TAP CENTERER METHOD AND STRUCTURE FOR COHERENT OPTICAL RECEIVER

Abstract

A coherent optical receiver includes equalizer circuitry having a plurality of taps, the equalizer circuitry being configured to receive an input signal and compensate for polarization mode dispersion affecting the input signal to generate a compensated input signal. The coherent optical receiver further includes error evaluation circuitry configured to calculate a determinant of a frequency-domain (FD) coefficient-based matrix using a plurality of tap signals from among the plurality of taps, adjust an error of convergence of the compensated input signal to generate an adjusted input signal, and iteratively adjust the determinant of the FD coefficient-based matrix based on the adjusted input signal to minimize the error of convergence.

Claims

1. A coherent optical receiver, comprising: equalizer circuitry having a plurality of taps, the equalizer circuitry being configured to (i) receive an input signal and (ii) compensate for polarization mode dispersion affecting the input signal to generate a compensated input signal; and error evaluation circuitry configured to calculate a determinant of a frequency-domain (FD) coefficient-based matrix using a plurality of tap signals from among the plurality of taps, adjust an error of convergence of the compensated input signal to generate an adjusted input signal, and iteratively adjust the determinant of the FD coefficient-based matrix based on the adjusted input signal to minimize the error of convergence.

2. The coherent optical receiver of claim 1, wherein the equalizer circuitry includes a polarization mode dispersion equalizer configured to compensate for the polarization mode dispersion.

3. The coherent optical receiver of claim 2, wherein the polarization mode dispersion equalizer includes an adaptive frequency-domain equalizer.

4. The coherent optical receiver of claim 2, further comprising a chromatic dispersion equalizer configured to compensate for chromatic dispersion affecting the input signal.

5. The coherent optical receiver of claim 4, wherein the polarization mode dispersion equalizer is configured to compensate for the polarization mode dispersion subsequent to the chromatic dispersion equalizer compensating for the chromatic dispersion.

6. The coherent optical receiver of claim 1, wherein the error evaluation circuitry is configured to adjust the determinant according to:
ρ.sub.k+1(Ω.sub.m,0)=ρ.sub.k(Ω.sub.m,0)e.sup.jαΣ.sup.i=0.sup.k.sup.Δ{circumflex over (n)}.sup.d.sup.(i) where ρ.sub.k(Ω.sub.m, 0)=e.sup.−j4Δn.sup.d.sup.Ω.sup.me.sup.−jN.sup.fft.sup.Ω.sup.m(P(Ω.sub.m)P(−Ω.sub.m)).sup.2, where $\Delta n_d=n_d-\frac{N_{\mathrm{fft}}}{4},$ where n.sub.d is a group delay at impedance Ω=0, where N.sub.fft/2 is the number of the plurality of taps, and where P(Ω.sub.m) is a real positive number related to the magnitude of the frequency response of the input signal.

7. The coherent optical receiver of claim 6, wherein the evaluation circuitry is configured to estimate the group delay n.sub.d based on the plurality of taps.

8. The coherent optical receiver of claim 1, wherein the plurality of taps includes a plurality of polarization mode dispersion taps configured to compensate for the polarization mode dispersion.

9. The coherent optical receiver of claim 1 further comprising circuitry configured to increase a sampling rate of the compensated input signal.

10. The coherent optical receiver of claim 1, further comprising fast Fourier transform circuitry configured to calculate a Fourier transform of the input signal, wherein the equalizer circuitry is configured to receive the Fourier transform of the input signal.

11. The coherent optical receiver of claim 10, further comprising inverse Fourier transform circuitry configured to calculate an inverse Fourier transform of the compensated input signal.

12. The coherent optical receiver of claim 1, wherein the error evaluation circuitry includes interpolated timing recovery circuitry configured to retime the compensated input signal.

13. The coherent optical receiver of claim 1, wherein the error evaluation circuitry includes slicer circuitry configured to derive a data stream from the compensated input signal.

14. The coherent optical receiver of claim 1, further comprising least mean square circuitry configured to filter the input signal.

15. The coherent optical receiver of claim 1, wherein the error evaluation circuitry includes a phase error module configured to adjust the error of convergence.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more fully understand the present invention, reference is made to the accompanying drawings. Understanding that these drawings are not to be considered limitations in the scope of the invention the presently described embodiments and the presently understood best mode of the invention are described with additional detail through the use of the accompanying drawings in which:

[0015] FIG. 1 is a simplified set of graphs illustrating the impulse response and group delay according to an example of the present invention.

