Abstract:
Aspects of the present invention provide techniques for compensating nonlinear impairments of a signal traversing an optical communications system. A parallel array of linear convolutional filters are configured to process a selected set of samples of the signal to generate an estimate of a nonlinear interference field. The predetermined set of samples comprises a first sample and a plurality of second samples. A processor applies the estimated nonlinear interference field to the first sample to least partially compensate the nonlinear impairment.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based on, and claims benefit of, U.S. provisional Patent Application No. 61/875,381 filed Sep. 9, 2013, the entire content of which is hereby incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to optical communications systems, and more particularly to methods and systems for reduced complexity nonlinear compensation in an optical communications system. 
     BACKGROUND 
     The integrity of an optical signal transmitted over long lengths of optical fiber degrades due to nonlinear interference (NLI) generated between different spectral components of the modulated optical field. Following detection, the NLI results in a noise-like degradation of the received signal that, in addition to amplified spontaneous emission (ASE) and other possible sources of noise in the receiver, increases the bit error rate and limits transmission reach. Unlike ASE, however, NLI is in principle predictable, and can be estimated for a known transmit data sequence and from the physical characteristics of the optical fiber link. If the complexity of an NLI estimation algorithm is sufficiently low, NLI can be compensated through appropriate digital signal processing (DSP) at either the transmitter or receiver of a coherent optical transmission system. For coherent transmission, the complexities of present methods of nonlinear pre-compensation grow rapidly with channel memory (in unit intervals, UI, or, equivalently, signalling intervals) to make implementation impractical for systems with channel memories exceeding 100 UI. 
     Techniques for addressing this limitation are desired. 
     SUMMARY 
     Aspects of the present invention provide techniques for compensating nonlinear impairments of a signal traversing an optical communications system. A parallel array of linear convolutional filters are configured to process a selected set of samples of the signal to generate an estimate of a nonlinear interference field. The predetermined set of samples comprises a first sample and a plurality of second samples. A processor applies the estimated nonlinear interference field to the first sample to at least partially compensate the nonlinear impairment. 
     The present techniques reduce the computational complexity of estimating the NLI developed within the bandwidth of a single optical channel, i.e. intra-channel nonlinearity such as self-phase modulation (SPM), and the NLI developed between adjacent subcarriers of a multiple subcarrier optical modulation, i.e. inter-sub-channel nonlinearity such as cross-phase modulation (XPM) and cross-polarization modulation (XPoIM). Time and frequency domain variants of this NLI compensation algorithm are described. The resulting low-complexity NLI compensation scheme may improve the maximum achievable system margin for 16QAM modulation by ˜1 to 2 dB over transmission distances of 1000-2000 km of G.0.652 fiber for a single optical channel comprised of one or more optical subcarriers. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features and advantages of the present invention will become apparent from the following detailed description, taken in combination with the appended drawings, in which: 
         FIG. 1  is a block diagram schematically illustrating elements of an optical communications system in which methods in accordance with the present invention may be implemented; 
         FIGS. 2A and 2B  are block diagrams schematically illustrating pre-compensation and post-compensation of nonlinear impairments in accordance with representative embodiments of the present invention; 
         FIG. 3  is a block diagram schematically illustrating elements of an evaluator of  FIG. 2A ; 
         FIGS. 4A and 4B  are charts schematically illustrating representative performance of the embodiments of  FIGS. 2-3  as a function of number of parallel convolutional filters for polarization multiplexed 16QAM modulation (PM-16QAM) over 1040 km standard single mode fiber (SMF); 
         FIGS. 5A and 5B  are charts schematically illustrating the representative performance of a variant of the embodiments of  FIGS. 2-3 , as a function of the number of additions employed in the NLI compensation, for PM-16QAM transmitted over 1040 km SMF; 
         FIG. 6  is a chart illustrating a representative comparison between the Relative System Margin (RSM) for PM-16QAM transmitted over 1040 km SMF without NLI compensation; full complexity NLI compensation; and the simplified NLI compensation in accordance with embodiments of the present invention; 
         FIG. 7  is a diagram illustrating a link divided into two sub-links by means of partial electronic compensation of chromatic dispersion (CD) at the optical transmitter; 
         FIG. 8  is a chart illustrating the representative effect of partial electronic compensation of chromatic dispersion (CD) at the optical transmitter, in combination with simplified NLI compensation in accordance with embodiments of the present invention for a single PM-16QAM optical subcarrier with 131.25 GHz modulation bandwidth transmitted over 1040 km SMF; 
         FIGS. 9A and 9B  are diagrams schematically illustrating frequency domain pre-compensation of NLI in accordance with another representative embodiment of the present invention; 
         FIG. 10  is a block diagram schematically illustrating elements of an evaluator of  FIG. 9A ; 
         FIG. 11  is a chart illustrating the RSM of the embodiment of  FIGS. 9 and 10  as a function of optical Launch Power, for PM-16QAM modulation with and without full complexity NLI compensation, for a single optical carrier with 131.25 GHz modulation bandwidth transmitted over 1040 km SMF; 
         FIG. 12  is a chart illustrating the Maximum RSM (MSM) of the embodiment of  FIGS. 9 and 10 , as a function of the number of PM-16QAM modulated optical subcarriers occupying a 131.25 GHz modulation bandwidth, with and without full complexity, frequency domain NLI compensation, transmitted over 1040 km SMF; 
         FIG. 13  is a chart illustrating a second representative MSM of the embodiment of  FIGS. 9 and 10 , as a function of the number of PM-16QAM modulated optical subcarriers occupying a 131.25 GHz modulation bandwidth, with and without full complexity, frequency domain NLI compensation, transmitted over 1040 km TrueWave Classic (TWc) fiber; 
         FIG. 14  is a chart illustrating a third representative MSM of the embodiment of  FIGS. 9 and 10 , as a function of the number of PM-16QAM modulated optical subcarriers occupying a 131.25 GHz modulation bandwidth, with and without full complexity, frequency domain NLI compensation, transmitted over 1040 km SMF with 90% inline optical dispersion compensation; 
     
