Blind estimation of residual chromatic dispersion and carrier frequency offset

Upon receiving a communications signal conveying symbols at a symbol period T, a receiver applies filter coefficients to a digital representation of the communications signal, thereby generating filtered signals characterized by a substantially raised cosine shape in the frequency domain with a roll-off factor α, where components of the filtered signals correspond to angular frequencies The receiver calculates first-order components from a first phase derivative of the components at a first differential distance, second-order components from a second phase derivative of the first-order components at a second differential distance, and composite second-order components from an average of the second-order components over multiple time intervals. Using the composite second-order components, the receiver calculates at least one of (i) an estimate of residual chromatic dispersion (CD) associated with the filtered signals, and (ii) an estimate of carrier frequency offset (CFO) associated with the filtered signals.

TECHNICAL FIELD

This document relates to the technical field of optical communications.

BACKGROUND

In an optical communication network, an optical transmitter may transmit an optical signal over a communication channel to an optical receiver, where the signal is representative of digital information in the form of symbols or bits. The receiver may process the signal received over the communication channel to recover estimates of the symbols or bits. Various components of the optical communication network may contribute to signal degradation, such that the signal received at the receiver comprises a degraded version of the signal that was generated at the transmitter. Degradation or distortion may be caused by polarization mode dispersion (PMD), polarization dependent loss or gain (PDL or PDG), amplified spontaneous emission (ASE), wavelength-dependent dispersion or chromatic dispersion (CD), and other effects.

CD causes a signal to broaden as it travels down a length of fiber. This broadening is the result of different velocities experienced by different spectral components of the signal. A technique known as dispersion compensation may be used to compensate for the net CD in a given link. Dispersion compensation is achieved by providing a negative dispersion to oppose the positive dispersion in the link. Dispersion compensation may be performed using some combination of pre-compensation at the transmitter and post-compensation at the receiver. The sum of the CD pre-compensation and post-compensation should always be substantially equal to the net CD of the link but with the opposite sign, so as to substantially compensate for the net CD of the link. The amount of CD detected in a communications signal at the receiver may be referred to as the residual CD, and is equivalent to any remaining portion of the net CD in the link that was not already compensated for by the CD pre-compensation applied at the transmitter.

In general, the carrier frequency generated at a transmitter oscillator differs from the carrier frequency generated at a receiver oscillator. The difference between the two carrier frequencies may be referred to as carrier frequency offset (CFO). The ability to accurately estimate the CFO at the receiver is important for decoding digital information from coherent optical signals.

SUMMARY

According to a broad aspect, a receiver apparatus comprises a communication interface configured to receive a communications signal conveying symbols at a symbol period T, where T is a positive real number. The receiver apparatus comprises circuitry configured to apply filter coefficients to a digital representation of the communications signal, thereby generating filtered signals characterized by a substantially raised cosine shape in the frequency domain with a roll-off factor α, wherein a is a real number satisfying 0<α≤1, and where components of the filtered signals correspond to a range of angular frequencies

ω=-π⁡(1+α)T⁢…-π⁡(1-α)T,+π⁡(1-α)T⁢…+π⁡(1+α)T.
The receiver apparatus comprises circuitry configured to calculate first-order components from a first phase derivative of the components of the filtered signals at a first differential distance. The receiver apparatus comprises circuitry configured to calculate second-order components from a second phase derivative of the first-order components at a second differential distance. The receiver apparatus comprises circuitry configured to calculate composite second-order components from an average of the second-order components over a plurality of time intervals. The receiver apparatus comprises circuitry configured to calculate, using the composite second-order components, at least one of (i) an estimate of residual chromatic dispersion (CD) associated with the filtered signals, and (ii) an estimate of carrier frequency offset (CFO) associated with the filtered signals.

According to one example, the receiver apparatus comprises circuitry configured to calculate updated filter coefficients using the estimate of the residual CD, and circuitry configured to apply the updated filter coefficients to the digital representation of the communications signal, thereby generating updated filtered signals.

According to another example, the estimate of the residual CD comprises an initial estimate based on an initial value of the second differential distance, and the receiver apparatus comprises circuitry configured to calculate a subsequent estimate of the residual CD associated with the updated filtered signals based on a subsequent value of the second differential distance, where the subsequent value of the second differential distance exceeds the initial value of the second differential distance, and where the initial estimate of the residual CD exceeds the subsequent estimate of the residual CD.

According to another example, the receiver apparatus comprises circuitry configured to calculate third-order components from a third phase derivative of the composite second-order components at a third differential distance, and circuitry configured to calculate the estimate of the residual CD using an average of the third-order components over a plurality of frequencies.

According to another example, the receiver apparatus comprises circuitry configured to calculate a timing bin corresponding to a peak magnitude of an inverse Fourier transform of the composite second-order components, and circuitry configured to calculate the estimate of the residual CD using the timing bin.

According to another example, the receiver apparatus comprises circuitry configured to perform carrier recovery on the filtered signals using the estimate of the CFO.

According to another example, the receiver apparatus comprises circuitry configured to calculate either a frequency corresponding to a peak magnitude of the composite second-order components or a frequency corresponding to a center of gravity of the magnitudes of the composite second-order components, and circuitry configured to calculate the estimate of the CFO using the frequency.

According to another example, the receiver apparatus comprises circuitry configured to calculate a frequency shift of an expected shape of the composite second-order components that maximizes a correlation between the composite second-order components and the expected shape, and circuitry configured to calculate the estimate of the CFO using the frequency shift.

According to another example, the receiver apparatus comprises circuitry configured to calculate an autocorrelation of an inverse Fourier transform of the composite second-order components, and circuitry configured to calculate the estimate of the CFO using the autocorrelation.

According to another example, the estimate of the CFO is calculated using only the composite second-order components having magnitudes equal to or greater than a predefined threshold.

DETAILED DESCRIPTION

FIG. 1illustrates an example communication network100, in accordance with some examples of the technology disclosed herein.

The communication network100may comprise at least one transmitter device102and at least one receiver device104, where the transmitter device102is capable of transmitting signals over a communication channel, such as a communication channel106, and where the receiver device104is capable of receiving signals over a communication channel, such as the communication channel106. According to some examples, the transmitter device102is also capable of receiving signals. According to some examples, the receiver device104is also capable of transmitting signals. Thus, one or both of the transmitter device102and the receiver device104may be capable of acting as a transceiver. According to one example, the transceiver may comprise a modem. The signals transmitted in the communication network100may be representative of digital information in the form of symbols or bits.