[0016] FIG. 2 is a simplified set of graphs illustrating simulation results of the center of gravity and the group delay according to an example of the present invention.

[0017] FIG. 3 is a simplified diagram illustrating an error evaluation module according to an example of the present invention.

[0018] FIG. 4 is a simplified set of graphs illustrating simulation results of the center of gravity and the group delay according to an example of the present invention.

[0019] FIG. 5 is a simplified diagram illustrating a block diagram of a coherent optical receiver according to an example of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0020] The present invention relates to communication systems and integrated circuit (IC) devices. More particularly, the present invention provides for improved methods and devices for optical communication.

[0021] The following description is presented to enable one of ordinary skill in the art to make and use the invention and to incorporate it in the context of particular applications. Various modifications, as well as a variety of uses in different applications will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to a wide range of embodiments. Thus, the present invention is not intended to be limited to the embodiments presented, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0022] In the following detailed description, numerous specific details are set forth in order to provide a more thorough understanding of the present invention. However, it will be apparent to one skilled in the art that the present invention may be practiced without necessarily being limited to these specific details. In other instances, well-known structures and devices are shown in block diagram form, rather than in detail, in order to avoid obscuring the present invention.

[0023] The reader's attention is directed to all papers and documents which are filed concurrently with this specification and which are open to public inspection with this specification, and the contents of all such papers and documents are incorporated herein by reference. All the features disclosed in this specification, (including any accompanying claims, abstract, and drawings) may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, each feature disclosed is one example only of a generic series of equivalent or similar features.

[0024] Furthermore, any element in a claim that does not explicitly state “means for” performing a specified function, or “step for” performing a specific function, is not to be interpreted as a “means” or “step” clause as specified in 35 U.S.C. Section 112, Paragraph 6. In particular, the use of “step of” or “act of” in the Claims herein is not intended to invoke the provisions of 35 U.S.C. 112, Paragraph 6.

[0025] Please note, if used, the labels left, right, front, back, top, bottom, forward, reverse, clockwise and counter clockwise have been used for convenience purposes only and are not intended to imply any particular fixed direction. Instead, they are used to reflect relative locations and/or directions between various portions of an object.

[0026] The center of gravity (CG) of the filter coefficients can be used to evaluate a proper convergence of a time-domain adaptive equalizer. Examples of the present invention provide for structures and methods of estimating the CG directly from the FD equalizer taps and compensate for an error of convergence based off of the estimated CG. The derivation of the relevant algorithms is provided below.

I. Evaluation of the Center of Gravity

[0027] Let f(n) be the discrete time, causal, impulse response of the fractional spaced equalizer. The CG of f(n) is defined as follows:

[00001]$\begin{array}{cc}{c}_g=\frac{\underset{n=0}{\overset{\infty }{.Math.}}n{.Math.f\left(n\right).Math.}^2}{\underset{n=0}{\overset{\infty }{.Math.}}{.Math.f\left(n\right).Math.}^2}& \left(1\right)\end{array}$

This equation can be used as a measure of the proper convergence of the equalizer. The following derivations produce a simple method to estimate CD based on the taps of the frequency domain equalizer.

A. Evaluation of the CG in the Presence of Chromatic Dispersion (CD)

[0028] In the presence of chromatic dispersion (CD), the Fourier transform (FT) of f(n) can be defined as follows:

F(Ω)=|F(Ω)|e.sup.jn.sup.d.sup.Ω-jβΩ.sup.2  (2)

where n.sub.d is the group delay at Ω=0 and β is the CD parameter. Without loss of generality, it can be assumed that |F(Ω)| is the magnitude of an ideal low-pass filter (i.e., a rectangular pulse in the frequency domain).

[0029] Let x(n) be a sequence with FT given by X(Ω). Then, it is verified that the FT of nx(n) results in

[00002]$j\frac{dX\left(\Omega \right)}{d\Omega }.$

The real function x(n) is defined as follows:

x(n)=n|f(n)|.sup.2  (3)

with FT given by the following:

[00003]$\begin{array}{cc}X\left(\Omega \right)=j\frac{d}{d\Omega }\left[\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F\left(\Theta \right)F^{\star }\left(\Theta -\Omega \right)d\Theta \right]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F\left(\Theta \right)\left[j\frac{d}{d\Omega }{F}^{\star }\left(\Theta -\Omega \right)\right]d\Theta & \left(4\right)\end{array}$

with the FT of |f(n)|.sup.2 being

[00004]$\left[\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F\left(\Theta \right)F^{\star }\left(\Theta -\Omega \right)d\Theta \right].$

since X(Ω)=Σ.sub.n x(n)e.sup.−jΩn, then X(0) is as follows:

[00005]$\begin{array}{cc}X\left(0\right)=\underset{n}{.Math.}x\left(n\right)=\underset{n=0}{\overset{\infty }{.Math.}}n{.Math.f\left(n\right).Math.}^2& \left(5\right)\end{array}$

[0030] Next, the FT of the sequence x(n)=n|f(n)|.sup.2 at Ω=0 (i.e., X(0)). Since |F(Ω)| is assumed to have an ideal low-pass response (i.e., its derivative is zero at Ω=0; this assumption is also valid for practical filters such as raised cosine pulses), the result is as follows:

[00006]$\begin{array}{cc}\underset{\Omega .fwdarw.0}{\lim }j\frac{d}{d\Omega }\left\{-j\left[2\beta \left(\Theta -\Omega \right)+n_d\right].Math.F\left(\Theta -\Omega \right).Math.e^{j{n}_d\left(\Theta -\Omega \right)+j{\beta \left(\Theta -\Omega \right)}^2}\right\}=\left(2\beta \Theta +n_d\right).Math.F\left(\Theta \right).Math.e^{j{n}_d\Theta +j\beta {\Theta }^2}& \left(6\right)\end{array}$

Replacing (6) in (4), and taking into account that |F(Θ)|.sup.2 is an even function, the following is obtained:

[00007]$\begin{array}{cc}X\left(0\right)=\underset{n=0}{\overset{\infty }{.Math.}}n{.Math.f\left(n\right).Math.}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\left(2\beta \Theta +n_d\right){.Math.F\left(\Theta \right).Math.}^{2}d\Theta =n_d\left[\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{.Math.F\left(\Theta \right).Math.}^{2}d\Theta \right]=n_d\underset{n=0}{\overset{\infty }{.Math.}}{.Math.f\left(n\right).Math.}^2& \left(7\right)\end{array}$

Finally, the center of gravity (1) reduces to the following:

[00008]$\begin{array}{cc}\begin{array}{c}{c}_g=\frac{\underset{n=0}{\overset{\infty }{.Math.}}n{.Math.f\left(n\right).Math.}^2}{\underset{n=0}{\overset{\infty }{.Math.}}{.Math.f\left(n\right).Math.}^2}\\ =\frac{X\left(0\right)}{\underset{n=0}{\overset{\infty }{.Math.}}{.Math.f\left(n\right).Math.}^2}\\ =n_d\end{array}& \left(8\right)\end{array}$

[0031] From (8), the CG of the time-domain impulse response f(n) can be easily derived from the group delay of F(Ω) at Ω=0.

B. Numerical Results

[0032] FIG. 1 shows the impulse response (graphs 101 and 103) and the group delay (graphs 102 and 104 for two optical channels with chromatic dispersion: 850 ps/nm (50 km) in graphs 101 and 102; and 3400 ps/nm (200 km) in graphs 103 and 104. Baud rate is 32 GBd. A raised cosine filter with roll-off factor of 20% is used. Here, it is verified that the GD at Ω=0 (˜11 and 22 samples for 50 and 200 km, respectively) agrees very well with the center of the impulse response (i.e., ˜the center of gravity).

[0033] FIG. 2 provides simulation results of the center of gravity and the group delay Ω=0 for a frequency domain, multiple-input multiple-output, frequency spreading equalizer (FD-MIMO-FSE) with quadrature phase shifting keying (QPSK) modulation (graph 201). FIG. 2 also depicts the received constellation at the equalizer output (graph 202). An FS-MIMO-FSE with 128 taps, 50% overlap (i.e., N.sub.fft=256), and oversampling (OS) of 4/3 is considered. A tap leakage algorithm is used. The baud rate is 1/T=32 GBd and the optical signal to noise ratio (OSNR) is 14 dB. The simulation results consider an optical channel with variable differential group delay (DGD) between 0 and 468 ps with a low-pass filter for different values of the low-pass filter (LPF) parameter (β). In FIG. 2, the LPF parameter is β=2.sup.−12. The evolution of the DGD and CG is similar in all cases. Also, the total CG remains approximately constant around 64. On the other hand, the CG for a given polarization follows the variation of the DGD very well (i.e., 234 ps is ˜10 samples at T/OS). Further, the fluctuations of the GD estimation can be mitigated by reducing the bandwidth of the low-pass filter at the expense of higher latency.