    
    
     It will be noted that throughout the appended drawings, like features are identified by like reference numerals. 
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a representative optical communication system in which techniques in accordance with the present disclosure may be implemented. In the optical communication system of  FIG. 1 , a transmitter  2  generally comprise an encoder  4  for encoding a pair of data signals (Sx and Sy) using an N d -dimensional symbol constellation to generate a drive signal  6 . The drive signal is then supplied to a dual-polarization modulator  8  for modulating respective dimensions of a continuous wave (CW) optical carrier in accordance with the drive signal. In the example of  FIG. 1 , a pair of data signals (Sx and Sy) may be encoded as (N d =4)-dimensional symbols {right arrow over (d)}(k)=[x(k),y(k)], where x(k) and y(k) are complex-valued components of the k:th transmitted symbol vector {right arrow over (d)}(k). Each of the components x(k) and y(k) of the symbol vector {right arrow over (d)}(k) are processed by a driver circuit  10  to generate samples {right arrow over (A)}(t j )=[x(t j ), y(t j )], where x(t j ) and y(t j ) are complex-valued components of {right arrow over (A)}(t j ) at the j:th sampling instant t j . Each of the components x(t j ) and y(t j ) of the sample vector {right arrow over (A)}(t j ) are converted into respective analog drive signals  6  by an Analog-to-digital (A/D) converter block  12  and modulated onto two dimensions (e.g. phase and amplitude, or I and Q) of a respective one of the two orthogonal polarizations of the CW optical carrier. The CW carrier is typically generated by a laser  14  in a manner known in the art, and the dual-polarization modulator  8  may be implemented using a polarization beam splitter  15  and any of a variety of known modulator devices, such as phase modulators, variable optical attenuators, Mach-Zehnder interferometers, etc. The modulated optical signal appearing at the output of the dual-polarization modulator  8  is transmitted through an optical fiber link  16  to a coherent receiver  18 . 
     A receiver  18  configured to receive and detect the transmitted data signals may be provided as a coherent receiver, which includes a polarization beam splitter  20  for splitting the received optical signal into received X and Y polarizations, an optical hybrid  22  for separately mixing the X and Y polarizations with a local oscillator, and a set of photodetectors  24  for detecting the optical power of each of the mixing products generated by the optical hybrid  22 . An Analog-to-Digital (A/D) converter block  26  samples each photodetector current, and the resulting sample streams—each of which is related to the modulated dimensions of the optical carrier field—are processed by a Digital Signal Processor  28  in accordance with the N d -dimensional constellation to generate recovered signals Rx and Ry that correspond with the transmitted data signals Sx and Sy. 
     During transmission through the link  16 , the modulated optical signal is subject to distortions due to linear intra-channel effects such as chromatic dispersion, polarization dependent loss, polarization mode dispersion, and nonlinear intra-channel effects such as self-phase modulation, among others. As such, the j:th received sample vector {right arrow over (R)}(t j ) detected by the receiver  18  can be related to its corresponding transmitted sample vector {right arrow over (A)}(t j ) as
 
 {right arrow over (R)} ( t   j )≈ {right arrow over (A)} ( t   j )+Δ {right arrow over (a)} ( t   j )+ {right arrow over (n)} ( t   j )  Eq.1
 
where Δ{right arrow over (a)}(t j ) is the intra-channel nonlinear interference (NLI) field and {right arrow over (n)}(t j ) represents any additional sources of additive noise or distortion, such as, for example, ASE.
 
