Backward propagation with compensation of some nonlinear effects of polarization mode dispersion

Abstract

An optical transport system configured to compensate nonlinear signal distortions using a backward-propagation algorithm in which some effects of polarization mode dispersion on the nonlinear signal distortions are accounted for by employing two or more different approximations of said effects within the bandwidth of the optical communication signal. In an example embodiment, the corresponding digital signal processor (DSP) is configured to switch between different approximations based on a comparison, with a fixed threshold value, of a difference between frequencies of various optical waves contributing to the nonlinear signal distortions, e.g., through four-wave-mixing processes. In different embodiments, the backward-propagation algorithm can be executed by the transmitter's DSP or the receiver's DSP.

Claims

1. An apparatus comprising: an optical transmitter that includes a laser, a plurality of optical modulators, associated electrical drivers, and a digital signal processor, the optical transmitter being configured to modulate a digital symbol stream onto two polarization components of an optical carrier generated by the laser in a manner that pre-compensates some intra-channel nonlinear optical distortions produced by transmission of a resulting modulated optical carrier over an optical fiber; wherein the digital signal processor is electrically connected to control the associated electrical drivers and comprises a digital circuit configured to estimate intra-channel nonlinear optical distortions during the transmission of the modulated optical carrier in a polarization-mode-dispersion dependent manner that gives a lower weight to some contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of larger intra-channel frequency difference than to other contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of smaller intra-channel frequency difference; and wherein the digital signal processor is configured to generate an estimate of the intra-channel nonlinear optical distortions for the two polarization components of the modulated optical carrier based on a comparison, with a fixed threshold value, of a difference between frequencies of the frequency components of the modulated optical carrier contributing to the intra-channel nonlinear optical distortions.

2. The apparatus of claim 1, wherein the digital signal processor is configured to generate the estimate using two or more different approximations of an effect of polarization mode dispersion on the intra-channel nonlinear optical distortions within a bandwidth of the modulated optical carrier.

3. The apparatus of claim 2, wherein the digital signal processor is further configured to generate the estimate using a backward-propagation algorithm, the backward-propagation algorithm being configured to cause a pre-distortion of the modulated optical carrier that tends to be removed by the transmission thereof over the optical fiber.

4. The apparatus of claim 2, wherein the digital signal processor is further configured to switch between the two or more different approximations based on the comparison.

5. The apparatus of claim 4, wherein the digital signal processor is further configured to: apply a first of the two or more approximations if an absolute value of a difference between frequencies of a first frequency component and a second frequency component is smaller than the fixed threshold value; and apply a second of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is greater than the fixed threshold value.

6. The apparatus of claim 5, wherein the digital signal processor is further configured to apply a third of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is smaller than the fixed threshold value, and an absolute value of a difference between frequencies of a third frequency component and the second frequency component is greater than the fixed threshold value.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Other aspects, features, and benefits of various disclosed embodiments will become more fully apparent, by way of example, from the following detailed description and the accompanying drawings, in which:

(2) FIG. 1 shows a block diagram of an optical transport system according to an embodiment;

(3) FIG. 2 shows a block diagram of a DSP that can be used in the optical transport system of FIG. 1 according to an embodiment; and

(4) FIG. 3 shows a block diagram of a backward-propagation module that can be used in the DSP of FIG. 2 according to an embodiment;

(5) FIG. 4 graphically shows different spectral regions in which different approximations of the effect of PMD on nonlinear interference noise (NLIN) can be used according to an embodiment;

(6) FIG. 5 shows a block diagram of an NLIN-compensation module that can be used in the backward-propagation module of FIG. 3 according to an embodiment;

(7) FIG. 6 shows a block diagram of an NLIN estimator that can be used in the NLIN-compensation module of FIG. 5 according to an embodiment; and

(8) FIGS. 7A-7B show flowcharts of signal processing methods that can be used to implement the NLIN-compensation module of FIG. 5 according to an embodiment.

DETAILED DESCRIPTION

(9) FIG. 1 shows a block diagram of an optical transport system 100 according to an embodiment. System 100 has an optical transmitter 110 and a coherent optical receiver 190 connected by way of an optical transport link 140. In an example embodiment, optical transport link 140 can be implemented using one or more sections 142 of optical fiber or fiber-optic cable. For illustration purposes and without any implied limitations, optical transport link 140 is shown as (i) having N sections 142 that are labeled in FIG. 1 as 142.sub.1-142.sub.N and (ii) being an amplified optical link having a plurality of optical amplifiers 144 located between adjacent sections 142 and configured to amplify the optical signals that are being transported through the optical transport link, e.g., to counteract signal attenuation. In an alternative embodiment, optical transport link 140 that has only one or even no optical amplifiers 144 can similarly be used. In some embodiments, optical transport link 140 may also include one or more inline dispersion-compensation modules (e.g., dispersion-compensating fiber (DCF)).

(10) In operation, transmitter 110 receives a digital electrical input stream 102 of payload data and applies it to a digital signal processor (DSP) 112. DSP 112 processes input data stream 102 to generate digital signals 114.sub.1-114.sub.4. In an example embodiment, DSP 112 may perform, inter alia, one or more of the following: (i) de-multiplex input stream 102 into two sub-streams, each intended for optical transmission using a respective one of orthogonal (e.g., X and Y) polarizations of an optical output signal 130; (ii) encode each of the sub-streams using a suitable code, e.g., to prevent error propagation and enable error correction at receiver 190; (iii) convert each of the two resulting sub-streams into a corresponding sequence of constellation symbols; and (iv) perform digital signal pre-distortion, e.g., to mitigate the adverse effects imposed by an electrical-to-optical (E/O) converter (also sometimes referred to as a front-end circuit) 116 of transmitter 110, optical transport link 140, and/or a front-end circuit 172 of receiver 190. In each signaling interval (also referred to as a symbol period or time slot), signals 114.sub.1 and 114.sub.2 carry digital values that represent the in-phase (I) component and quadrature (Q) component, respectively, of a corresponding (possibly pre-distorted) constellation symbol intended for transmission using a first (e.g., X) polarization of light. Signals 114.sub.3 and 114.sub.4 similarly carry digital values that represent the I and Q components, respectively, of the corresponding (possibly pre-distorted) constellation symbol intended for transmission using a second (e.g., Y) polarization of light.

(11) E/O converter 116 operates to transform digital signals 114.sub.1-114.sub.4 into a corresponding modulated optical output signal 130. More specifically, drive circuits 118.sub.1 and 118.sub.2 transform digital signals 114.sub.1 and 114.sub.2, as known in the art, into electrical analog drive signals I.sub.X and Q.sub.X, respectively. Drive signals I.sub.X and Q.sub.X are then used, in a conventional manner, to drive an I-Q modulator 124.sub.X. In response to drive signals I.sub.X and Q.sub.X, I-Q modulator 124.sub.X operates to modulate an X-polarized beam 122.sub.X of light supplied by a laser source 120 as indicated in FIG. 1, thereby generating a modulated optical signal 126.sub.X.