The communication network100may comprise additional elements not illustrated inFIG. 1. For example, the communication network100may comprise one or more additional transmitter devices, one or more additional receiver devices, one or more controller devices, and one or more other devices or elements involved in the communication of signals in the communication network100.

According to some examples, the signals that are transmitted and received in the communication network100may comprise any combination of electrical signals, optical signals, and wireless signals. For example, the transmitter device102may comprise a first optical transceiver, the receiver device104may comprise a second optical transceiver, and the communication channel106may comprise an optical communication channel. According to one example, one or both of the first optical transceiver and the second optical transceiver may comprise a coherent modem.

Each optical communication channel in the communication network100may include one or more links, where each link may comprise one or more spans, and each span may comprise a length of optical fiber and one or more optical amplifiers.

Where the communication network100involves the transmission of optical signals, the communication network100may comprise additional optical elements not illustrated inFIG. 1, such as wavelength selective switches, optical multiplexers, optical de-multiplexers, optical filters, and the like. Frequency division multiplexing (FDM) may be used to digitally divide up the modulated optical spectrum into a plurality of subcarriers, each with a different center frequency, such that each subcarrier may be used to transmit a signal that is representative of a different stream of symbols. In this manner, a plurality of symbol streams may be simultaneously communicated, in parallel, over the optical communication channel106. FDM is possible when the frequencies of the subcarriers are sufficiently separated that the bandwidths of the signals do not significantly overlap. Each different subcarrier corresponds to a different FDM sub-band, also referred to as a FDM channel. Wavelength division multiplexing (WDM) may be used to transmit a plurality of data streams in parallel, over a respectively plurality of carriers, where each carrier is generated by a different laser.

According to some examples, a signal generated at the transmitter device102may be representative of a stream of symbols to be transmitted at times set by a transmitter symbol clock, where the frequency of the transmitter symbol clock is set to be, for example, a certain ratio or fraction of a transmitter sampling frequency which may be set by a voltage controlled oscillator (VCO) and associated clocking circuits at the transmitter device102. The frequency of the symbol clock at the transmitter device102may be referred to as the transmitter symbol frequency or symbol rate or baud rate (which may be denoted herein by fSin Hertz or ωSin radians/sec). At the receiver device104, estimates of the symbols may be recovered by sampling the received signal at times set by a receiver sample clock, where the frequency of the receiver sample clock may be set by a VCO at the receiver device104. The frequency of the sample clock at the receiver device104may be referred to as the receiver sample frequency or sample rate (which may be denoted herein by FSin Hertz). The receiver sample rate may be selected to satisfy the Nyquist criterion for the highest anticipated transmitter symbol rate. For example, if the transmitter symbol rate is expected to be 10 GBaud, then the receiver sample rate may be set to 20 GHz. U.S. Pat. No. 7,701,842 to Roberts et al. describes using a fractional sample rate that is less than double the symbol rate. A symbol clock at the receiver device104may be set to be a certain ratio or fraction of the receiver sample rate and, after initial processing, the information stream may be resampled to a receiver symbol rate that is equal to the transmitter symbol rate. Herein, the term “samples” is generally used to refer to samples taken at the receiver symbol rate, or at some oversampling rate, depending upon the context.

Various elements and effects in the communication network100may result in the degradation of signals transmitted between different devices. Thus, a signal received at the receiver device104may comprise a degraded version of a signal transmitted by the transmitter device102. For example, where the communication channel106is an optical communication channel, the signal transmitted by the transmitter device102may be degraded by polarization mode dispersion (PMD), polarization dependent loss or gain (PDL or PDG), state of polarization (SOP) rotation, amplified spontaneous emission (ASE) noise, and wavelength-dependent dispersion or chromatic dispersion (CD), and other effects. The degree of signal degradation may be characterized by signal-to-noise ratio (SNR), or alternatively by noise-to-signal ratio (NSR).

Any remaining portion of the net CD in the link that was not already compensated for by CD pre-compensation applied at the transmitter may be referred to as the residual CD, and should be compensated for by CD post-compensation applied at the receiver.

The first stages of start-up of a receiver, including clock recovery, carrier recovery, and linear equalizer training, may depend on a priori knowledge of the residual CD in the communications signal. Where the net CD of the link and the amount of CD pre-compensation applied at the transmitter are known, the residual CD may be inferred. However, where one or both of the net CD and amount of CD pre-compensation are unknown, then it may be necessary to perform measurements to estimate the residual CD.

Various techniques for estimation of the residual CD in a link are known. A common technique involves applying a filter at the receiver that is intended to invert or reverse the residual CD in the link (i.e., a CD post-compensation filter), where the filter is characterized by as-yet-undefined coefficients. According to some examples, the filter is applied while sweeping through a series of different values of the coefficients, and the values that result in a peak in the strength of the Godard clock signal (or variant thereof) are used to estimate the residual CD in the link. Examples of this technique are described by Hauske et al. in “Precise, Robust and Least Complexity CD estimation,” in Optical Fiber Communication Conference, Paper JWA032, 2011, and by Sui et al. in “Fast and Robust Blind Chromatic Dispersion Estimation Using AutoCorrelation of Signal Power Waveform for Digital Coherent Systems,”Journal of Lightwave Technology, Volume 31, Issue 2, 2013. The requirement to scan through multiple values of filter coefficients makes this a slow technique for CD estimation, adding significant delay to the start-up of the receiver device (also referred to as the acquisition procedure). Furthermore, the methods described by Hauske et al. and Sui et al. are sensitive to clock frequency offset between the transmitter and the receiver, and are also sensitive to fiber optical impairments such as differential group delay (DGD), SOP rotation, and PDL.

Conventional methods for estimating carrier frequency offset (CFO) rely on the insertion of synchronization symbols or pilot symbols among the data symbols conveyed by the communication signal. This is described, for example, by Magarini et al. in “Pilot-Symbols-Aided Carrier-Phase Recovery for 100-G PM-QPSK Digital Coherent Receivers,”IEEE Photonics Technology Letters, vol. 24, no. 9, pp. 739-741, May 2012, and by Spalvieri and Barletta in “Pilot-Aided Carrier Recovery in the Presence of Phase Noise,”IEEE Transactions on Communications, vol. 59, no. 7, pp 1966-1974, July 2011. However, the inclusion of these synchronization symbols reduces the effective data transmission rate. Furthermore, CFO estimation based on synchronization symbols may be dependent on clock recovery, as well as the equalization of various channel impairments such as CD, group delay (GD), and PMD. Consequently, using synchronization symbols to estimate CFO may not be possible during the first stages of receiver modem start-up, when the circuits are not yet functioning properly.