     The intra-channel NLI field, Δ{right arrow over (a)}(t j ) can be estimated as a sum:
 
Δ {right arrow over (a)} ( t   j )≈Σ m=−m   M Σ n=−N   N   C   mn   {right arrow over (A)}   H ( t   j+m+n ) {right arrow over (A)} ( t   j+n ) {right arrow over (A)} ( t   j+m )  Eq.2
 
involving interactions between the transmitted sample vector {right arrow over (A)}(t j )=[x(t j ), y(t j )], weighted by a complex number, C mn , that characterizes the efficiency of the nonlinear interaction. In this formulation, the superscript H denotes the Hermitian conjugate (i.e. the adjoint) of {right arrow over (A)}(t j ), while the summation ranges over the number of samples, 2·max(M, N)+1, that interact through chromatic dispersion (i.e. the channel memory). The coefficients C mn  quantify all relevant details of the dispersion map and transmitter pulse shape.
 
     As is known in the art, the sample frequency (or rate) will typically be higher than the symbol rate. For example, in conventional Nyquist sampling, the sample frequency (or rate) will typically be twice the bandwidth of a bandlimited signal. Other sampling techniques are known in which the ratio between the sample rate and the signal bandwidth is not equal to 2. Accordingly, in the general case, the present technique will normally be implemented in the “sample domain”, with signals being processed on a sample-by-sample basis at the appropriate sample rate. However, for the purpose of understanding the present technique, it is convenient to consider the special case in which the sample rate is equal to that of the symbol rate, and the sample timing corresponds with the symbol timing. In this case, the sample vectors {right arrow over (A)}(t j )=[x(t j ), y(t j )] correspond with the symbol vectors {right arrow over (d)}(k)=[x(k), y(k)], and Eq. 2 above can be reformulated as:
 
Δ {right arrow over (a)} ( k )≈Σ m=−M   M Σ n=−N   N   C   mn   {right arrow over (d)}   H ( k+m+n ) {right arrow over (d)} ( k+n ) {right arrow over (d)} ( k+m )  Eq.2A
 
     The following description will use this special case for the purposes of explaining the present technique, it being understood that it will be appropriate to implement many embodiments in the sample domain, using the corresponding sample-domain formulations. 
     Referring to  FIG. 2A , pre-compensation of the nonlinear interference can be realized by subtracting the estimate Δ{right arrow over (a)}(k) from the transmit data sequence through appropriate digital signal processing at the transmitter:
 
 {right arrow over (d)}   precomp ( k )= {right arrow over (d)} ( k )−Δ {right arrow over (a)} ( k )  Eq.3
 
     In the arrangement of  FIG. 2A , each successive symbol vector {right arrow over (d)}(k)=[x(k), y(k)] generated by the encoder  4  is supplied to an evaluator  30  which operates to calculate the intra-channel NLI field Δ{right arrow over (a)}(k) based on both the k:th symbol vector {right arrow over (d)}(k) and a plurality of other symbol vectors. The computed value of Δ{right arrow over (a)}(k) is then subtracted from its corresponding symbol {right arrow over (d)}(k) to yield {right arrow over (d)} precomp (k), which can then be supplied to the drive circuit  10  and modulators  8  for transmission. 
     As may be seen in  FIG. 2B , post-compensation of NLI can be realized by subtracting the estimate Δ{right arrow over (a)}(k) from the received data sequence through appropriate DSP at the receiver
 
 {right arrow over (r)}   postcomp ( k )= {right arrow over (r)} ( k )−Δ {right arrow over (a)} ( k )  Eq.4
 