(12) Drive circuits 118.sub.3 and 118.sub.4 similarly transform digital signals 114.sub.3 and 114.sub.4 into electrical analog drive signals I.sub.Y and Q.sub.Y, respectively. In response to drive signals I.sub.Y and Q.sub.Y, an I-Q modulator 124.sub.Y operates to modulate a Y-polarized beam 122.sub.Y of light supplied by laser source 120 as indicated in FIG. 1, thereby generating a modulated optical signal 126.sub.Y. A polarization beam combiner 128 operates to combine modulated optical signals 126.sub.X and 126.sub.Y, thereby generating optical output signal 130, this optical output signal being a polarization-division-multiplexed (PDM) signal. Optical output signal 130 is then applied to optical transport link 140.

(13) After propagating through optical transport link 140, optical signal 130 becomes optical signal 130′, which is applied to receiver 190. Optical signal 130′ may differ from optical signal 130 because optical transport link 140 typically adds noise and imposes various linear and nonlinear signal distortions, such as those caused by the above-mentioned Kerr effect, CD, PMD, SPM, XPM, and FWM.

(14) Front-end circuit 172 of receiver 190 comprises an optical hybrid 160, light detectors 161.sub.1-161.sub.4, analog-to-digital converters (ADCs) 166.sub.1-166.sub.4, and an optical local-oscillator (OLO) source 156. Optical hybrid 160 has (i) two input ports labeled S and R and (ii) four output ports labeled 1 through 4. Input port S receives optical signal 130′ from optical transport link 140. Input port R receives an OLO signal 158 generated by OLO source 156. OLO signal 158 has an optical-carrier wavelength (frequency) that is sufficiently close to that of signal 130′ to enable coherent (e.g., intradyne) detection of the latter signal. OLO signal 158 can be generated, e.g., using a relatively stable tunable laser whose output wavelength (frequency) is approximately the same as the carrier wavelength (frequency) of optical signal 130.

(15) In an example embodiment, optical hybrid 160 operates to mix input signal 130′ and OLO signal 158 to generate different mixed (e.g., by interference) optical signals (not explicitly shown in FIG. 1). Light detectors 161.sub.1-161.sub.4 then convert the mixed optical signals into four electrical signals 162.sub.1-162.sub.4 that are indicative of complex values corresponding to two orthogonal-polarization components of signal 130′. For example, electrical signals 162.sub.1 and 162.sub.2 may be an analog I signal and an analog Q signal, respectively, corresponding to a first (e.g., horizontal, h) polarization component of signal 130′. Electrical signals 162.sub.3 and 162.sub.4 may similarly be an analog I signal and an analog Q signal, respectively, corresponding to a second (e.g., vertical, v) polarization component of signal 130′. Note that the orientation of the h and v polarization axes at receiver 190 may not coincide with the orientation of the X and Y polarization axes at transmitter 110.

(16) Each of electrical signals 162.sub.1-162.sub.4 is converted into digital form in a corresponding one of ADCs 166.sub.1-166.sub.4. Optionally, each of electrical signals 162.sub.1-162.sub.4 may be amplified in a corresponding electrical amplifier (not explicitly shown) prior to the resulting signal being converted into digital form. Digital signals 168.sub.1-168.sub.4 produced by ADCs 166.sub.1-166.sub.4 are then processed by a DSP 170 to recover the data of the original input data stream 102 applied to transmitter 110. In an example embodiment, DSP 170 may perform, inter alia, one or more of the following: (i) signal processing directed at dispersion compensation; (ii) signal processing directed at compensation of nonlinear distortions; (iii) electronic polarization de-multiplexing; and (iv) error correction based on the data encoding applied at DSP 112. Example embodiments of DSP 170 are described in more detail below in reference to FIGS. 2-3.

(17) In some embodiments, at least some of the signal processing directed at dispersion compensation and/or compensation of nonlinear distortions can be performed at DSP 112 instead of being performed at DSP 170. In this case, this signal processing can be used to pre-distort optical output signal 130 in a manner that causes optical signal 130′ to be less distorted than in the absence of this pre-distortion.

(18) FIG. 2 shows a block diagram of DSP 170 (FIG. 1) according to an embodiment. Digital signals 168.sub.1-168.sub.4 and output data stream 102 are also shown in FIG. 2 to better illustrate the relationship between the circuits shown in FIGS. 1 and 2.

(19) Ideally, digital signals 168.sub.1 and 168.sub.2 represent the I and Q components, respectively, of the horizontal polarization component of optical signal 130′, and digital signals 168.sub.3 and 168.sub.4 represent the I and Q components, respectively, of the vertical polarization component of that optical signal. However, various transmission impairments, front-end implementation imperfections, and configuration inaccuracies generally cause each of digital signals 168.sub.1-168.sub.4 to be a convoluted signal that has various linear and nonlinear distortions and/or contributions from different signal components originally generated at transmitter 110 (FIG. 1). The train of signal processing implemented in DSP 170 is generally directed at reducing the adverse effects of the signal distortions and de-convolving digital signals 168.sub.1-168.sub.4 so that the transmitted data can be properly recovered to generate output data stream 102 with an acceptably low BER.

(20) In an example embodiment, DSP 170 comprises a signal-pre-processing module 210 configured to receive digital signals 168.sub.1-168.sub.4. One of the functions of module 210 may be to adapt the signal samples received via digital signals 168.sub.1-168.sub.4 to a form that is more suitable for the signal-processing algorithms implemented in the downstream modules of DSP 170. For example, module 210 may be configured to (i) resample digital signals 168.sub.1-168.sub.4 such that each of these signals carries two samples per symbol period and/or (ii) convert real-valued signal samples into the corresponding complex-valued signal samples. The resulting complex-valued digital signals generated by signal-pre-processing module 210 are labeled 212.sub.1-212.sub.2.

(21) DSP 170 further comprises a backward-propagation (BP) module 220 that converts digital signals 212.sub.1 and 212.sub.2 into digital signals 222.sub.1 and 222.sub.2, respectively. In an example embodiment, BP module 220 implements signal processing directed at causing digital signals 222.sub.1 and 222.sub.2 to provide a good approximation of the undistorted optical signal 130 applied by transmitter 110 to optical transport link 140 (see FIG. 1). In particular, BP module 220 operates to (i) carry out dispersion-compensation processing and (ii) at least partially remove nonlinear-interference (NLI) noise from digital signals 222.sub.1 and 222.sub.2. Example embodiments of BP module 220 are described in more detail below in reference to FIGS. 3-7.

(22) Digital signals 222.sub.1 and 222.sub.2 are applied to a 2×2 MIMO (multiple-input/multiple-output) equalizer 230 for MIMO-equalization processing therein, and the resulting equalized signals are complex-valued digital signals 232.sub.X and 232.sub.Y. In an example embodiment, equalizer 230 can be a butterfly equalizer configured to perform electronic polarization demultiplexing and reduce some effects of inter-symbol interference. Example 2×2 MIMO equalizers that can be used to implement equalizer 230 are disclosed, e.g., in U.S. Pat. No. 9,020,364 and U.S. Patent Application Publication No. 2015/0372764, both of which are incorporated herein by reference in their entirety.

(23) Digital signals 232.sub.X and 232.sub.Y generated by equalizer 230 are applied to a carrier-recovery module 240 that is configured to perform signal processing generally directed at (i) compensating the frequency mismatch between the carrier frequencies of OLO signal 158 and optical input signal 130′ and/or (ii) reducing the effects of phase noise. Various signal-processing techniques that can be used to implement the corresponding signal processing in carrier-recovery module 240 are disclosed, e.g., in U.S. Pat. Nos. 7,747,177, 8,073,345, and 9,112,614, all of which are incorporated herein by reference in their entirety.