II. Center-Tap Algorithm

A. Timing Recovery Based on the Taps of Adaptive FD Equalizers

[0034] Let F(Ω.sub.m) be the frequency domain coefficient of the MIMO-FSE at a certain frequency Ω.sub.m such that 0<Ω.sub.mOS/T<π/T. The MIMO FD coefficient can be expressed as follows:

F(Ω.sub.m)=e.sup.−jn.sup.d.sup.Ω.sup.m.sup.-jτΩ.sup.m.sup.-jβΩ.sup.m.sup.2e.sup.jϕP(Ω.sub.m)/(Ω.sub.m)  (9)

where τ is the sampling phase error, n.sub.d is the group delay at Ω=0 and τ=0 (i.e., no sampling phase error; also, from (8), assume c.sub.g=n.sub.d), β is the CD parameter, ϕ is an arbitrary phase, P(Ω.sub.m) is a real positive number related to the magnitude of the frequency response of the impulse response of the electrical filter used for both polarizations, while J(Ω.sub.m) is a 2×2 unitary Jones matrix. Let e.sup.jθ(Ω)G(Ω) be the frequency response of a filter with G(Ω) and θ(Ω) denoting the magnitude and the phase response, respectively. The zero-forcing equalizer response results in F(Ω.sub.m)=e.sup.−jθ(Ω)P(Ω) with P(Ω)=1/G(Ω).

[0035] Note the following equation:

F(−Ω.sub.m)=e.sup.−jn.sup.d.sup.Ω.sup.m.sup.-jτΩ.sup.m.sup.+jβΩ.sup.m.sup.2e.sup.−jϕP(−Ω.sub.m)J.sup.H(−Ω.sub.m)  (10)

where .sup.H denotes transpose and complex conjugation. From (9) and (10), a 2×2 matrix M.sub.f(Ω.sub.m) can be defined as follows:

M.sub.f(Ω.sub.m)=F(Ω.sub.m)F.sup.H(−Ω.sub.m)  (11)

=e.sup.−j2n.sup.d.sup.Ω.sup.m.sup.-j2τΩ.sup.mP(Ω.sub.m)J(Ω.sub.m)J.sup.H(−Ω.sub.m)  (12)

The determinant of M.sub.f(Ω.sub.m) results in the following:

ρ(Ω.sub.m)=det{M.sub.f(Ω.sub.m)}=e.sup.−j4n.sup.d.sup.Ω.sup.m.sup.-j4τΩ.sup.m(Ω.sub.m)  (13)

where (Ω.sub.m)=(P(Ω.sub.m)P(−Ω.sub.m)).sup.2 is real and positive. In general, the sampling phase changes with time, therefore the determinant can be rewritten as follows:

ρ(Ω.sub.m)=e.sup.−j4n.sup.d.sup.Ω.sup.m.sup.-j4τ(t)Ω.sup.m(Ω.sub.m)  (14)

[0036] Without loss of generality, it can be assumed that the sampling phase error at t=0 is zero (i.e., ρ(Ω.sub.m, 0)=e.sup.−j4n.sup.d.sup.Ω.sup.m(Ω.sub.m)) Thus, the angle of the product is as follows:

ρ(Ω.sub.m,t)ρ*(Ω.sub.m,0)=e.sup.−j4τ(t)Ω.sup.m.sup.2(Ω.sub.m)  (15)

Here, (15) provides an estimate of the sampling phase error at instant t, which can be used for timing recovery.

B. Center-Tap Algorithm

[0037] Next, it is assumed that the FD equalization is achieved by using an overlap-and-save technique. Without loss of generality, we also assume that the overlap factor is 50%; therefore, the time domain impulse response has N.sub.fft/2 taps. In an ideal situation, the center of gravity should be half the number of taps, that is, n.sub.d=N.sub.fft/4 taps. However, as a result of an imperfect start-up procedure (e.g., interaction between the timing recovery stage and the adaptive equalizer), the CG of the time-domain equalizer response may be shifted to a certain side. The latter effect may cause performance degradation; therefore, an algorithm to center the equalizer taps is required.