     In the arrangement of  FIG. 2B , sample streams generated by the A/D converter block  26  are processed by a detector  32  to recover symbol vectors {right arrow over (r)}(k), which are supplied to a decision block  33  that operates to generate estimates of the transmitted symbol vectors, {circumflex over (d)}(k). The transmitted symbol estimates {circumflex over (d)}(k) are then supplied to an evaluator  34  which operates to calculate the intra-channel NLI field Δ{right arrow over (a)}(k) (which may be computed according to Eq. 2A above by replacing {right arrow over (d)}(k) with the estimated {right arrow over (d)}(k)) based on both the k:th transmitted symbol estimate {circumflex over (d)}(k) and a plurality of other transmitted symbol estimates. The computed value of Δ{right arrow over (a)}(k) is then subtracted from its corresponding symbol {right arrow over (r)}(k) to yield {right arrow over (r)} postcomp (k), which can then be supplied to a decoder  35  to determine appropriate values of the recovered signals Rx and Ry. 
     In both pre- and post-compensation scenarios, the practicality of compensating intra-channel interference (and especially NLI such as self-phase modulation) largely depends upon the computational complexity involved in the evaluation of Δ{right arrow over (a)}(k). This, in turn, is a function of the channel memory. For example, it can be shown that for practical optical fiber links of between 1000 and 2000 km in length, the evaluation of Eq. 2A above may require the summation of over 10000 terms. 
     In accordance with the present technique, Δ{right arrow over (a)}(k) can be evaluated using a parallel series of linear convolutional filters operating on the product {right arrow over (d)} H  (k+m+n){right arrow over (d)}(k+n) of transmit symbol vectors {right arrow over (d)}(k). With this arrangement, an equivalent of 15 complex multiplies per symbol to evaluate Δ{right arrow over (a)}(k) can be achieved for channel memories of ˜100 UI through appropriate quantization of the C mn  coefficients. This represents a reduction in complexity equivalent to a factor of 13 relative to the calculation of the double summation involved in Eq. 2A above. 
       FIG. 3  schematically illustrates an embodiment of the evaluator  30  in which parallel linear convolutional filters are used to evaluate Δ{right arrow over (a)}(k) in the time-domain. In the embodiment of  FIG. 3 , the evaluator  30  comprises a set of N filter  linear filters, each of which includes N tap  filter taps. Each filter tap computes the product C mn {right arrow over (d)} H (k+m+n){right arrow over (d)}(k+n) for a respective value of (m, n), and the results are then summed together and multiplied by {right arrow over (d)}(k+m). The outputs of the N filter  linear filters are then summed to calculate Δ{right arrow over (a)}(k). With this architecture, the correction field Δ{right arrow over (a)}(k+1) at the next symbol index k+1 requires only (N filter −1)/2+1 additional evaluations of the product {right arrow over (d)} H (k+1+m+n){right arrow over (d)}(k+1+n); that is, for m=0, 1, . . . , (N filter −1)/2. In the embodiment of  FIG. 3 , the index “m” identifies a respective one of the linear convolutional filters, and the index “n” identifies a respective tap within a given linear convolutional filter. 
     In the typical case where the coefficients have C mn =−C* m,−n  symmetry, the required number of linear filters is reduced by a factor of ˜2. 
     In embodiments in which the coefficients C mn  are quantized to a sufficiently small number of discrete levels, the linear convolutional filters can be replaced by summations without significantly reducing system performance. This technique can yield satisfactory performance when the coefficients C mn  are quantized to 4 discrete levels or less. 
     In dispersion uncompensated networks, the memory length of the optical channel is minimized with 50% electronic CD pre-compensation applied through appropriate DSP at the transmitter. In this case, the required number of linear filters is reduced by a factor of ˜2. Further, when the electronic CD pre-compensation is optimized in systems with arbitrary optical dispersion compensation, the coefficients C mn  are (approximately) imaginary valued. Under these conditions, the required number of complex operations is also reduced by a factor of ˜2. 
     In networks with a high degree of optical dispersion compensation, N filter  may range from N filter =1 to N filter =4. 
     As may be appreciated, the above techniques may be implemented individually or in combination. In some embodiments, Δ{right arrow over (a)}(k) may be evaluated in the optical transmitter, and used to pre-compensate the optical signal prior to transmission. In other embodiments, Δ{right arrow over (a)}(k) may be evaluated in the optical receiver, and used to post-compensate the optical signal received through the link  12 . 
       FIGS. 4A and 4B  illustrate the performance of pre-compensation using Δ{right arrow over (a)}(k) as calculated with the evaluator  30  of  FIG. 3 , as a function of the number N filter  of linear filters, for polarization multiplexed 16QAM modulation (PM-16QAM) over 1040 km dispersion uncompensated standard single mode fiber (SMF). As may be seen in these figures, both the nonlinear signal-to-noise ratio (SNR) and the required optical signal-to-noise ratio (ROSNR), evaluated at an uncoded bit-error rate (BER) of 0.034, improve with an increasing number of filters, with the majority of the improvement occurring with N filter &lt;10 filters. This implies that reasonable performance can be obtained using an evaluator having N filter =10 linear filters, which, using the simplification techniques described above, implies an evaluator having 6 taps per filter for a total of 32 complex multipliers and 21 adders. Increasing the number of linear filters beyond N filter =10 can yield further increases in performance, but the rate of improvement is significantly lower than that obtained with N filter &lt;10. 
     As noted above, the filter taps can be realized by additions instead of multiplications. For example, if C mn  is constant for a given index m, then a next successive filter output can be obtained by subtracting the final filter input (e.g. {right arrow over (d)} B (k−4){right arrow over (d)}(k−2)) and adding the new input (e.g. {right arrow over (d)} H (k+1){right arrow over (d)}(k+3)). The number of additions required to perform this operation depends on the quantization level of C mn . As noted above, satisfactory results may be obtained by using values of C mn  having up to four quantization levels.  FIGS. 5A and 5B  illustrate performance of pre-compensation using Δ{right arrow over (a)}(k) evaluated using this technique, as a function of the number of additions for PM-16QAM transmitted over 1040 km dispersion uncompensated SMF. As may be seen in these figures, performance improves with an increasing number of additions, with the majority of the improvement occurring with &lt;300 additions. This implies that reasonable performance can be obtained when the number of quantization levels of Δ{right arrow over (a)}(k) is selected to yield ˜300 additions. Increasing the number of quantization levels (and therefore the number of additions) can yield further increases in performance, but the rate of improvement is lower than that obtained when the number of additions is &lt;300. 
       FIG. 6 , illustrates the relative system margin (RSM) versus Launch Power in an optical link using full complexity NL pre-compensation (i.e. following Equation 2A above), simplified Time-Domain NL pre-compensation using the methods described above with reference to  FIGS. 3-5 , and with no NL pre-compensation, for PM-16QAM over 1040 km dispersion uncompensated SMF. Here, the RSM, RSM dB =P dBm −RSNR dB , in units of decibels, is taken to be the difference between the channel&#39;s average power, P dBm , and the required electrical signal-to-noise ratio, RSNR dB , at an uncoded BER of 0.034. As may be seen in  FIG. 6 , the full complexity NL pre-compensation can ideally achieve a 2.7 dB Maximum RSM (MSM) improvement as compared to no NL pre-compensation. By comparison, the simplified NL pre-compensation technique (using 32 complex multipliers, 21 adders and 6 taps per filter) can achieve a 2 dB MSM improvement as compared to the case of no NL precompensation, but with dramatically less processing complexity (and thus cost) than is required for full complexity NL pre-compensation. 
     The paragraphs above describe techniques for pre- and post-compensation of nonlinear impairments using a simplified time-domain evaluation of the intra-channel nonlinear interference field, Δ{right arrow over (a)}(k). In some embodiments, these techniques may be used in conjunction with electronic Chromatic Dispersion (CD) pre- and/or post-compensation. In some embodiments, approximately 50% of the link CD may be compensated in each of the transmitter and receiver. In particular, in the case of dispersion uncompensated systems with 50% electronic CD pre-compensation, the fiber link can be considered to be the concatenation of two constituent sub-links as shown in  FIG. 7 . 
     For the case in which the total length of the link is 1040 km, the C mn  coefficients for the entire link can be calculated as:
 