(24) Digital signals 242.sub.X and 242.sub.Y generated by carrier-recovery module 240 are applied to a symbol-detection module 250. In an example embodiment, symbol-detection module 250 is configured to use the complex values conveyed by digital signals 242.sub.X and 242.sub.Y to appropriately map each complex value onto an operative constellation to determine the corresponding received symbol and, based on this mapping, determine the corresponding bit-word encoded by the symbol. Symbol-detection module 250 then concatenates the determined bit-words to generate data streams 252.sub.X and 252.sub.Y.

(25) In some embodiments, data streams 252.sub.X and 252.sub.Y can be applied to an optional forward-error-correction (FEC) decoder 260 configured to perform digital signal processing that implements error correction based on data redundancies (if any) in optical signal 130. FEC decoder 260 appropriately multiplexes the resulting error-corrected data streams to generate output data stream 102.

(26) In some embodiments, the use of symbol-detection module 250 configured to determine the likelihood of the corresponding received symbol can alternatively be used, e.g., in embodiments employing soft FEC.

(27) FIG. 3 shows a block diagram of BP module 220 (FIG. 2) according to an embodiment. Digital signals 212.sub.1-212.sub.2 and 222.sub.1-222.sub.2 are also shown in FIG. 3 to better illustrate the relationship between the circuits shown in FIGS. 2 and 3.

(28) BP module 220 comprises N processing stages labeled 310.sub.1-310.sub.N, where N is a positive integer. In operation, each of processing stages 310.sub.1-310.sub.N is configured to carry out dispersion compensation and NLI-noise (NLIN) reduction corresponding to a respective section 142 of link 140. In the embodiment shown in FIG. 3, the number N is at least three (i.e., N≥3). However, alternative embodiments in which N=1 or N=2 are also possible.

(29) A processing stage 310; comprises a dispersion-compensation module 320; and an NLIN-compensation module 330.sub.1, where i=1, 2, . . . , N. Dispersion-compensation module 320.sub.1 is configured to receive digital signals 212.sub.1-212.sub.2. NLIN-compensation module 330.sub.N is configured to output digital signals 222.sub.1-222.sub.2. NLIN-compensation module 330.sub.j (where j=1, 2, . . . , N−1) is configured to output digital signals 312.sub.ja-312.sub.jb, which are then applied to dispersion-compensation module 320.sub.j+1. The resulting dispersion-compensated signals generated by dispersion-compensation module 320.sub.j+1 are applied to NLIN-compensation module 330.sub.j+1.

(30) Example digital circuits that can be used to implement dispersion-compensation module 320.sub.i are disclosed, e.g., in U.S. Pat. Nos. 8,260,154, 7,636,525, 7,266,310, all of which are incorporated herein by reference in their entirety.

(31) In operation, NLIN-compensation module 330.sub.i uses a BP algorithm that carries out frequency-domain perturbation-based digital backward propagation configured to at least partially remove NLI noise from the corresponding signals. In an example embodiment, the used BP algorithm represents a modified version of Volterra-series digital backward propagation, with some of the modifications being directed at (at least) partially accounting for the influence of PMD on NLI noise in link 140. The theoretical basis of the BP algorithm used in NLIN-compensation module 330.sub.i is described in more detail below in reference to Eqs. (1)-(11). An example signal-processing method that can be used to implement this BP algorithm in NLIN-compensation module 330.sub.i is shown in and described in reference to FIG. 7. Example digital circuits that can be used in NLIN-compensation module 330; for this purpose are shown in and described in reference to FIGS. 5-6.

(32) In some embodiments, the positions of dispersion-compensation module 320.sub.i and NLIN-compensation module 330; may be swapped in each or some of processing stages 310.sub.i. In such embodiments, dispersion-compensation module 320.sub.i is located after (downstream from) NLIN-compensation module 330.sub.i.

(33) PMD can affect the way in which nonlinearities accumulate during the forward propagation of optical signal 130 through link 140. For example, PMD can induce frequency-dependent polarization rotations onto optical signal 130 that evolve stochastically during the forward propagation through link 140. The exact evolution of these polarization rotations along link 140 is typically unknown due to the stochastic nature thereof.

(34) Conventional digital BP algorithms typically ignore the effects of PMD on NLI noise, thereby disadvantageously impairing the accuracy of the algorithm and/or limiting the achievable BP-equalizer gain.

(35) These and some other related problems in the state of the art can be addressed, e.g., using embodiments of the BP algorithm in which at least some effects of PMD on NLI noise are accounted for using the signal processing implemented in BP module 220. Experiments and computer simulations indicate that BP module 220 configured in this manner beneficially enables DSP 170 to provide an additional signal-to-noise-ratio (SNR) gain of up to ˜2.5 dB compared to a configuration of the same DSP in which one or more NLIN-compensation modules 330 are disabled or removed from the corresponding chain of signal processing. In some embodiments, the obtained SNR gain can be significantly higher than that provided by a comparable conventional digital BP algorithm in which the effects of PMD on NLI noise are ignored.

(36) In the presence of amplified-spontaneous-emission (ASE) noise, a spectral component r(ω) that represents the optical signal at the output of a corresponding fiber section can be expressed using Eq. (1):
r(ω)=J(ω,L)(a(ω)+Δa(ω))+Z(ω)  (1)
where ω is the angular frequency; Z(ω) is the ASE noise at the angular frequency co accumulated in the corresponding fiber section; J(ω, L) is a 2×2 matrix that describes the accumulated polarization rotation at angular frequency ω and distance L; a(ω) and Δa(ω) are the two-component column vectors that represent the optical signal at the input of the corresponding fiber section and the NLI noise accumulated in the fiber section, respectively, in the two polarizations of the optical signal.

(37) Based on the first-order perturbation analysis, the accumulated NLI noise Δa(ω) can be expressed as follows:
Δa(ω)=∫∫ψ(ω.sub.1,ω.sub.2,ω.sub.3)dω.sub.1dω.sub.2  (2)
where ω.sub.3=ω−ω.sub.1+ω.sub.2; and the function ψ(ω.sub.1,ω.sub.2,ω.sub.3) describes the FWM processes that nonlinearly couple the spectral components a(ω.sub.1), a(ω.sub.2), and a(ω.sub.3). For a fiber section of length L, the function ψ(ω.sub.1,ω.sub.2,ω.sub.3) can be expressed as follows:
ψ(ω.sub.1,ω.sub.2,ω.sub.3)=.sup.L∫.sub.0ρ(ω.sub.1,ω.sub.2,ω.sub.3,z)J(ω,z).sup.HJ(ω.sub.3,z)a(ω.sub.3)a(ω.sub.2).sup.HJ(ω.sub.2,z).sup.HJ(ω.sub.1,z)a(ω.sub.1)dz  (3)
where J(ω,z) is a 2×2 matrix that describes the accumulated polarization rotation at angular frequency ω and distance z; H denotes the conjugate transpose; and the function ρ is expressed as follows:

(38) $\begin{array}{cc}\rho \left({\omega }_1,{\omega }_2,{\omega }_3,z\right)=\frac{8}{9}j\gamma f\left(z\right)\exp \left(j{\beta }_2\left({\omega }_3-{\omega }_2\right)\left({\omega }_2-{\omega }_1\right)z\right)& \left(4\right)\end{array}$
where f(z), γ, and β.sub.2 represent the power profile of the optical signal in the fiber, the nonlinear coefficient of the fiber, and the dispersion coefficient of the fiber, respectively.