FIG. 2illustrates an example receiver device200, in accordance with some examples of the technology disclosed herein. The receiver device200is an example of the receiver device104. The receiver device200may comprise additional components that are not described in this document.

The receiver device200is configured to receive an optical signal204, which may comprise a degraded version of an optical signal generated by a transmitter device, such as the transmitter device102. According to some examples, a laser of the transmitter device may generate a continuous wave (CW) optical carrier, which is split by a polarizing beam splitter into polarized components. The polarized components may be modulated by electrical-to-optical (E/O) modulators of the transmitter device to produce modulated polarized optical signals that are combined by a beam combiner, thus yielding the optical signal. The optical signal generated by the transmitter device may be representative of information bits (also referred to as client bits) which are to be communicated to the receiver device200. The client bits may be mapped to symbols such that optical signal is representative of a stream of symbols having a symbol rate fs. According to some examples, the transmitter device may be configured to apply forward error correction (FEC) encoding to the client bits to generate FEC-encoded bits, which may then be mapped to one or more streams of data symbols. The transmitter device may be configured to apply processing to the one or more streams of symbols, including digital up-sampling of the symbols, followed by operations such as pulse shaping, FDM subcarrier multiplexing, distortion pre-compensation, and CD pre-compensation. The processing may include the application of one or more filters, which may involve the application of one or more Fast Fourier Transforms (FFTs) and one or more corresponding inverse FFTs (IFFTs). The FFT operations described throughout this document may alternatively be performed using discrete Fourier transform (DFT) operations. Similarly, the IFFT operations described throughout this document may alternatively be performed using inverse DFT (IDFT) operations.

At the receiver device200, a polarizing beam splitter206is configured to split the received optical signal204into polarized components208. According to one example, the polarized components208may comprise orthogonally polarized components corresponding to an X polarization and a Y polarization. An optical hybrid210is configured to process the components208with respect to an optical signal212produced by a laser214, thereby resulting in optical signals216,218,220,222corresponding to the dimensions XI, XQ, YI, YQ, where XI and XQ denote the in-phase and quadrature components of the X polarization, respectively, and YI and YQ denote the in-phase and quadrature components of the Y polarization, respectively. Photodetectors224,226,228,230are configured to convert the optical signals216,218,220,222output by the optical hybrid210to respective analog signals232,234,236,238. Together, elements such as the beam splitter206, the laser214, the optical hybrid210and the photodetectors224,226,228,230may form a communication interface configured to receive optical signals from other devices in a communication network.

The receiver device200may comprise an application-specific integrated circuit (ASIC)240. The ASIC240may comprise analog-to-digital converters (ADCs)242,244,246,248which are configured to sample the analog signals232,234,236,238, respectively, and to generate respective digital signals250,252,254,256. Although illustrated as comprised in the ASIC240, in an alternate implementation the ADCs242,244,246,248or portions thereof may be separate from the ASIC240. The ADCs242,244,246,248sample the analog signals232,234,236,238periodically at a sample rate, where the sample rate may be based on a signal received from a VCO at the receiver device200(not shown).

The ASIC240is configured to apply digital signal processing258to the digital signals250,252,254,256, which ultimately results in corrected client bits202. In general, the processing258may comprise the application of one or more filters to the digital signals250,252,254,256, which may involve the application of one or more FFTs and one or more corresponding IFFTs. The processing258may also include down-sampling, FDM subcarrier de-multiplexing, distortion post-compensation, and CD post-compensation. The processing258may comprise the application of an adaptive linear equalizer to compensate for low-memory linear imperfections of the fiber, that is, any linear impairment whose time-domain impulse response is short, such as DGD, PDL, and SOP rotation. The processing258may further comprise clock recovery, carrier recovery, and decoding. Where the optical signal204is representative of symbols comprising FEC-encoded bits generated as a result of applying FEC encoding to client bits at the transmitter, the processing258may comprise FEC decoding to recover the corrected client bits202.

The CD post-compensation and the carrier recovery implemented as part of the processing258depend on accurate estimates of residual CD and CFO, respectively. As previously noted, existing methods for estimating residual CD and CFO suffer from various limitations. In particular, the known methods for estimating residual CD and CFO may be sensitive to clock frequency offset, DGD, GD, SOP rotation, PDL, and PMD. Consequently, these methods may be unsuitable for “blind” implementation (i.e., during modem start-up, prior to compensation of various signal impairments).

This document proposes technology for blind estimation of residual CD and CFO that may avoid one or more disadvantages of existing methods. The proposed techniques may be used for “blind estimation” of residual CD and/or CFO in that they do not require any prior knowledge of transmitter signal patterns, and are independent of clock phase/frequency offset, and optionally independent of DGD, SOP rotation, and PDL. Accordingly, the techniques may be applied during the first stages of receiver modem start-up, when the DSP circuit(s) of the receiver are not yet functioning properly.

FIG. 3illustrates example digital signal processing300for blind estimation of residual CD and CFO in accordance with some examples of the technology disclosed herein. The digital signal processing300is an example of the digital signal processing258.

Digital signals302and304corresponding to the X and Y polarizations are input to the digital signal processing300. The digital signal302may represent, for example, the signals250,252output by the ADCs242,244, respectively. The digital signal304may represent, for example, the signals254,256output by the ADCs246,248, respectively.

A timing alignment operation306may be applied to the digital signals302,304, thereby resulting in respective signals308,310. According to some examples, the transmitted data is encapsulated into frames, where each frame consists of multiple slices (or blocks), and where each slice has a number of samples equivalent to the IFFT size at the transmitter and the FFT size at the receiver. At the transmitter, the slice timing may be aligned with the IFFT output. At the receiver, due to GD and DGD, the slice may not be perfectly aligned with the FFT input. The timing aligner306is employed to adjust the timing of each slice by an integer number of samples to the beginning of the FFT input. The timing aligner306may be implemented, for example, as two barrel shifters, each of which applies a different amount of adjustment on the X and Y polarizations. Although not explicitly illustrated inFIG. 3, the timing aligner306may be followed by an overlap-and-save (OAS) operation.