 C   mn (1040 km)− C   mn (520 km,−β 2 )+ C   mn (520 km,β 2 )
 
Ignoring the effects of attenuation, the coefficients of the two constituent sub-links satisfy C mn (520 km, −β 2 )=−C* mn  (520 km, β 2 ) and, consequently, the final C mn (1040 km) coefficients are purely imaginary. Moreover, C mn (1040 km) are concentrated near the m=0 and n=0 axes, implying that the required number of terms in Eq. 2A can be significantly reduced while maintaining equivalent system performance. Further, with optimum electronic CD pre-compensation, the C mn  coefficients can be quantized to ≦4 discrete levels with &lt;0.2 dB reduction in system performance.
 
       FIG. 8  illustrates the effect of the above-noted simplified NL pre-compensation in combination with 50% electronic CD pre-compensation for PM-16QAM transmitted over 1040 km dispersion uncompensated SMF. For ease of comparison, the performance curves from  FIG. 6  are replicated in dashed lines. In the illustrated example, the simplified NL pre-compensation is implemented with an evaluator  30  comprising only 10.5 complex multipliers and 26 additions. As may be seen in  FIG. 8 , 50% electronic CD pre-compensation alone without NL pre-compensation yields a small, 0.5 dB improvement in MSM. The combination of simplified NL pre-compensation and 50% electronic CD pre-compensation reduces the MSM by ˜0.6 dB as compared to full complexity NL pre-compensation ( FIG. 6 ). This implies that the use of electronic CD pre-compensation in combination with the present simplified NL pre-compensation techniques achieves a performance comparable to the full complexity implementation. 
     The above paragraphs describe embodiments in which the NLI field, Δ{right arrow over (a)}(k), is evaluated in the time domain using 2·(M+N)+1 successive dual-polarization transmit symbol vectors, {right arrow over (d)}(k). If desired, the NLI field may be evaluated using an array of band-pass filters that separate the modulated optical field, {right arrow over (A)}(t), into a number of spectral sub-bands, N SB . The intra-sub-band and inter-sub-band NLI fields are evaluated for each sub-band and the results recombined to obtain the total NLI field estimate associated with {right arrow over (A)}(t). Letting {right arrow over (A)}(ω) be the frequency-domain representation of {right arrow over (A)}(t), the modulated optical field is decomposed into N SB  sub-bands, {right arrow over (B)} l (ω), so that
 
 {right arrow over (A)} (ω)=Σ l=1   N     SB     {right arrow over (B)}   l (ω−Ω l ).  Eq. 5
 