(39) When the effects of PMD on NLI noise can be neglected, each of the terms J(ω,z).sup.HJ(ω.sub.3,z) and J(ω.sub.2,z).sup.HJ(ω.sub.1,z) in Eq. (3) can be replaced by the identity matrix, which removes the PMD dependence from the corresponding FWM terms in Eqs. (2)-(3).

(40) When the effects of PMD on NLI noise need to be taken into account, it is useful to note that the polarization-rotation matrices J(ω, z) at frequencies co and ω+Δω tend to lose mutual coherence at sufficiently large values of Δω. In this case, the terms J(ω,z).sup.HJ(ω.sub.3,z) and J(ω.sub.2,z).sup.HJ(ω.sub.1,z) in Eq. (3) differ from the identity matrix, which causes the corresponding FWM terms in Eqs. (1)-(3) to depend on the (unknown) PMD evolution along the length of the fiber.

(41) From an algorithmic perspective, one can approximate the above-described behavior of the polarization-rotation matrices J(ω, z) using a step-like autocorrelation function according to which: (i) the matrices J(ω, z) and J(ω+Δω, z) are statistically independent if |Δω|>Δω.sub.0, where Δω.sub.0 is a fixed threshold value that can be one of the parameters of the corresponding algorithm; and (ii) the matrices J(ω, z) and J(ω+Δω, z) are identical, i.e., J(ω, z)=J(ω+Δω, z), if |Δω|≤Δω.sub.0. Based on this autocorrelation function, the function ψ(ω.sub.1,ω.sub.2,ω.sub.3) can be expressed using three different approximations, the use of which depends on the relationship between the frequencies ω, ω.sub.1, ω.sub.2, and ω.sub.3. The following three sets of inequalities can be used to define three respective spectral regions in a multidimensional frequency space representing the signal bandwidth, in which the three approximations can be applied:

(42) (A) |ω.sub.1−ω.sub.2|=|ω.sub.3−ω|≤Δω.sub.0;

(43) (B) |(ω.sub.1−ω.sub.2|=|ω.sub.3−ω|>Δω.sub.0 and |ω.sub.1−ω|=|(ω.sub.2−ω.sub.3|≤Δω.sub.0; and

(44) (C) all other ω, ω.sub.1, ω.sub.2, and ω.sub.3 that do not satisfy (A) or (B).

(45) A person of ordinary skill in the art will understand that a similar analysis can be performed for any suitable auto-correlation function, including more complex auto-correlation functions constructed to more accurately represent the spectral characteristics of PMD.

(46) FIG. 4 graphically shows spectral regions A-C for a fixed frequency co according to an embodiment. The spectral regions A-C shown in FIG. 4 correspond to the above-shown sets of inequalities (A)-(C), respectively, and are parts of a multidimensional frequency space representing the bandwidth of optical signal 130.

(47) In the spectral region A, Eq. (3) can be approximated using Eq. (5):
ψ(ω.sub.1,ω.sub.2,ω.sub.3)=.sup.L∫.sub.0ρ(ω.sub.1,ω.sub.2,ω.sub.3,z)a(ω.sub.3)a(ω.sub.2).sup.Ha(ω.sub.1)dz  (5)
In Eq. (5) the terms containing matrices J(ω, z) are replaced by identity matrices. As a result, the corresponding approximations of the function ψ(ω.sub.1,ω.sub.2,ω.sub.3) and NLI noise do not depend on PMD.

(48) In the spectral region B, Eq. (3) can be approximated using Eq. (6):

(49) $\begin{array}{cc}\psi \left({\omega }_1,{\omega }_2,{\omega }_3\right)=\frac{1}{2}\underset{0}{\overset{L}{\int }}\rho \left({\omega }_1,{\omega }_2,{\omega }_3,z\right)a\left({\omega }_1\right){a\left({\omega }_2\right)}^{H}a\left({\omega }_3\right)dz& \left(6\right)\end{array}$
Compared to Eq. (5), the spectral components a(ω.sub.1) and a(ω.sub.3) are interchanged in Eq. (6), and the magnitude of the function ψ(ω.sub.1,ω.sub.2,ω.sub.3) is smaller by a factor of ½. As a result, Eq. (6) approximately takes into account the statistical average of the terms containing matrices J(ω, z) for configurations in which the NLI noise is somewhat sensitive to PMD. This sensitivity is such that it causes the statistical average of the terms containing matrices J(ω, z) to differ from zero in a non-negligible manner.

(50) In the spectral region C, Eq. (3) can be approximated using Eq. (7):
ψ(ω.sub.1,ω.sub.2,ω.sub.3)=0  (7)
In this spectral region, NLI noise is significantly more sensitive to PMD than in the spectral region B, which causes the statistical average of the terms containing matrices J(ω, z) to be sufficiently close to zero to make the approximation expressed by Eq. (7) sufficiently accurate.

(51) FIG. 5 shows a block diagram of NLIN-compensation module 330.sub.i (FIG. 3) according to an embodiment. Digital input signals 502.sub.ia and 502.sub.ib applied to NLIN-compensation module 330.sub.i are generated by dispersion-compensation module 320.sub.i (also see FIG. 3). For i=1, 2, . . . , N−1, digital output signals 504.sub.ia and 504.sub.ib generated by NLIN-compensation module 330.sub.i are signals 312.sub.ia and 312.sub.ib, respectively (see FIG. 3). For i=N, digital output signals 504.sub.ia and 504.sub.ib generated by NLIN-compensation module 330.sub.i are signals 222.sub.1 and 222.sub.2, respectively (see FIGS. 2-3).

(52) In operation, NLIN-compensation module 330.sub.i carries out calculations that are based on discrete versions of Eqs. (1)-(7), e.g., as further explained below. An example embodiment of an NLIN estimator 540 used in NLIN-compensation module 330.sub.i is described in more detail below in reference to FIG. 6.

(53) Digital signals 502.sub.ia and 504.sub.ia correspond to the first (e.g., horizontal) polarization of optical signal 130′. A fast-Fourier-transform (FFT) module 510.sub.a operates to transform a sequence of time-domain samples supplied by digital signal 502.sub.ia into a corresponding set of frequency components {tilde over (x)}.sup.(n), where the index n represents the frequency.

(54) Digital signals 502.sub.ib and 504.sub.ib correspond to the second (e.g., vertical) polarization of optical signal 130′. An FFT module 510.sub.b operates to transform a sequence of time-domain samples supplied by digital signal 502.sub.ib into a corresponding set of frequency components {tilde over (y)}.sup.(n).

(55) Using the sets {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)} generated by FFT modules 510.sub.a and 510.sub.b, NLIN estimator 540 computes estimates Δ{tilde over (x)}.sup.(n) and Δ{tilde over (y)}.sup.(n) of NLI noise for the two polarizations. These computations take into account the effects of PMD on NLI noise in accordance with Eqs. (2)-(7). Accordingly, NLIN estimator 540 is configured to use a fixed threshold value Δω.sub.0 to delineate the spectral regions A-C graphically illustrated in FIG. 4.