Each of the time-domain signals308,310may undergo a FFT operation312of length N to generate respective frequency-domain signals314,316corresponding to the X and Y polarizations, respectively, where N is a positive integer. According to one example, N=400. The frequency-domain signals314,316are made up of FFT blocks (also referred to as slices), where each FFT block corresponds to a different time interval.

The processing300comprises a filter318(herein referred to as a “C filter”) which is designed to at least partially compensate for slowly changing channel impairments, such as residual CD. Where the sample rate at the receiver satisfies the Nyquist criterion, application of a Nyquist-pulse shape has the desirable effects of achieving zero inter-symbol interference (ISI) in the time domain and minimum noise bandwidth. A well-known example of a Nyquist pulse shape is a raised cosine filter. It is common practice to split the Nyquist filter, such as the raised cosine filter, between a transmitter and receiver by applying, for example, a root-raised cosine filter at each device, also known as matched filters. For example, the C filter318may comprise a root-raised cosine filter that matches a root-raised cosine filter applied at the transmitter. The shape of the raised cosine achieved by the pair of matched filters is characterized by a roll-off factor α, where a is a real number satisfying 0<α≤1. As the value of the roll-off factor α approaches zero, the shape of the raised cosine becomes closer to a rectangle function in the frequency domain.

Using either convolution in the time domain, or multiplication in the frequency domain, the C filter318may apply first compensation coefficients319to the signals314,316, thereby resulting in respective signals320,322. As will be described in more detail herein, the first compensation coefficients319may be calculated such that the C filter318at least partially compensates for the residual CD in the signals314,316. The C filter318may be referred to as “static” because the updating of the first compensation coefficients319may be relatively infrequent. For example, the first compensation coefficients319may be updated once every second, such that the C filter318is able to track and compensate for relatively slow changes in the channel response, such as changes in CD, which are typically at a rate on the order of <1 Hz.

The processing300further comprises a filter324(herein referred to as an “adaptive filter”) which is designed to at least partially compensate for relatively fast changes in the channel response, such as SOP changes, PMD changes, PDL changes, small amounts of residual CD, and analog characteristics of the transmitter and receiver, which change at a rate on the order of kHz. For example, the adaptive filter324may compensate for impairments varying at a rate of approximately 100 kHz. According to some examples, the adaptive filter324may rely on a Least Mean Squares (LMS) feedback loop or other equalization techniques, such as adaptive Wiener filtering using a constant modulus algorithm (CMA) or an affine projection algorithm (APA) or a recursive least squares (RLS) algorithm. Techniques for LMS equalization in the frequency domain are described, for example, in U.S. Pat. No. 8,005,368 to Roberts et al., U.S. Pat. No. 8,385,747 to Roberts et al., U.S. Pat. No. 9,094,122 to Roberts et al., and U.S. Pat. No. 9,590,731 to Roberts et al.

Using either convolution in the time domain, or multiplication in the frequency domain, the adaptive filter324may apply second compensation coefficients325to the signals320,322, thereby resulting in respective signals326,328. As will be described in more detail herein, the second compensation coefficients may be calculated so as to at least partially compensate for residual impairments in the signals320,322. The second compensation coefficients may be periodically and incrementally adjusted so as to minimize the errors on the symbols that are currently being decoded.

A carrier recovery operation330may be applied to the signals326,328, thereby resulting in respective signals332,334which are at least partially compensated for CFO. The carrier recovery operation330may undo the effect caused laser frequency offset between the transmitter laser and the receiver laser, such as the laser214. The carrier recovery operation330may apply a phase rotation to each symbol in the signals326,328, where the amount of phase rotation applied may be linearly increased or decreased from symbol to symbol. Although not shown inFIG. 3, the signals326,328may undergo additional operations such as down-sampling and IFFT and discard prior to the carrier recovery operation330.

A decision circuit may apply a decoding operation336to the signals332,334to recover bit estimates which are represented by signal390. According to some examples, the decoding operation336may comprise soft decoding. Although not explicitly illustrated, the signal390may subsequently undergo FEC decoding.

The second compensation coefficients325applied by the adaptive filter324may be calculated using an adaptive coefficient calculation operation342which is dependent on feedback from the carrier recovery operation330and feedback from the decoding operation336, as denoted by feedback signals338,340, respectively. For example, in the case of LMS equalization, the signal338generated by the carrier recovery operation330may be representative of signals that have been at least partially compensated for laser frequency offset and linewidth, while the signal340generated by the decoding operation336may be representative of symbol estimates (corresponding to the bit estimates represented by signal390). The adaptive coefficient calculation342may include a calculation of the difference between the signals338and340(i.e., calculation of an error), and calculations that result in second coefficients325that are designed to reduce to this error in a subsequent time interval.

FIG. 4illustrates an example representation of a single polarization of a communications signal in the frequency domain following the application of a C filter. For example, the plot inFIG. 4may represent the signal320(or the signal322) following the application of the C filter318. As previously described, the C filter318may comprise a root-raised cosine filter that matches a root-raised cosine filter applied at the transmitter. The full raised cosine transfer function may be denoted by r cos(ω), where ω denotes the angular frequency. When plotted in the frequency domain, as illustrated inFIG. 4, the C-filtered signal comprises a negative roll-off region (also referred to as a lower side band) and a positive roll-off region (also referred to as an upper side band). The lower side band is defined by those angular frequencies ω within the range

ω=-π⁡(1+α)T⁢…-π⁡(1-α)T,
while the upper side band is defined by those angular frequencies ω within the range

ω=+π⁡(1-α)T⁢…+π⁡(1+α)T,
where T denotes the symbol period, and where a denotes the roll-off factor of the raised cosine. Collectively, the lower side band and the upper side band may be referred to as the Godard band (or alternatively the zeroth-order Godard band). The centres of the two side bands are separated by an angular symbol frequency

FIG. 4represents the magnitude of one polarization of a C-filtered signal as a function of angular frequency ω for a single time interval. In the case of a dual-polarization communications signal, the amplitude of the X polarization may be denoted by X(ω), the amplitude of the Y polarization may be denoted by Y(ω), and the received optical field may be expressed as

A plurality of instantaneous Godard correlation matrices may be defined over a range of angular frequencies

G⁡(ω)=R→⁡(ω)⁢R→H⁡(ω+2⁢πT),[1]
where G(ω) denotes a 2×2 matrix, where {right arrow over (R)}(ω) denotes the received optical field over the X and Y polarizations, where the superscript H denotes Hermitian conjugation, and where T denotes the symbol period. Thus, each instantaneous Godard correlation matrix is defined by multiplying first components in the lower side band with the Hermitian conjugate of second components in the upper side band, where the second components are separated from the first components by the angular symbol frequency ωS. Collectively, the plurality of instantaneous Godard correlation matrices may herein be referred to as the first-order Godard band. The information contained in the first-order Godard band may be used for clock phase detection, for example, as described by Godard in “Passband timing recovery in an all-digital modem receiver,”IEEE Trans. Commun.26(5), 517-523, 1978.