[0038] We define the error of convergence as follows:

Δn.sub.d=n.sub.d−N.sub.fft/4  (16)

Note that the optimal convergence is experienced when the CG (or n.sub.d) is N.sub.fft/4, that is, when Δn.sub.d=0. From (16), the determinant (14) at instant t=0 can be expressed as follows:

ρ(Ω.sub.m,0)=e.sup.−j4Δn.sup.d.sup.Ω.sup.m.sup.-jN.sup.fft.sup.Ω.sup.m(Ω.sub.m)  (17)

[0039] A timing recovery stage based on (15) seeks to keep to zero the phase error with respect to the reference (17). Therefore, in order to minimize the “convergence error” Δn.sub.d, the reference (35) is iteratively adjusted by using the following:

ρ.sub.k+1(Ω.sub.m,0)=ρ.sub.k(Ω.sub.m,0)e.sup.jαΔ{circumflex over (n)}.sup.d.sup.(k)  (18)

=ρ.sub.k(Ω.sub.m,0)e.sup.jαΣ.sup.i=0.sup.k.sup.{circumflex over (n)}.sup.d.sup.(i)  (19)

where α is a small positive gain and Δ{circumflex over (n)}.sub.d(k) is the error of convergence at the k-th iteration (Δ{circumflex over (n)}.sub.d(0)=Δn.sub.d) given by the following:

Δ{circumflex over (n)}.sub.d(k)={circumflex over (n)}.sub.d(k)−N.sub.fft/4  (20)

with {circumflex over (n)}.sub.d(k) being the group delay at Ω=0 at the k-th iteration, which is estimated as described in Section I. From (17) note that (19) can be thought of as a first-order PLL designed to compensate a (constant) phase error of −4Δn.sub.dΩ.sub.m (see FIG. 3). Then, it is verified that

[00009]$\underset{k.fwdarw.\infty }{\lim }\Delta {\hat{n}}_d\left(k\right)=0\mathrm{while}\underset{k.fwdarw.\infty }{\lim }{\rho }_k\left({\Omega }_m,0\right)=e^{-j{N}_{\mathrm{fft}}{\Omega }_m}𝒫\left({\Omega }_m\right).$

[0040] As a result of the high latency in the “phase error” computation block of FIG. 3 due to the equalizer adaptation, the centered process (18) should be done slowly (e.g., α<<1). As shown, error evaluation module 300 includes an iterator module 310 coupled in a loop to a phase error module 320, a loop filter module 330, and a feedback module 340. In an example, the error evaluation module 300, which can be implemented in a timing recovery module, is configured to estimate the group delay n.sub.d from the plurality of PMD taps. In a specific example, estimating the group delay n.sub.d is accomplished using two of the plurality of PMD taps. In a specific example, these modules implement the algorithms described previously. The iterator module is configured to compute ρ.sub.k+1(Ω.sub.m, 0) and the phase error module is configure dot adjust the error of convergence Δn.sub.d of the input signal resulting in an adjusted input signal. The loop filter is configured to filter the adjusted input signal and the feedback module is configured to provide the adjusted input signal to the iterator module. Those of ordinary skill in the art will recognize other variations, modifications, and alternatives.

[0041] FIG. 4 shows two sets of graphs (401/402 and 403/404), each with the group delay and CG on one graph and the received constellation on another graph. FIG. 4 shows an example with N.sub.fft=256 where the equalizer taps are initialized with an “error of convergence” (Δn.sub.d) of ˜4 (graphs 401 and 402) and −3 (graphs 403 and 404). The update process (18) was carried out every 4000 data symbols with α=2.sup.−7. In both cases, it is verified that the center of gravity at regime tends to the optimal value n.sub.d=N.sub.fft/4=64.

[0042] FIG. 5 is a simplified diagram illustrating a block diagram of a coherent optical receiver according to an example of the present invention. As shown, device 500 can include an input signal; a first fast Fourier transform (FFT) module receiving the input signal, the first FFT module 511 being configured to compute a first discrete Fourier transform (DFT) of the input signal; a chromatic dispersion (CD) equalizer module 520 coupled to the first FFT module 511, the CD equalizer module 520 being configured to compensate for CD affecting the input signal; a polarization mode dispersion (PMD) equalizer module 530 coupled to the CD equalizer and a least means square (LMS) module 580, the PMD equalizer module 530 being configured to compensate for PMD affecting the input signal following compensation by the CD equalizer module. The PMD equalizer module 530 includes a plurality of PMD taps. In a specific example, the CD equalizer module includes a non-adaptive frequency-domain (FD) equalizer, and the PMD equalizer module includes an adaptive FD equalizer. Further, the input can be a dual polarization input with an x-type and y-type inputs.