(56) In an example embodiment, the computations performed by NLIN estimator 540 can be based on the following mathematical expressions:

(57) $\begin{array}{cc}\left(\begin{array}{c}\Delta {\tilde{x}}^{\left(n\right)}\\ \Delta {\tilde{y}}^{\left(n\right)}\end{array}\right)=\underset{h}{.Math.}\underset{k}{.Math.}\underset{m}{.Math.}{\tilde{\rho }}_{h-n,k-n,m-n}\left(\begin{array}{c}{\tilde{x}}^{\left(h\right)}\\ {\tilde{y}}^{\left(h\right)}\end{array}\right){\left(\begin{array}{c}{\tilde{x}}^{\left(k\right)}\\ {\tilde{y}}^{\left(k\right)}\end{array}\right)}^H\left(\begin{array}{c}{\tilde{x}}^{\left(m\right)}\\ {\tilde{y}}^{\left(m\right)}\end{array}\right)& \left(8\right)\end{array}$
{tilde over (ρ)}.sub.h,k,m=ρ.sub.h,k,m, if |h−k|≤Δω.sub.0 and |m−k|≤Δω.sub.0  (9a)
{tilde over (ρ)}.sub.h,k,m=ρ.sub.h,k,m+0.5ρ.sub.m,k,h, if |m−k|≤Δω.sub.0 and |h−k|>Δω.sub.0  (9b)
ρ.sub.h,k,m=0, for all other combinations of h,k, and m  (9c)
Eq. (8) represents a discrete version of Eqs. (2)-(4). Eqs. (9a)-(9c) represent discrete versions of Eqs. (5)-(7), respectively, adapted for an example embodiment of NLIN estimator 540.

(58) An adder 520.sub.a operates to generate a set of corrected frequency components {tilde over (X)}.sup.(n) by subtracting the estimates Δ{tilde over (x)}.sup.(n) from the corresponding frequency components {tilde over (x)}.sup.(n). An adder 520.sub.b similarly operates to generate a set of corrected frequency components {tilde over (Y)}.sup.(n) by subtracting the estimates Δ{tilde over (y)}.sup.(n) from the corresponding frequency components {tilde over (y)}.sup.(n). Mathematical expressions of the corresponding operations are given by Eqs. (10a)-(10b):
{tilde over (X)}.sup.(n)={tilde over (x)}.sup.(n)−Δ{tilde over (x)}.sup.(n)  (10a)
{tilde over (Y)}.sup.(n)={tilde over (y)}.sup.(n)−Δ{tilde over (y)}.sup.(n)  (10b)

(59) An inverse FFT (IFFT) module 530.sub.a operates to transform the set {{tilde over (X)}.sup.(n)} received from adder 520.sub.a into a corresponding sequence of corrected time-domain samples and outputs the latter by way of digital signal 504.sub.ia. An IFFT module 530.sub.b similarly operates to transform the set {{tilde over (Y)}.sup.(n)} received from adder 520.sub.b into a corresponding sequence of corrected time-domain samples and outputs the latter by way of digital signal 504.sub.ib.

(60) FIG. 6 shows a block diagram of NLIN estimator 540 according to an example embodiment. As shown, NLIN estimator 540 includes a triplet generator 610, an array 620 of multipliers 622, a memory 630, and a sum generator 640 interconnected as indicated in FIG. 6.

(61) Memory 630 has stored therein the values of {tilde over (ρ)}.sub.h,k,m for all possible combinations of the frequency indices h, k, and m, for a total of M.sup.3 values, where M is the size of the sets {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)}. The values of {tilde over (ρ)}.sub.h,k,m can be calculated and saved in memory 630 during the initial configuration setup of NLIN estimator 540, e.g., using the selected value of the threshold Δω.sub.0 and Eqs. (9a)-(9c).

(62) Triplet generator 610 is configured to compute M.sup.3 different triplets (h, k, m) using the following expression:

(63) $\begin{array}{cc}\left(\begin{array}{c}{\tilde{x}}^{\left(h\right)}\\ {\tilde{y}}^{\left(h\right)}\end{array}\right){\left(\begin{array}{c}{\tilde{x}}^{\left(k\right)}\\ {\tilde{y}}^{\left(k\right)}\end{array}\right)}^H\left(\begin{array}{c}{\tilde{x}}^{\left(m\right)}\\ {\tilde{y}}^{\left(m\right)}\end{array}\right)& \left(11\right)\end{array}$
where h=1, 2, . . . , M; k=1, 2, . . . , M; and m=1, 2, . . . , M.

(64) Array 620 and sum generator 640 are configured to use the M.sup.3 different triplets (h, k, m) received from triplet generator 610 and the corresponding values of {tilde over (ρ)}.sub.h,k,m retrieved from memory 630 to compute the estimates Δ{tilde over (x)}.sup.(n) and Δ{tilde over (y)}.sup.(n) in accordance with Eq. (8). More specifically, each of multipliers 622 is configured to generate a product of the respective triplet and the corresponding value of {tilde over (ρ)}.sub.h-n,k-n,m-n. In an example embodiment, the latter values can be obtained using the values of {tilde over (ρ)}.sub.h,k,m stored in memory 630 and one or more conventional shift registers (not explicitly shown in FIG. 6) configured to apply a shift corresponding to different values of n. The products obtained in this manner are summed by sum generator 640 to generate the estimates Δ{tilde over (x)}.sup.(n) and Δ{tilde over (y)}.sup.(n).

(65) FIGS. 7A-7B show flowcharts of signal processing methods 700 and 720, respectively, each of which can be used to implement NLIN-compensation module 330.sub.i according to an embodiment. Steps 702-706 of method 700 can be performed, e.g., at the factory, during the initial configuration setup of NLIN-compensation module 330.sub.i. Steps 708-714 of method 720 can be executed, e.g., after the deployment in the field, during the subsequent transmission of payload data 102 in system 100. In some embodiments, methods 700 and 720 can be executed by different entities, e.g., the equipment manufacturer/installer and the operator of the corresponding optical transport system, respectively. In some embodiments, methods 700 and 720 can be executed by the same entity, e.g., the installer/operator of the corresponding optical transport system and, as such, can be combined into a single method by an optional logical link between steps 706 and 708 shown by the dashed arrow.

(66) At step 702 of method 700 (FIG. 7A), the pertinent parameters of the corresponding fiber-optic section 142 may be entered into the configuration file(s) of NLIN-compensation module 330.sub.i. Such parameters may include but are not limited to: (i) the length L of the optical fiber used in the corresponding fiber-optic section; (ii) the optical input power; (iii) the power profile f(z) of the optical signal along the fiber; (iv) the nonlinear coefficient γ of the fiber; and (v) the dispersion coefficient β.sub.2 of the fiber (also see Eq. (4)).

(67) At step 704, the value of the threshold Δω.sub.0 is selected and saved in the configuration file(s). In an example embodiment, the value of Δω.sub.0 can be selected using an iterative optimization procedure during which the BP-processing gain is evaluated for different values of Δω.sub.0 based on (i) the signal bandwidth and (ii) the link parameters specified at step 702. A value of Δω.sub.0 that provides an acceptable (e.g., relatively high) BP-processing gain may then be selected for the subsequent use at step 706.