Provided that root-raised cosine matched filtering is used at the transmitter and the receiver, Equation 1 may be expressed as

G⁡(ω)=g⁡(ω)⁢g*⁡(ω+2⁢πT)·T⁡(ω)⁢TH⁡(ω+2⁢πT),[2]
where g(ω)=r cos(ω)·exp(jϕ(ω)), where “r cos” denotes a raised cosine function, where ϕ(ω) denotes a frequency-dependent phase function, where the superscript * denotes complex conjugation, and where T(ω) is a 2×2 matrix denoting the multiple-input multiple-output (MIMO) channel response at the angular frequency ω. The assumption that g(ω) has the raised cosine shape r cos(ω) is reasonable, since the magnitude of the CFO is typically small.

The phase function ϕ(ω), which is also denoted by ∠g(ω), may be expressed as

ϕ⁡(ω)=(β0+β1⁢ω+β22⁢ω2+β36⁢ω3+…),[3]
where β0denotes a zeroth-order phase term indicative of a phase error due to the fact that the transmitter and the receiver are not typically phase-locked, where β1denotes a first-order phase term indicative of overall timing phase/temporal delay (caused, for example, by C filter delay, GD, DGD, and clock slip), where β2denotes a second-order phase term indicative of residual CD, and where β3denotes a third-order phase term indicative of higher-order residual CD. According to some examples, clock recovery may be achieved based on an estimate of β1, and CD compensation may be achieved based, at least in part, on an estimate of β2. In the event that the third-order phase term β3is significant relative to β2, CD compensation may additionally be achieved based on an estimate of β3.

Given the characteristics of the phase function ϕ(ω) as expressed in Equation 3, it may be shown that the phase terms β0, β1, β2, and β3are obtainable using a series of derivative calculations with back-substitutions. For example, a calculation of the third-order derivative of the phase function ϕ(ω), that is

∂3⁢ϕ⁡(ω)∂ω3,
may be used to obtain an estimate of the third-order phase term β3. In addition, a calculation of the second-order derivative of the phase function ϕ(ω), that is

∂2⁢ϕ⁡(ω)∂ω2,
may be used together with a back-substitution of the estimate of β3to obtain an estimate of the second-order phase term β2. In addition, a calculation of the first-order derivative of the phase function ϕ(ω), that is

∂ϕ⁡(ω)∂ω,
may be used together with back-substitutions of the estimates of β3and β2to obtain an estimate of the first-order phase term β1. Finally, back-substitutions of the estimates of β3, β2, and β1into Equation 3 may be used to obtain an estimate of the zeroth-order phase term β0.

For each one of the phase terms β3, β2, β1, and β0, a plurality of unique estimates may be calculated at a respective plurality of angular frequencies ω within a frequency band of interest, such as the band

-πT⁢(1+α)≤ω≤-πT⁢(1-α)
over which the first-order Godard band is defined. The plurality of unique phase term estimates may be averaged to obtain an average estimate. For example, the average estimate of the third-order phase term β3over a range of angular frequencies ωa≤ω≤ωbmay be obtained using integration as follows:

The technique of using phase derivatives to estimate the phase terms β3, β2, β1, and β0may be implemented in the discrete frequency domain using correlation of complex samples and summation over frequency bins, as will now be described. In the following examples, X[n, k] denotes the amplitude of the X polarization of the C-filtered signal at the nthfrequency bin and the kthFFT block, while Y[n, k] denotes the amplitude of the Y polarization of the C-filtered signal at the nthfrequency bin and the kthFFT block, where n and k are integers. Referring to the C-filtered signals320and322, the index n satisfies n∈{0, 1, N−1}, and where N denotes the size of the FFT operation312.

Referring again toFIG. 3, an operation344may be applied to the signals320,322to extract those portions of the signals320,322that are within the Godard band. The resulting signals346,348are collectively referred to as the Godard band components (or zeroth-order Godard band components), and, at the kthFFT block, are expressed as

G0,x⁡[n,⁢k]=X⁡[n+N2-Ng,k]⁢⁢G0,x⁡[n,⁢k]=Y⁡[n+N2-Ng,k],[5]
where n∈{0, 1, 2Ng−1}, where 2Ngdenotes the size of the extracted Godard band, where

Ng=(L-1)⁢NL,
where N denotes the FFT size, and where L is a real number greater than one (L>1) which denotes an up-sampling factor. The up-sampling factor L may be selected to be large enough to cover the frequency band of interest

-πT⁢(1+α)≤ω≤-πT⁢(1-α)
without aliasing. In other words, L≥αmax+1, where αmaxdenotes a maximum roll-off factor supported by the system. For a system with a fixed value of α, the most efficient choice for the up-sampling factor is L=α+1.

The signals G0,x[n, k], G0,y[n, k] may be respectively referred to as the X polarization components of the Godard band and the Y polarization components of the Godard band. Each polarization of the Godard band is expected to have a raised cosine shape with an envelope R0[n] expressed as

FIG. 5illustrates an example plot of the expected envelope R0[n] of the extracted Godard band.

Returning toFIG. 3, the extracted Godard band signals346,348may undergo a first-order Godard band calculation350, thereby resulting in signals352which are representative of four first-order correlation signals G1,xx[n, k], G1,xy[n, k], G1,yx[n, k], G1,yy[n, k]. The first-order correlation signals352are collectively referred to as the first-order Godard band components, and, at the kthFFT block, are expressed as
G1,xx[n,k]=G0,x[n,k]G0,x*[n+Ng,k]
G1,xy[n,k]=G0,x[n,k]G0,y*[n+Ng,k]
G1,yx[n,k]=G0,y[n,k]G0,x*[n+Ng,k]
G1,yy[n,k]=G0,y[n,k]G0,y*[n+Ng,k],  [7]
where n∈{0, 1, . . . , Ng−1}, and where the superscript * denotes complex conjugation. The first-order Godard band components are equivalent to a phase derivative of the extracted Godard band components at a first differential distance Δ1=Ng.