[0043] In an example, the device can also include an inverse FFT (IFFT) module 540 coupled to the PMD equalizer module 530, the IFFT module being configured to compute an inverse DFT of the input signal; an interpolated timing recovery (ITR), slicer, and error evaluation module 550 coupled to the IFFT module 540. The ITR, slicer, and the error evaluation can be separate modules, the ITR module being configured to retime the input signal, the slicer module being configured to derive the data stream, and the error evaluation module being configured to retime the input signal. The error evaluation module can include a structure and function similar to that shown in FIG. 3.

[0044] In an example, the device can include a zero padding module 560 coupled to the slicer and error evaluation module 550, the zero padding module 560 being configured to increase a sampling rate of the input signal; and a second FFT module 570 coupled to the zero padding module 560, the second FFT module 570 being configured to compute a second DFT of the input signal. In an example, the LMS module 580 is coupled to the second FFT module 570, the CD equalizer module 520, and the PMD equalizer module 530. The LMS module 580 outputs to the PMD equalizer module 530 and is configured to filter the input signal. Those of ordinary skill in the art will recognize other variations, modifications, and alternatives.

[0045] The reduction of complexity results from not having to use separate FD BCD and FFE equalizers. As shown in FIG. 5, only one (dual polarization) FFT and only one (dual polarization) IFFT are required in the signal path. An architecture using separate FD blocks for the BCD and FFE requires extra FFTs and IFFTs as a result of going back and forth from the frequency domain to the time domain. Further, the LMD update can be viewed as computing a correlation between the input signal and the error. Here, this correlation is computed in the frequency domain, which reduces complexity the same way as in convolution computations.

[0046] With this architecture, an interaction problem arises when TR is achieved after the adaptive equalizer (i.e., PMD equalizer). This problem occurs because the adaptation algorithm of the equalizer and the timing-synchronizer use the same (equalized) signal as their input. The equalizer tries to compensate the misadjustment of the discrete time impulse response due to the sampling phase error, while the TR tries to equalize the distortion of the impulse response by changing the sampling phase. As a consequence, the timing phase and the equalizer taps are drifting. Conventional solutions to this problem have severe drawbacks in (time variant) coherent optical channels. Making the timing loop much faster than the equalizer can mitigate this interaction problem, but the timing phase may still drift slowly over long periods of time.

[0047] According to an example of the present invention, a tap centering algorithm can be used to estimate CG directly from the FD equalizer taps and compensate for an error of convergence based off of the estimated CG. This estimation method and associated device architecture is able to achieve an excellent tradeoff between accuracy and complexity.

[0048] In an example, the present invention provides a method of operating a coherent optical receiver device. The method can include providing an input signal; computing, by a first fast Fourier transform (FFT) module receiving the input signal, a first discrete Fourier transform (DFT) of the input signal. The method can include compensating, by a chromatic dispersion (CD) equalizer module coupled to the first FFT module, for CD affecting the input signal; and compensating, by a polarization mode dispersion (PMD) equalizer module coupled to the CD equalizer module and coupled to a least means square (LMS) module and having a plurality of PMD taps, for PMD affecting the input signal following the compensation by the CD equalizer module. Further, the method can include computing, by an inverse FFT (IFFT) module coupled to the PMD equalizer module, an inverse DFT of the input signal. In an example, the method includes filtering, by the LMS module coupled to the CD equalizer module and the second FFT module and the PMD equalizer module, the input signal.

[0049] In an example, the method includes iteratively adjusting, by an error evaluation module coupled to the ITR module, a determinant of a frequency-domain (FD) coefficient-based matrix to minimize an error of convergence. The iterative adjustment can include estimating, by the error evaluation module, the group delay n.sub.d from the plurality of PMD taps. In a specific example, the error evaluation module includes iterator module coupled in a loop to a phase error module, a loop filter module, and a feedback module; further, the iterative adjustment of the determinant of the FD coefficient-based matrix includes computing, by an iterator module, the iterative function ρ.sub.k+1(Ω.sub.m, 0); adjusting, by the phase error module, the error of convergence Δn.sub.d of the input signal resulting in an adjusted input signal; filtering, by the loop filter, the adjusted input signal; and providing, by the feedback module, the adjusted input signal to the iterator module.

[0050] While the above is a full description of the specific embodiments, various modifications, alternative constructions and equivalents may be used. Therefore, the above description and illustrations should not be taken as limiting the scope of the present invention which is defined by the appended claims.