     Here, Ω l  denotes the center frequency of the l:th sub-band, while, in what follows, {right arrow over (B)} l (t) is the time-domain representation of {right arrow over (B)} l (ω). Including both intra- and inter-sub-band nonlinear interactions, the NLI field, Δ{right arrow over (a)} l (t j ), of the l:th sub-band at sampling instant t j , is estimated as 
                     Δ   ⁢           ⁢         a   →     l     ⁡     (     t   j     )         ≈       ∑       l   1     =     (     1   -   l     )           N   SB     -   l       ⁢       ∑       l   2     =     (     1   -   l   -     l   1       )           N   SB     -   l   -     l   1         ⁢       ∑     m   =     -     M     l   1             M     l   1         ⁢       ∑     n   =     N     l   1           N     l   1         ⁢         C   mn     ⁡     (       l   1     ,     l   2       )       ⁢         B   →       l   +     l   1     +     l   2       H     ⁡     (     t     j   +   m   +   n       )       ⁢         B   →       l   +     l   2         ⁡     (     t     j   +   n       )       ⁢         B   →       l   +     l   1         ⁡     (     t     j   +   m       )                         Eq   .           ⁢   6               
involving {right arrow over (B)} l (t j ) of differing sub-bands, weighted by a complex number, C mn (l 1 , l 2 ), that characterizes the efficiency of the nonlinear interaction. For example, in Eq. 6 the intra-sub-band nonlinear field is described by those terms with (l 1 =0, l 2 =0), while the largest contributions to the inter-sub-band nonlinear field occur whenever (l 1 =0, l 2 ≠0) or (l 1 ≠0, l 2 =0). Terms appearing in Eq. 6 with (l 1 ≠, l 2 ≠0) are associated with four-wave-mixing processes, and are typically negligible in optical fibers with non-zero chromatic dispersion. The number of samples that interact through chromatic dispersion, described by M l  and N l , in general depend upon the sub-band index l.
 
     The total NLI field estimate, Δ{right arrow over (a)}(t j ), is obtained from Δ{right arrow over (a)} l (t j ) by evaluating Δ{right arrow over (a)} l (ω n ) at the discrete sample-frequency ω n  through an appropriate Discrete Fourier Transform (DFT) of Δ{right arrow over (a)}(t j ). A filter H l (ω n ) is selected to limit the spectral extent of Δ{right arrow over (a)} l (ω n ) and the total nonlinear field
 
Δ {right arrow over (a)} (ω n )=Σ l=1   N     SB     H   l (ω n −Ω l )Δ {right arrow over (a)}   l (ω n −Ω l )  Eq. 7
 
is computed. The time-domain received optical field {right arrow over (R)}(t j ), incident upon the receiver  14  is then related to the corresponding transmitted optical field, {right arrow over (A)}(t j ), through
 
 {right arrow over (R)} ( t   j )≈ {right arrow over (A)} ( t   j )+Δ {right arrow over (a)} ( t   j )+ {right arrow over (n)} ( t   j )  Eq. 8
 
where Δ{right arrow over (a)}(t j ) is the inverse-DFT of Δ{right arrow over (a)}(ω n ) and {right arrow over (n)}(t j ) represents any additional sources of additive noise or distortion.
 
     In accordance with the present technique, Eq. 6 can be evaluated using a parallel series of linear convolutional filters operating on the terms {right arrow over (B)} H   l+l     1     +l     2   (t j+m+n ){right arrow over (B)} l+l     2   (t j+n ). The complexity of the linear convolutional filters can be significantly reduced using the techniques outlined in the preceding discussion. 
     It is known in the art that modulating data onto multiple optical subcarriers, with Nyquist or near-Nyquist subcarrier frequency separation, can increase tolerance to nonlinear interference, particularly when the total modulation bandwidth exceeds ˜50 GHz. In this embodiment, {right arrow over (B)} l (t j ) and Δ{right arrow over (a)} l (t j ) in Eq. 6 are identified as the modulated optical field and NLI field of the l:th optical subcarrier, respectively. 
     For the special case in which the sampling rate is equal to the subcarrier symbol rate them at appropriate sampling phase, the sample vectors {right arrow over (B)} l (t j ) and Δ{right arrow over (a)} l (t j ) correspond with the symbol vectors {right arrow over (d)} l (k) and Δ{right arrow over (a)} l (k), respectively, and Eq. 6 above can be reformulated as: 
                     Δ   ⁢           ⁢         a   →     l     ⁡     (   k   )         ≈       ∑       l   1     =     (     1   -   l     )           N   SC     -   l       ⁢       ∑       l   2     =     (     1   -   l   -     l   1       )           N   SC     -   l   -     l   i         ⁢       ∑     m   =     -     M     l   1             M     l   1         ⁢       ∑     n   =     -     N     l   1             N     l   1         ⁢         C   mn     ⁡     (       l   1     ,     l   2       )       ⁢         d   →       l   +     l   1     +     l   2       H     ⁡     (     k   +   m   +   n     )       ⁢         d   →       l   +     l   2         ⁡     (     k   +   n     )       ⁢         d   →       l   +     l   1         ⁡     (     k   +   m     )                         Eq   .           ⁢   9               
where N SC  is the number of modulated subcarriers. The received symbol vector {right arrow over (r)} l (k) of the l:th subcarrier detected by the receiver  14  can be related to its corresponding transmitted symbol vector {right arrow over (d)} l (k) as
 