(68) At step 706, the values of {tilde over (ρ)}.sub.h,k,m corresponding to different sets of the indices h, k, and m are computed using Eqs. (9a)-(9c) and saved, e.g., in memory 630 (FIG. 6). The number of non-zero values of {tilde over (ρ)}.sub.h,k,m depends, inter alia, on the value of Δω.sub.0 selected at step 704 and the number M of digital samples used in the FFT operations carried out by FFT modules 510.sub.a and 510.sub.b (FIG. 5). The number M is a parameter of the BP algorithm that can be selected, e.g., for good performance at an acceptable processing delay (latency) and to meet applicable power-consumption limitations.

(69) At step 708 of method 720 (FIG. 7B), NLIN-compensation module 330.sub.i operates to generate the sets {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)} of frequency components by applying an FFT operation to the corresponding sequences of time-domain samples provided by digital input signals 502.sub.ia and 502.sub.ib, respectively. In an example embodiment, step 708 can be performed using FFT modules 510.sub.a and 510.sub.b (FIG. 5).

(70) At step 710, NLIN-compensation module 330.sub.i operates to generate the sets {Δ{tilde over (x)}.sup.(n)} and {Δ{tilde over (Y)}.sup.(n)} of NLIN estimates using the sets {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)} of frequency components generated at step 708 and the values of {tilde over (ρ)}.sub.h,k,m generated at step 706. In an example embodiment, step 710 can be carried out using NLIN estimator 540 (FIGS. 5-6).

(71) At step 712, NLIN-compensation module 330.sub.i operates to generate the sets {{tilde over (X)}.sup.(n)} and {{tilde over (Y)}.sup.(n)} of corrected frequency components using the sets {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)} of frequency components generated at step 708 and the sets {Δ{tilde over (x)}.sup.(n)} and {Δ{tilde over (y)}.sup.(n)} of NLIN estimates generated at step 710. In an example embodiment, step 712 can be performed using adders 520.sub.a and 520.sub.b (FIG. 5).

(72) At step 714, NLIN-compensation module 330.sub.i operates to generate two sequences of time-domain samples corresponding to the sets {{tilde over (X)}.sup.(n)} and {{tilde over (Y)}.sup.(n)}, respectively, of corrected frequency components generated at step 712. In an example embodiment, step 714 can be performed using IFFT modules 530.sub.a and 530.sub.b, and the two resulting sequences of corrected time-domain samples can be provided to downstream circuits by way of digital output signals 504.sub.ia and 504.sub.ib, respectively, (see FIG. 5).

(73) The processing of method 700 can then be directed back to step 708, e.g., to process a next block of time-domain samples supplied by digital input signals 502.sub.ia and 502.sub.ib.

(74) According to an example embodiment disclosed above in reference to FIGS. 1-7, provided is an apparatus (e.g., 100, FIG. 1) comprising: a coherent optical receiver (e.g., 190, FIG. 1) that includes an optical hybrid (e.g., 160, FIG. 1), a plurality of light detectors (e.g., 161, FIG. 1), a plurality of analog-to-digital converters (e.g., 166, FIG. 1), and a digital signal processor (e.g., 170, FIG. 1), the analog-to-digital converters being configured to output a stream (e.g., 168, FIG. 1) of digitalizations of measurements of two polarization components of a modulated optical carrier (e.g., 130′, FIG. 1), the light detectors being configured to make the measurements on interferences of the modulated optical carrier produced by the optical hybrid; and wherein the digital signal processor comprises a first digital circuit (e.g., 220, FIG. 2) configured to estimate, from the stream of digitalizations, intra-channel nonlinear optical distortions of the modulated optical carrier in a polarization-mode-dispersion dependent manner that gives a lower weight to some contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of larger intra-channel frequency difference than to other contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of smaller intra-channel frequency difference.

(75) In some embodiments of the above apparatus, the digital signal processor is configured to generate an estimate of the intra-channel nonlinear optical distortions for the two polarization components of the modulated optical carrier, the estimate being generated using two or more different approximations of an effect of polarization mode dispersion on the intra-channel nonlinear optical distortions within a bandwidth of the modulated optical carrier.

(76) In some embodiments of any of the above apparatus, the digital signal processor is further configured to recover respective data encoded in the two polarization components of the modulated optical carrier, using the estimate.

(77) In some embodiments of any of the above apparatus, the digital signal processor is further configured to switch between the two or more different approximations based on a comparison, with a fixed threshold value (e.g., Δω.sub.0, FIG. 4), of a difference between frequencies of the frequency components of the modulated optical carrier contributing to the intra-channel nonlinear optical distortions.

(78) In some embodiments of any of the above apparatus, the digital signal processor is further configured to: apply a first of the two or more approximations if an absolute value of a difference between frequencies of a first frequency component and a second frequency component is smaller than the fixed threshold value; and apply a second of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is greater than the fixed threshold value.

(79) In some embodiments of any of the above apparatus, the digital signal processor is further configured to apply a third of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is smaller than the fixed threshold value, and an absolute value of a difference between frequencies of a third frequency component and the second frequency component is greater than the fixed threshold value.

(80) In some embodiments of any of the above apparatus, the digital signal processor is configured to use three different approximations of the effect of polarization mode dispersion the intra-channel nonlinear optical distortions within the bandwidth of the modulated optical carrier.

(81) In some embodiments of any of the above apparatus, the first digital circuit comprises: a second digital circuit (e.g., 510, FIG. 5) configured to apply a Fourier transformation to a sequence of signal samples (e.g., 502, FIG. 5) generated using the digitalizations to generate a corresponding set of frequency components e.g., {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)}, FIG. 5); and a third digital circuit (e.g., 540, FIG. 5) configured to estimate the intra-channel nonlinear optical distortions of the modulated optical carrier using the corresponding set of frequency components.

(82) In some embodiments of any of the above apparatus, the first digital circuit further comprises a fourth digital circuit (e.g., 520, FIG. 5) configured to generate a set of corrected frequency components using an estimate of the intra-channel nonlinear optical distortions and the corresponding set of frequency components; and wherein the digital signal processor is configured to recover respective data encoded in the two polarization components of the modulated optical carrier, using the set of corrected frequency components.

(83) In some embodiments of any of the above apparatus, the first digital circuit further comprises a fifth digital circuit (e.g., 530, FIG. 5) configured to apply an inverse Fourier transformation to the set of corrected frequency components to generate a sequence of corrected signal samples; and wherein the digital signal processor is configured to recover the respective data using the corrected signal samples.

(84) In some embodiments of any of the above apparatus, the third digital circuit is operatively connected to a memory (e.g., 630, FIG. 6) configured to store therein digital values representing two or more different approximations of an effect of polarization mode dispersion on the intra-channel nonlinear optical distortions within a bandwidth of the modulated optical carrier; and wherein the third digital circuit is further configured to estimate the intra-channel nonlinear optical distortions of the modulated optical carrier using the digital values stored in the memory.

(85) In some embodiments of any of the above apparatus, the first digital circuit is further configured to estimate the intra-channel nonlinear optical distortions of the modulated optical carrier using a backward-propagation algorithm corresponding to the modulated optical carrier.

(86) In some embodiments of any of the above apparatus, the first digital circuit comprises two or more serially connected processing stages (e.g., 310.sub.1-310.sub.N, FIG. 3), each configured to perform backward-propagation processing corresponding to a respective fiber-optic section traversed by the modulated optical carrier.