An expected envelope R1[n] for each one of the first-order correlation signals in Equation 7 may be expressed as

R1⁡[n]=R0⁡[n]⁢R0⁡[n+Ng]=18⁢(1-cos⁡(2⁢π⁢nNg)),[8]
where R0[n] is defined in Equation 6, and where n∈{0, 1, . . . , Ng−1}. Thus, the envelope of the first-order Godard band components represented by the signals352follow a raised cosine response with α=1.

FIG. 6illustrates an example plot of the expected envelope R1[n] of the first-order Godard band components.

The phase ϕ1,xx[n, k] of the first-order correlation signal G1,xx[n, k], which is also denoted by ∠G1,xx[n, k], may be expressed in radians as

ϕ1,xx⁡[n,⁢k]=θ1,xx+(β2⁢2⁢πL+β3⁡(2⁢πL)2)⁢ωn+β3⁢2⁢πL⁢ωn2,[9]
where n∈{0, 1, . . . , Ng−1}, where θ1,xxdenotes a frequency-independent phase amount that is a function α, β1, β2, β3, and where ωndenotes the discrete angular frequency in units of radians per cycle, which is expressed as

As a result of performing a first-order phase derivative on the extracted Godard band components, ϕ1,xx[n, k] is not dependent on the zeroth-order phase term β0. However, as is apparent from Equation 9, the phase ϕ1,xx[n, k] has a quadratic dependency on the frequency ωn. Additionally, the value of θ1,xxmay vary from block to block due to PMD and clock slip. Corresponding expressions to that of Equation 9 may be derived for each of the phases ϕ1,xy[n, k], ϕ1,yx[n, k], and ϕ1, [n, k] of the first-order correlation signals G1,xy[n, k], G1,yx[n, k], and G1,yy[n, k], respectively.

Returning toFIG. 3, the signals352representing the first-order Godard band components may undergo a second-order Godard band calculation354comprising the calculation of four second-order correlation signals356denoted by G2,xx[n, k], G2,xy[n, k], G2,yx[n, k], G2,yy[n, k]. The second-order correlation signals356are collectively referred to as the second-order Godard band components, and, at the kthFFT block, are expressed as
G2,xx[n,k]=G1,xx[n,k]G1,xx*[n+Δ2,k]
G2,xy[n,k]=G1,xy[n,k]G1,xy*[n+Δ2,k]
G2,yx[n,k]=G1,yx[n,k]G1,yx*[n+Δ2,k]
G2,yy[n,k]=G1,yy[n,k]G1,yy*[n+Δ2,k],  [11]
where n∈{0, 1, . . . , Ng−Δ2−1}, where Δ2is a positive integer, and where the superscript * denotes complex conjugation. The phase term of the second-order Godard band components are equivalent to a phase derivative of the first-order Godard band components at a second differential distance equal to Δ2.

An expected envelope R2[n] for each one of the second-order correlation signals in Equation 11 may be expressed as
R2[n]=R1[n]R1[n+Δ2],  [12]
where R1[n] is defined in Equation 8, and where n∈{0, 1, . . . , Ng−Δ2−1}.

FIG. 7illustrates an example plot of the expected envelope R2[n] of the second-order Godard band components obtained using three different values for the second differential distance, namely Δ2=1, Δ2=10, and Δ2=20.

The phase ϕ2,xx[n, k] of the second-order correlation signal G2,xx[n, k], which is also denoted by ∠G2,xx[n, k], may be expressed in radians as

A comparison of Equations 9 and 13 demonstrates that, unlike the phase ϕ1,xx[n, k] of the first-order correlation signal G1,xx[n, k], the phase ϕ2,xx[n, k] of the second-order correlation signal G2,xx[n, k] is independent of the first-order phase term β1. Accordingly, it may be shown that the phases ϕ2,xx[n, k], ϕ2,xy[n, k], ϕ2,yx[n, k], ϕ2,yy[n, k] of the respective second-order correlation signals G2,xx[n, k], G2,xy[n, k], G2,yx[n, k], G2,yy[n, k] are identical to one another, and are insensitive to both DGD and clock slip, as well as the temporal delay reflected by the phase terms β0and β1. It follows that the four second-order correlation signals may be accumulated as a running average over time to generate composite second-order Godard band components denoted by G2[n] which may be expressed as
G2[n]=Σk(G2,xx[n,k]+G2,xy[n,k]+G2,yx[n,k]+G2,yy[n,k]),  [14]
where Σkdenotes summation over a plurality of FFT blocks, and where n∈{0, 1, . . . , Ng−Δ2−1}. Averaging over multiple FFT blocks using the summation operation Σkmay mitigate noise in the composite second-order Godard band components G2[n]. According to some examples, the averaging may be performed over 500-1000 FFT blocks. InFIG. 3, this averaging is implemented by applying a summation operation358to the signals356, thereby resulting in a signal360which represents the composite second-order Godard band components G2[n]. According to some examples, G2[n] may be stored in firmware.

It is noted that the phase ϕ2,xx[n, k] expressed in Equation 13 (which, as discussed, is the same as the phases ϕ2,xy[n, k], ϕ2,yx[n, k], ϕ2,yy[n, k]) has a linear dependency on the angular frequency ωn. It is contemplated that this linear dependency may be exploited to obtain an estimate of the third-order phase term β3.

From Equations 11, 12, and 13, it may be shown that the composite second-order Godard band components G2[n] may be expressed as
G2[n]=R2[n]exp(j(γ2+γ3ωn)),  [15]
where ωndenotes the discrete angular frequency defined in Equation 10, where γ2denotes an amount of residual dispersion expressed as

γ2=-β2⁢4⁢π2⁢Δ2L⁢N-β3⁢8⁢π3⁢Δ2L⁢N⁢(1L-Δ2N),[16]
and where γ3denotes a dispersion slope expressed as

As illustrated inFIG. 3, a residual CD calculation362may be performed on the signal360representing the composite second-order Godard band components G2[n]. According to one example, the residual CD calculation362may comprise using the signal360to calculate third-order correlation signals G3[n], referred to as the third-order Godard band components, and expressed as
G3[n]=G2[n]G2*[n+Δ3],  [18]
where n∈{0, 1, . . . , Ng−Δ2−Δ3−1}, where Δ2and Δ3are positive integers, and where the superscript * denotes complex conjugation. The phase of the third-order Godard band components G3[n] is equivalent to a phase derivative of the composite second-order Godard band components G2[n] at a third differential distance equal to Δ3.