 {right arrow over (r)}   l ( k )≈ {right arrow over (d)}   l ( k )+Δ {right arrow over (a)}   l ( k )+{right arrow over (n)} l ( k )  Eq. 10
 
where {right arrow over (n)} l (k) represents any additional sources of additive noise or distortion.
 
     Pre-compensation or post-compensation of intra- and inter-subcarrier NLI is realized by subtracting the estimate Δ{right arrow over (a)} l ((k) from either the transmit, {right arrow over (d)} l (k), or received, {right arrow over (r)} l (k), symbol vectors, respectively. 
     The above paragraphs describe embodiments in which Δ{right arrow over (a)}(k) is evaluated in the time domain, using 2·(M+N)+1 successive dual-polarization transmit symbol vectors {right arrow over (d)}(k). If desired, the nonlinear interference may also be evaluated in the frequency domain.  FIG. 9A  illustrates an example embodiment that may be implemented in the transmitter  2 . In this case, the stream of {right arrow over (A)}(t j ) output from the encoder  4  are supplied to a Fourier Transform (FT) block, which computes a corresponding series of signal vectors {right arrow over (A)}(ω k ) representing the DFT of {right arrow over (A)}(t j ). The vectors {right arrow over (A)}(ω k ) are supplied to an evaluator which operates to compute the intra-channel nonlinear interference field, Δ{right arrow over (A)}(ω k ), which is subtracted from {right arrow over (A)}(ω k ) to yield a pre-compensated signal vector {right arrow over (A)} precomp (ω k ). The pre-compensated vector {right arrow over (A)} precomp (ω k ) is then supplied to an inverse Fourier Transform block, which computes the inverse-DFT of {right arrow over (A)} precomp (ω k ) to generate the time domain pre-compensated signal vector {right arrow over (a)} precomp (t k ). In some embodiments, the intra-channel nonlinear interference field Δ{right arrow over (A)}(ω k ) can be computed by the evaluator according to
 
Δ {right arrow over (A)} (ω k )≈Σ m=−M   M Σ n=−N   N   {tilde over (C)}   mn   {right arrow over (A)}   H (ω k+m+n ) {right arrow over (A)} (ω k+n ) {right arrow over (A)} (ω k+m )  Eq. 11
 
in which {right arrow over (A)} H  is the Hermitian conjugate (i.e. the adjoint) of {right arrow over (A)} and ω k  is the k:th sample frequency of {right arrow over (A)}(ω) or {right arrow over (A)} H (ω), as applicable. The coefficient, {tilde over (C)} mn , is a complex number characterizing the efficiency of the frequency domain nonlinear interaction. The summation ranges over the frequency bins within a (provisionable) bandwidth Δω. The summation can be implemented as a parallel series of linear filters, as may be seen in  FIG. 10 . As in the embodiment of  FIG. 3 , in the embodiment of  FIG. 10  the evaluator  30  comprises a set of N filter  linear filters, each of which comprises a set of N tap  filter taps. Additionally, the index “m” identifies a respective one of the linear convolutional filters and the index “n” identifies a respective tap within a given linear convolutional filter. In the embodiment of  FIG. 10 , however, each filter tap computes the product {tilde over (C)} mn {right arrow over (A)} H (ω k+m+n ){right arrow over (A)}(ω k+n ) for a respective value of (m, n), and the results are then summed together and multiplied by {right arrow over (A)}(ω k+n ).
 
     The performance of the frequency domain NL pre-compensation improves with increasing bandwidth Δω and approaches a limiting value for some Δω NL , termed the nonlinear bandwidth, which is related to fiber type and link length. For high net dispersion systems, Δω NL  is generally less than 2πF S  where F S  is the transmitter modulation bandwidth. It follows that the NL pre-compensation can be applied within a window of spectral width Δω NL  centered on the k:th discrete sample-frequency, ω k , as shown in  FIG. 9B . In this ease, the complexity of the NL pre-compensation implementation increases linearly with the modulation bandwidth, implying that the computational complexity per discrete frequency sample ω k  is constant. Accordingly, in some embodiments the evaluator may operate to compute Δ{right arrow over (A)}(ω k ) for each frequency sample ω k  using only those frequency samples lying within the frequency range 
               ω   k     ±       1   2     ⁢   Δ   ⁢           ⁢       ω   NL     .             
This can reduce the number of linear filters required to implement the evaluator to match the number of frequency samples lying within the window. With this arrangement, Δ{right arrow over (A)}(ω k ) may be calculated by “sliding” the window across the vector {right arrow over (A)}(ω k ) output from the FT block, and determining a respective value of Δ{right arrow over (A)}(ω k ) for each k.
 