(87) In some embodiments of any of the above apparatus, each of the two or more serially connected processing stages comprises: a respective dispersion-compensation module (e.g., 320.sub.i, FIG. 3) configured to perform dispersion-compensation processing on signal samples corresponding to the two polarization components of the modulated optical carrier; and a respective nonlinear-interference-noise-compensation module (e.g., 330.sub.i, FIG. 3) operatively connected to the respective dispersion-compensation module and configured to at least partially compensate the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier corresponding to the two polarization components thereof.

(88) According to another example embodiment disclosed above in reference to FIGS. 1-7, provided is an apparatus (e.g., 100, FIG. 1) comprising: an optical transmitter (e.g., 110, FIG. 1) that includes a laser (e.g., 120, FIG. 1), a plurality of optical modulators (e.g., 124, FIG. 1), associated electrical drivers (e.g., 118, FIG. 1), and a digital signal processor (e.g., 112, FIG. 1), the optical transmitter being configured to modulate a digital symbol stream (e.g., 114, FIG. 1) onto two polarization components of an optical carrier generated by the laser in a manner that pre-compensates some intra-channel nonlinear optical distortions produced by transmission of a resulting modulated optical carrier over an optical fiber; and wherein the digital signal processor is electrically connected to control the associated electrical drivers and comprises a digital circuit (e.g., analogous to 330, FIG. 3) configured to estimate intra-channel nonlinear optical distortions during the transmission of the modulated optical carrier in a polarization-mode-dispersion dependent manner that gives a lower weight to some contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of larger intra-channel frequency difference than to other contributions to the intra-channel nonlinear optical distortions caused by interactions between frequency components of the modulated optical carrier of smaller intra-channel frequency difference.

(89) In some embodiments of the above apparatus, the digital signal processor is configured to generate an estimate (e.g., {Δ{tilde over (x)}.sup.(n)} and {Δ{tilde over (y)}(n)}, Eq. (8)) of the intra-channel nonlinear optical distortions for the two polarization components of the modulated optical carrier, the estimate being generated using two or more different approximations of an effect of polarization mode dispersion on the intra-channel nonlinear optical distortions within a bandwidth of the modulated optical carrier.

(90) In some embodiments of any of the above apparatus, the digital signal processor is further configured to generate the estimate using a backward-propagation algorithm, the backward-propagation algorithm being configured to cause a pre-distortion of the modulated optical carrier that tends to be removed by the transmission thereof over the optical fiber.

(91) In some embodiments of any of the above apparatus, the digital signal processor is further configured to switch between the two or more different approximations based on a comparison, with a fixed threshold value (e.g., Δω.sub.0, FIG. 4), of a difference between frequencies of the frequency components of the modulated optical carrier contributing to the intra-channel nonlinear optical distortions.

(92) In some embodiments of any of the above apparatus, the digital signal processor is further configured to: apply a first of the two or more approximations if an absolute value of a difference between frequencies of a first frequency component and a second frequency component is smaller than the fixed threshold value; and apply a second of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is greater than the fixed threshold value.

(93) In some embodiments of any of the above apparatus, the digital signal processor is further configured to apply a third of the two or more approximations if the absolute value of the difference between the frequencies of the first frequency component and the second frequency component is smaller than the fixed threshold value, and an absolute value of a difference between frequencies of a third frequency component and the second frequency component is greater than the fixed threshold value.

(94) According to yet another example embodiment disclosed above in reference to FIGS. 1-7, provided is an apparatus (e.g., 190, FIG. 1) comprising: an optical front-end circuit (e.g., 172, FIG. 1) configured to receive an optical communication signal (e.g., 130′, FIG. 1); and a digital signal processor (e.g., 172, FIG. 1) operatively connected to the optical front-end circuit and configured to generate (e.g., at 710, FIG. 7) an estimate (e.g., {Δ{tilde over (x)}.sup.(n)} and {Δ{tilde over (y)}(n)}, Eq. (8)) of nonlinear interference noise for two orthogonal polarizations of the optical communication signal, the estimate being generated using two or more different approximations (e.g., in accordance with Eqs. (9a)-(9c)) of an effect of polarization mode dispersion on the nonlinear interference noise within a bandwidth of the optical communication signal.

(95) In some embodiments of the above apparatus, the digital signal processor is further configured to recover data (e.g., 102, FIG. 1) encoded in the two orthogonal polarizations of the optical communication signal, using the estimate (e.g., at 712/714, FIG. 7).

(96) In some embodiments of any of the above apparatus, the digital signal processor is further configured to: generate (e.g., at 712/714, FIG. 7) corrected signal samples (e.g., 504, FIG. 5) corresponding to the two orthogonal polarizations of the optical communication signal using the estimate; and recover data (e.g., 102, FIG. 1) encoded in the two orthogonal polarizations of the optical communication signal, using the corrected signal samples.

(97) In some embodiments of any of the above apparatus, the digital signal processor is further configured to switch between the two or more different approximations based on a comparison, with a fixed threshold value (e.g., Δω.sub.0, FIG. 4), of a difference between frequencies of optical waves contributing to the nonlinear interference noise.

(98) In some embodiments of any of the above apparatus, the digital signal processor is further configured to: apply a first (e.g., in accordance with Eq. (9a)) of the two or more approximations if an absolute value of a difference between frequencies of a first optical wave and a second optical wave is smaller than the fixed threshold value; and apply a second (e.g., in accordance with Eq. (9c)) of the two or more approximations if the absolute value of the difference between the frequencies of the first optical wave and the second optical wave is greater than the fixed threshold value.

(99) In some embodiments of any of the above apparatus, the digital signal processor is further configured to apply a third (e.g., in accordance with Eq. (9b)) of the two or more approximations if the absolute value of the difference between the frequencies of the first optical wave and the second optical wave is smaller than the fixed threshold value, and an absolute value of a difference between frequencies of a third optical wave and the second optical wave is greater than the fixed threshold value.

(100) In some embodiments of any of the above apparatus, the digital signal processor is configured to use three different approximations (e.g., defined by Eqs. (9a)-(9c)) of the effect of polarization mode dispersion on the nonlinear interference noise within the bandwidth of the optical communication signal.

(101) In some embodiments of any of the above apparatus, the digital signal processor comprises: a first digital circuit (e.g., 510, FIG. 5) configured to apply a Fourier transformation (e.g., at 708, FIG. 7) to a sequence of signal samples (e.g., 502, FIG. 5) to generate a corresponding set of frequency components (e.g., {{tilde over (x)}.sup.(n)} and {{tilde over (y)}.sup.(n)}, FIG. 5); and a second digital circuit (e.g., 540, FIG. 5) configured to generate the estimate using the corresponding set of frequency components.

(102) In some embodiments of any of the above apparatus, the digital signal processor further comprises a third digital circuit (e.g., 520, FIG. 5) configured to generate (e.g., at 712, FIG. 7) a set of corrected frequency components using the estimate and the corresponding set of frequency components; and wherein the digital signal processor is configured to recover data (e.g., 102, FIG. 1) encoded in the two orthogonal polarizations of the optical communication signal, using the set of corrected frequency components.