The phase ϕ3[n, k] of the third-order correlation signal G3[n], which is also denoted by ∠G3[n, k]=∠G3[n], may be expressed in radians as

It follows that an estimate of the third-order phase term β3is calculable from the phase of the average of the third-order Godard band components G3[n] according to the following expression:

β3=-L⁢N28⁢π3⁢Δ2⁢Δ3⁢∠⁢∑n⁢G3⁡[n],[20]
where Σndenotes a summation over all the frequency bins satisfying n∈{0, 1, . . . , Ng−Δ2−Δ3−1}. Averaging over the frequency bins using the summation operation Σnmay reduce the noise in the estimate of β3.

According to another example, an estimate of β3may be obtained by calculating the location of a peak of the inverse Fourier transform of the signal360. The size of the IFFT for obtaining an estimate of β3does not need to be same as the size of G2[n]. In particular, an M-point IFFT of the composite second-order Godard band G2[n] may be expressed as

It may be shown that the peak magnitude of g2[m] is achieved when all of the components inside the summation are phase correlated, meaning that the phase term inside the summation is close to zero for all values of n. This condition is expressed as

-β3⁢4⁢π2⁢Δ2L⁢N+mp⁢e⁢a⁢kM≈0,∀n[22]
where mpeakdenotes the value of m at which the magnitude of g2[m] is at its peak.

Thus, by rearranging Equation 22, it is possible to calculate an estimate of the third-order phase term β3as follows

β3≈LNmp⁢e⁢a⁢k4⁢π2⁢Δ2⁢M.[23]
In other words, an estimate of the residual CD may be calculated using the timing bin, mpeak, that corresponds to a peak magnitude of the inverse Fourier transform of the composite second-order Godard band components, g2[m].

Given the estimate of the third-order phase term β3, an estimate of the dispersion slope S in units of ps/nm2may be calculated according to the following expression:

S=β3⁢4⁢π2⁢c2λ4⁢1⁢e⁢3Fs3,[24]
where c=299792458 m/s denotes the speed of light in a vacuum, where λ denotes wavelength in nm, and where FSdenotes the signal sampling rate in Giga-samples per second (Gsps). Equation 24 represents a scaling of β3into easier-to-understand units of ps/nm2. Additional conversion equations involving β2and β3are described, for example, by Asvial and Paramitha in “Analysis of high order dispersion and nonlinear effects in fiber optic transmission with Non Linear Schrodinger Equation model,” 2015 International Conference on Quality in Research (QiR), 2015. The estimate of the third-order phase term β3may be back-substituted into Equation 13, yielding

ϕ^2,xx⁡[n,k]=ϕ2,xx⁡[n,k]+β3⁢8⁢π3⁢Δ2L⁢N⁢(1L-Δ2N)+β3⁢4⁢π2⁢Δ2L⁢N⁢ωn=-β2⁢4⁢π2⁢Δ2L⁢N[25]
from which an estimate of the second-order phase term β2may be calculated according to the expression

According to Roberts et al. in “Performance of Dual-Polarization QPSK for Optical Transport Systems”, inJournal of Lightwave Technology, Vol. 27, No. 16, pp. 3546-3559, Aug. 15, 2009, the amount of residual dispersion in units of ps/nm, herein denoted by D, may be calculated from the second-order phase term β2according to the expression

The estimates of the third-order phase term β3and the second-order phase term β2may be used to calculate updated first coefficients according to

Cn⁢e⁢w⁡[n]=Co⁢l⁢d⁡[n]⁢exp⁡(-j⁢β22⁢ωn2-j⁢β36⁢ωn3),[28]
where Cnew[n] denotes the first coefficients to be used by the C filter at a current time interval, where Cold[n] denotes the first coefficients used by the C filter at a preceding time interval, and where n∈{0, 1, N−1}. Referring toFIG. 3, the β2and β3estimates may be represented by a signal364which is used by a first coefficient calculation operation366to generate the first coefficients319to be applied by the C filter318at the current time interval (i.e., Cnew[n]).

In the event of significant amounts of higher-order dispersion, the same logic may be used to estimate higher-order phase terms (such as β4, β5, etc.) by calculating higher-order Godard bands.

Where the range of the detected angle is [−π, +π), Equation 26 indicates that the maximum magnitude of β2that may be calculated is

β2,max=L⁢N4⁢πΔ2.
It follows that the maximum amount of residual dispersion, Dmax, that may be detected and compensated for in units of ps/nm may be expressed as

An example application is considered involving 400 Gbps optical transmission with α=0.25 and FS=75 GHz. Under these circumstances, selecting Δ2=2 would result in a maximum detectable residual dispersion of Dmax=2773 ps/nm. Since this particular application is expected to have a residual dispersion D of less than 2400 ps/nm, the choice of Δ2=2 is sufficient.

For applications where the residual dispersion is relatively small, a larger value of Δ2may be used to obtain a more accurate estimate of β2. The optimal value of Δ2is one that maximizes the SNR of β2as expressed in Equation 26. Assuming that the residual dispersion D is not high enough to cause the phase of the composite second-order Godard band components G2[n] to roll over (i.e., because the angle is outside of the range [−π, +π)), the signal power of the phase is proportional to (Δ2)2, as shown in Equation 13 (where the amplitude of ϕ2,xx[n, k] is proportional to Δ2). According to Berscheid in “FPGA-Based DOCSIS Upstream Demodulation, Section 4.4.2, Phase noise model”, University of Saskatchewan, 2011, the noise power may be shown to be proportional to the NSR of G2[n], which is expressed as

NSR=(Ng-Δ2)⁢N0(∑n⁢R2⁡[n])2,[30]
where Σndenotes a summation over all the frequency bins satisfying n∈{0, 1, . . . , Ng−Δ2−1}, and where N0is a positive real number that denotes the noise power spectral density. This expression derives from the fact that the summation in Equation 26 increases the noise power by a factor of (Ng−Δ2), which is equal to the amount of elements in the summation, while the signal power is increased by a factor of (ΣnR2[n])2, since all the elements of G2[n] have the same phase angle.

It follows that an optimized value of Δ2may be determined by maximizing a cost function C(Δ2) expressed as

For the 400-Gbps application, it may be shown that the cost function C(Δ2) is maximized when Δ2=24. However, according to Equation 28, using this value for Δ2would result in a maximum detectable residual dispersion of Dmax=231 ps/nm, which may be insufficient for this application.