     As may be appreciated, complexity of the Frequency Domain evaluator also depends on the length of the DFT employed within the FT block. For NL pre-compensation, the channel memory within each spectral window 
                 ω   k     -       1   2     ⁢   Δ   ⁢           ⁢     ω   NL         &lt;   ω   &lt;       ω   k     +       1   2     ⁢   Δ   ⁢           ⁢     ω   NL               
is linearly proportional to the modulation bandwidth. Accordingly, the DFT length in the FT block must also increase in proportion to the modulation bandwidth. However, since the number of linear filters required to implement NL pre-compensation is determined solely by Δω NL , the implementation complexity per frequency sample ω k  is again independent of modulation bandwidth.
 
       FIG. 11  compares the RSM with and without frequency domain NL pre-compensation, respectively. For PM-16QAM transmitted over 1040 km dispersion uncompensated SMF. This calculation employed a single optical subcarrier with F S =131.25 GHz modulation bandwidth, an 8192 sample DFT with 50% overlap-and-save, 50% electronic CD pre-compensation, Δω NL ≈33 GHz and a 256×256 coefficient matrix {tilde over (C)} mn . As shown in  FIG. 11 , the frequency domain NL pre-compensation in this case results in a maximum 1.8 dB improvement in MSM. 
       FIG. 12  illustrates the MSM improvement vs. Member of Subcarriers both with and without NL pre-compensation, respectively, under the same conditions as  FIG. 11 , except that the signal spectrum was comprised of a number of modulated optical subcarriers. As may be seen in  FIG. 12 , the use of 12 optical subcarriers in combination with full complexity NL precompensation yields a maximum 2.6 dB MSM improvement relative to a single subcarrier without NL pre-compensation. In embodiments in which a large number of optical subcarriers are used, the size of the DFT may be reduced. For example, in some embodiments, a DFT length of 2048 samples may be used without significantly reducing the MSM. 
       FIG. 13  illustrates the MSM improvement vs. Number of Subcarriers both with and without NL pre-compensation, respectively, under the same conditions as  FIG. 12 , except for 1040 km dispersion uncompensated True Wave Classic (TWc) fiber with a local dispersion of 2.8 ps/nm/km and a 1024 sample DFT with 50% overlap-and-save. As shown in  FIG. 13 , frequency domain NL pre-compensation with 5 optical subcarriers yields a 2.6 dB increase in MSM relative to a single optical subcarrier without NL pre-compensation. 
       FIG. 14  illustrates the MSM improvement vs. Number of Subcarriers both with and without NL pre-compensation, respectively, under the same conditions as  FIG. 12 , except for 1040 km SMF with 90% inline optical dispersion compensation and a 1024 sample DFT with 50% overlap-and-save. As shown in  FIG. 14 , frequency domain NL pre-compensation with 8 optical subcarriers yields a 4.0 dB increase in MSM relative to a single optical subcarrier without NL pre-compensation. 
     In the preceding description, for purposes of explanation, numerous details are set forth in order to provide a thorough understanding of the embodiments of the invention. However, it will be apparent to one skilled in the art that these specific details are not required in order to practice the invention. In other instances, well-known electrical and/or optical structures and circuits are shown in block diagram form in order not to obscure the invention. For example, specific details are not provided as to whether the embodiments of the invention described herein are implemented as a software routine, hardware circuit, firmware, or a combination thereof. 
     Embodiments of the invention can be represented as a software product stored in a machine-readable medium (also referred to as a computer-readable medium, a processor-readable medium, or a computer usable medium having a computer-readable program code embodied therein). The machine-readable medium can be any suitable tangible medium, including magnetic, optical, or electrical storage medium including a diskette, compact disk read only memory (CD-ROM), memory device (volatile or non-volatile), or similar storage mechanism. The machine-readable medium can contain various sets of instructions, code sequences, configuration information, or other data, which, when executed, cause a processor to perform steps in a method according to an embodiment of the invention. Those of ordinary skill in the art will appreciate that other instructions and operations necessary to implement the described invention can also be stored on the machine-readable medium. Software running from the machine-readable medium can interface with circuitry to perform the described tasks. 
     The above-described embodiments of the invention are intended to be examples only. Alterations, modifications and variations can be effected to the particular embodiments by those of skill in the art without departing from the scope of the invention, which is defined solely by the claims appended hereto.