(103) In some embodiments of any of the above apparatus, the digital signal processor further comprises a fourth digital circuit (e.g., 530, FIG. 5) configured to apply an inverse Fourier transformation (e.g., at 714, FIG. 7) to the set of corrected frequency components to generate a sequence of corrected signal samples (e.g., 504, FIG. 5); and wherein the digital signal processor is configured to recover the data using the corrected signal samples.

(104) In some embodiments of any of the above apparatus, the second digital circuit is operatively connected to a memory (e.g., 630, FIG. 6) configured to store therein digital values (e.g., {tilde over (ρ)}.sub.h,k,m, Eqs. (9a)-(9c)) representing the two or more different approximations; and wherein the second digital circuit is further configured to generate the estimate using the digital values stored in the memory (e.g., as shown in FIG. 6).

(105) In some embodiments of any of the above apparatus, the digital signal processor is further configured to generate the estimate using a backward-propagation algorithm (e.g., implemented using 220, FIG. 2) corresponding to the optical communication signal.

(106) In some embodiments of any of the above apparatus, the digital signal processor comprises two or more serially connected processing stages (e.g., 310.sub.1-310.sub.N, FIG. 3), each configured to perform backward-propagation processing corresponding to a respective fiber-optic section (e.g., 142.sub.i, FIG. 1) traversed by the optical communication signal, the backward propagation processing being configured to at least partially compensate the effect of polarization mode dispersion on the nonlinear interference noise, said effect being imposed by the respective fiber-optic section.

(107) In some embodiments of any of the above apparatus, each of the two or more serially connected processing stages comprises: a respective dispersion-compensation module (e.g., 320.sub.i, FIG. 3) configured to perform dispersion-compensation processing on signal samples (e.g., 212/312, FIG. 3) corresponding to the two orthogonal polarizations of the optical communication signal; and a respective nonlinear-interference-noise-compensation module (e.g., 330.sub.i, FIG. 3) operatively connected to the respective dispersion-compensation module and configured to at least partially compensate the nonlinear interference noise corresponding to the two orthogonal polarizations of the optical communication signal using the two or more different approximations.

(108) According to yet another example embodiment disclosed above in reference to FIGS. 1-7, provided is an apparatus (e.g., 110, FIG. 1) comprising: an optical front-end circuit (e.g., 116, FIG. 1) configured to transmit an optical communication signal (e.g., 130, FIG. 1); and a digital signal processor (e.g., 112, FIG. 1) operatively connected to the optical front-end circuit and configured to generate an estimate (e.g., {Δ{tilde over (x)}.sup.(n)} and {Δ{tilde over (y)}.sup.(n)}, Eq. (8)) of nonlinear interference noise for two orthogonal polarizations of the optical communication signal, the estimate being generated using two or more different approximations (e.g., in accordance with Eqs. (9a)-(9c)) of an effect of polarization mode dispersion on the nonlinear interference noise within a bandwidth of the optical communication signal.

(109) In some embodiments of the above apparatus, the optical front-end circuit is configured to apply the optical communication signal to an optical transport link (e.g., 140, FIG. 1) for propagation therethrough; and wherein the digital signal processor is further configured to generate the estimate using a backward-propagation algorithm, the backward-propagation algorithm being configured to cause a pre-distortion of the optical communication signal that tends to be removed by the propagation.

(110) In some embodiments of any of the above apparatus, the digital signal processor is further configured to switch between the two or more different approximations based on a comparison, with a fixed threshold value (e.g., Δω.sub.0, FIG. 4), of a difference between frequencies of optical waves contributing to the nonlinear interference noise.

(111) In some embodiments of any of the above apparatus, the digital signal processor is further configured to: apply a first (e.g., in accordance with Eq. (9a)) of the two or more approximations if an absolute value of a difference between frequencies of a first optical wave and a second optical wave is smaller than the fixed threshold value; and apply a second (e.g., in accordance with Eq. (9c)) of the two or more approximations if the absolute value of the difference between the frequencies of the first optical wave and the second optical wave is greater than the fixed threshold value.

(112) In some embodiments of any of the above apparatus, the digital signal processor is further configured to apply a third (e.g., in accordance with Eq. (9b)) of the two or more approximations if the absolute value of the difference between the frequencies of the first optical wave and the second optical wave is smaller than the fixed threshold value, and an absolute value of a difference between frequencies of a third optical wave and the second optical wave is greater than the fixed threshold value.

(113) In some embodiments of any of the above apparatus, the digital signal processor is configured to use three different approximations (e.g., defined by Eqs. (9a)-(9c)) of the effect of polarization mode dispersion on the nonlinear interference noise within the bandwidth of the optical communication signal.

(114) While this disclosure includes references to illustrative embodiments, this specification is not intended to be construed in a limiting sense. Various modifications of the described embodiments, as well as other embodiments within the scope of the disclosure, which are apparent to persons skilled in the art to which the disclosure pertains are deemed to lie within the principle and scope of the disclosure, e.g., as expressed in the following claims.

(115) Unless explicitly stated otherwise, each numerical value and range should be interpreted as being approximate as if the word “about” or “approximately” preceded the value or range.

(116) It will be further understood that various changes in the details, materials, and arrangements of the parts which have been described and illustrated in order to explain the nature of this disclosure may be made by those skilled in the art without departing from the scope of the disclosure, e.g., as expressed in the following claims.

(117) Although the elements in the following method claims, if any, are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence.

(118) Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the disclosure. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments necessarily mutually exclusive of other embodiments. The same applies to the term “implementation.”

(119) Also for purposes of this description, the terms “couple,” “coupling,” “coupled,” “connect,” “connecting,” or “connected” refer to any manner known in the art or later developed in which energy is allowed to be transferred between two or more elements, and the interposition of one or more additional elements is contemplated, although not required. Conversely, the terms “directly coupled,” “directly connected,” etc., imply the absence of such additional elements.

(120) The described embodiments are to be considered in all respects as only illustrative and not restrictive. In particular, the scope of the disclosure is indicated by the appended claims rather than by the description and figures herein. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope.

(121) The description and drawings merely illustrate the principles of the disclosure. It will thus be appreciated that those of ordinary skill in the art will be able to devise various arrangements that, although not explicitly described or shown herein, embody the principles of the disclosure and are included within its spirit and scope. Furthermore, all examples recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the disclosure and the concepts contributed by the inventor(s) to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the disclosure, as well as specific examples thereof, are intended to encompass equivalents thereof.

(122) The functions of the various elements shown in the figures, including any functional blocks labeled as “processors” and/or “controllers,” may be provided through the use of dedicated hardware as well as hardware capable of executing software in association with appropriate software. When provided by a processor, the functions may be provided by a single dedicated processor, by a single shared processor, or by a plurality of individual processors, some of which may be shared. Moreover, explicit use of the term “processor” or “controller” should not be construed to refer exclusively to hardware capable of executing software, and may implicitly include, without limitation, digital signal processor (DSP) hardware, network processor, application specific integrated circuit (ASIC), field programmable gate array (FPGA), read only memory (ROM) for storing software, random access memory (RAM), and non volatile storage. Other hardware, conventional and/or custom, may also be included. Similarly, any switches shown in the figures are conceptual only. Their function may be carried out through the operation of program logic, through dedicated logic, through the interaction of program control and dedicated logic, or even manually, the particular technique being selectable by the implementer as more specifically understood from the context.