π   ⁢           ⁢     mk   3       M             0           0         ⅇ     j   ⁢           ⁢         -   2     ⁢   π   ⁢           ⁢     mk   3       M               ]     ,               
where M is the cardinality, and {k 1 , k 2 , k 3 } are constant integers to be optimized. To achieve 2 bits per second per polarization, the cardinality shall be M=256, which is significantly reduced from the Grassmann constellations and the computational complexity is reduced as well. For that case, optimal integers are {k 1 , k 2 , k 3 }={104, 14, 0} to maximize the Euclidean distance. Since the fiber-optic channel is non-selective, the Euclidean distance is more important than the product distance. Note that every codeword U m  is unitary, i.e., U m U m   H =U m   H U m =I 2 .
 
     At the transmitter which employs unitary differential encoding, the digital data buffer  320  generates the codeword index m (m&lt;M) from 4q bits per 2 symbols. The unitary modulator  330  selects the m-th codeword U m  from the code set  310 . The x-/y-polarization signals are obtained by block-wise differential encoding  340 , wherein the transmitting DP signal block [S(2k), S(2k+1)] is differentially encoded  340  by a matrix multiplication  341  of the delayed version  342  and the codeword U m . The resulting x-polarization signal is therefore orthogonal to the y-polarization signal. It means that multiple unitary blocks become some sort of Grassmann constellations. Hence, differentially encoded unitary constellations are decoded by the GLRT process as well as differential decoding. 
     The Grassmann constellations and the unitary constellations are applicable to both wireless channels and optical channels. However, the fiber-optic channels have a unique characteristic distinguished from wireless communication channels, specifically, the optical channel H(k) is close to unitary since the polarization dependent loss (PDL) is low in practical fiber-optic systems. The embodiments of the invention provide an additional method to exploit the near-unitary property of the optical channels for performance improvement. 
       FIG. 4  shows a block diagram of the encoding and the decoding for a multi-dimensional crosstalk-free modulation, termed double-differential rotation codes which take advantage of the unitary property of the fiber-optic channels. At the transmitter  410 , a 2-dimensional modulation signal P(k)  421  is first differentially encoded in an element-wise manner  420 , i.e., x-/y-polarization signals are independently encoded in a conventional way. The differentially encoded vector V(k)  425  is then differentially encoded in a vector-wise manner  430  to obtain the transmitting DP signals S(k)  435 . For the element-wise differential encoding  420 , the signal P(k) is multiplied using an element-wise multiplier  423  by a vector V(k) with a delay  422  to obtain differential encoded signal V(k)  425 . For the vector-wise differential encoding  430 , the signal V(k) is multiplied  431  with a rotation matrix Q s    432 , which is generated at a rotator  433  feeding a delayed version  434  of the transmitting DP signals S(k). The rotation matrix is given as follows: 
               Q   S     =       1          S   ⁡     (     k   -   1     )              ⁡     [             S   x     ⁡     (     k   -   1     )             -       S   y   *     ⁡     (     k   -   1     )                     S   y     ⁡     (     k   -   1     )               S   x   *     ⁡     (     k   -   1     )             ]             
which becomes a unitary matrix for any non-zero signals S(k), namely, Q s Q s   H =Q s   H Q s =I 2 . Note that the crosstalk optical channel H(k) is also the same kind of such unitary rotation matrices except for a scalar term of A e jB .
 
     The double-differential rotation codes allow the receiver to decode the data in a differential manner even in the presence of a strong polarization crosstalk, whereas the conventional differential modulation per polarization does not accept crosstalk. At the receiver side  460 , the received DP signal R(k) is differentially decoded in a vector-wise fashion  470  to obtain an intermediate data V′(k)  476 , and it is further differentially decoded in an element-wise manner  480  to obtain an estimate of the desired data P′(k)  481 . 
     For the vector-wise differential decoding  470 , the received signal R(k) is multiplied with a rotation matrix Q R    473 , which is generated by a rotator  474  given the delayed version  475  of the received signal as follows: The rotation matrix is given as follows: 
     
       
         
           
             
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     Since the rotation matrix is unitary, there is no noise enhancement during the vector-wise differential decoding  470 . The differentially decoded data V′(k) is expressed as 
                       V   ′     ⁡     (   k   )       =       ⁢       Q   R   H     ⁢     R   ⁡     (   k   )                     =       ⁢         Q   R   H     ⁢     H   ⁡     (   k   )       ⁢     Q   S     ⁢     V   ⁡     (   k   )         +       Q   R   H     ⁢     N   ⁡     (   k   )                         =       ⁢         [           A   1         0           0         A   2           ]     ⁢     V   ⁡     (   k   )         +       N   ′     ⁡     (   k   )           ,               
where {A 1 , A 2 } are scalar values dependent on A e jB , and N′(k) is a residual noise. The above equation implies that any polarization crosstalk is automatically resolved by the differential rotation because the channel H(k) is close to a scaled unitary matrix.
 
     To deal with the unknown scalar values {A 1 , A 2 }, the element-wise differential decoding  480  is performed for the intermediate data V′(k)  476  with an element-wise division  482  of the own delayed version  483  to obtain the final decoded data P′(k)  481 . The rotation codes generally outperform the Grassmann constellations and the unitary constellations because the rotation codes exploit additional knowledge of the optical channel property. 
     The multi-dimensional crosstalk-free constellations are further enhanced by introducing additional error correction codes, such as low-density parity check codes and multi-dimensional trellis-coded modulations to achieve coding gains. 
     EFFECT OF THE INVENTION 
     The multi-dimensional crosstalk-free constellations according to embodiments of the invention have more than 3 dB improvement compared to the prior art DP constellations, at a high SNR and over a long fiber distance. In addition to the XPolM impairments, the method of the invention can deal with other nonlinear impairments of SPM and XPM as well as linear impairment of PMD, whereas the prior art schemes cannot do so. 
     The method of the invention increases the reach of the fiber distance by improving signal quality. Hence, the invention makes a considerable impact for the future long-haul optical communications which achieves beyond 100 Gbps data rate with DWDM and POLMUX. In telecommunication, the term “long-haul” pertains to switched circuits that span large distances, such as inter-state, and international communications. 
     The method offers a low-complexity receiver process which does not require polarization crosstalk cancellers and channel estimations. 
     The method achieves high quality factor because of the increased dimension of the constellations. Additional gains are available when multi-dimensional trellis-coded modulations and error correction codes are used in conjunction with the method of the invention. 
     Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications may be made within the spirit and scope of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.