According to some examples, the C filter may initially be configured to use coefficients calculated from a first estimate β2,1that was obtained using a low value of Δ2. For example, the first estimate β2,1may be calculated using Equation 24 with Δ2=2, where G2[n] represents the composite second-order Godard band components averaged over, for example, an initial 500 FFT blocks. The C filter may subsequently be configured to use coefficients calculated from a second estimate β2,2that was obtained using a higher value of Δ2than was used to calculate the first estimate β2,1. For example, the second estimate β2,2may be calculated using Equation 24 with Δ2=24, where G2[n] represents the composite second-order Godard band components averaged over, for example, a subsequent 500 FFT blocks following the initial 500 FFT blocks. The first estimate β2,1was relatively coarse due to the low value of Δ2. However, the coefficients calculated from this first estimate β2,1and used by the C filter may have achieved a significant reduction in the residual dispersion. The reduced residual dispersion enables the use of the higher value of Δ2to obtain the second estimate β2,2, which is finer and more precise than the first estimate β2,1, thereby resulting in updated coefficients that enable the C filter to achieve a further reduction in the residual dispersion.

FIGS. 8 and 9illustrate example plots of the phase and magnitude of composite second-order Godard band components G2[n] associated with a two-step process for blind estimation of residual CD. In each plot, the left axis shows the magnitude |G2[n]|, which is represented by a solid line, while the right axis shows the phase ∠G2[n], in units of π radians, which is represented by a dotted line.FIG. 8illustrates the initial properties of G2[n] that result from using an initial value of Δ2=2 for the second differential distance when calculating the phase derivative of initial first-order Godard band components (i.e., step 1). As shown inFIG. 8, the initial phase ∠G2[n] fluctuates around −0.85π, indicating a residual dispersion D of approximately 2400 ps/nm.FIG. 9illustrates the subsequent properties of G2[n] that result from using a subsequent value of Δ2=24 for the second differential distance when calculating the phase derivative of subsequent first-order Godard band components (i.e., step 2), where the subsequent first-order Godard band components are the result of using updated filter coefficients calculated based on the initial properties of G2[n] obtained in step 1. As is shown inFIG. 9, the phase ∠G2[n] during step 2 is much closer to 0 than during step 1 and also has significantly less fluctuation due to the lower NSR achieved with the higher value of Δ2. The phase ∠G2[n] during step 2 is approximately −0.067π, indicating a residual dispersion D of approximately 17 ps/nm. This small remaining residual dispersion may be compensated by the next update of the C filter coefficients.

In addition to enabling blind estimation of residual CD, the composite second-order Godard band components G2[n] may also be used for blind estimation of CFO. As illustrated inFIG. 3, a CFO calculation368may be performed on the signal360representing the composite second-order Godard band components G2[n]. A resulting estimate of the CFO, represented by signal370inFIG. 3, may be used by the carrier recovery operation330. For example, as previously described, the carrier recovery operation330may apply a phase rotation to each symbol of the signals326,328in order to reverse the effect of a laser frequency difference between the transmitter and the receiver. The slope at which the phase rotation changes is proportional to the signal370generated by the CFO calculation368.

Various techniques may be used to calculate an estimate of the CFO based on the composite second-order Godard band components G2[n]. Firstly it is noted that, with the existence of a non-zero CFO, Equation 15 may be expressed as

G2⁡[n]=R2⁡[n-Δn]⁢exp⁡(j⁡(γ2+γ3⁡(ωn-2⁢πΔnN))),[32]
where Δndenotes the value of the CFO in units of frequency bins and where γ2and γ3are defined in Equations 16 and 17, respectively. Considering G2[n] as a pulse defined in the frequency domain, a non-zero CFO value causes this pulse to shift by a number of bins Δn.

FIG. 10illustrates an example plot showing a frequency shift of the second-order Godard band components caused by CFO. In this example, the Godard band components are obtained with a sufficiently large averaging over FFT blocks to mitigate the effect of noise. The solid line represents |G2[n]| when Δn=0 bins, while the dashed line represents |G2[n]| when Δn=10 bins. The frequency shift of Δn=10 bins is reflected by the shift of the shape to the right. According to some examples, the value of Δnmay be determined by locating the peak location of |G2[n]|, that is, the number of bins Δnthat corresponds to the peak magnitude of G2[n], which may be expressed as

Δn=argmaxn⁢{G2⁡[n]}-Ng-Δ22[33]
where argmax denotes an operation that locates the argument (in this case, n) that gives the maximum value from the target function (in this case, |G2[n]|).

Referring again toFIGS. 8 and 9, |G2[n]| in these examples has been obtained from a noisy signal with non-sufficient averaging. The shape of the Godard band components is not smooth but rather there are small ripples due to noise. If Equation 33 is used to estimate Δnbased on the values of |G2[n]| plotted inFIG. 8orFIG. 9, the ripples at the peak of the shape may result in estimation error. More averaging may help to eliminate the ripples at the cost of extending the time for estimation of Δn. In the applications where acquisition time is critical, the amount of averaging may be limited. Accordingly, it may be of interest to have a CFO estimation method that is resilient to such noise ripples.

According to some examples, to mitigate the estimation error caused by noise ripples, the value of Δnmay be determined by finding the maximum correlation between |G2[n]| and the ideal shape of R2[n−Δn], which may be expressed as

Δn=argmaxɛ⁢{∑n⁢G2⁡[n]⁢R2⁡[n-ɛ]},[34]
where ε denotes a real number satisfying

The correlation expressed in Equation 34 has an equivalency in the time domain. In particular, a shift in the frequency domain is equivalent to a linear phase ramp in the time domain, for example, according to the expression

c2⁡[m]=I⁢D⁢F⁢T⁢{G2⁡[n]}·IDFT*⁢{R2⁡[n]}=a2⁡[m]⁢exp⁡(j⁢2⁢π⁢Δn⁢mNg-Δ2),[36]
for m∈{0, 1, . . . , Ng−Δ2−1}, where a2[m] are positive real numbers denoting magnitude of the multiplication, and where the superscript * denotes complex conjugation. Using Equation 36, the value of Δnmay alternatively be expressed as

Δn=Ng-Δ22⁢π⁢∠⁢∑m⁢c2*⁡[m]⁢c2⁡[m+1],[37]
where Σmdenotes a summation over all the timing bins satisfying m∈{0, 1, . . . , Ng−Δ2−2}, and where the superscript * denotes complex conjugation.

Equation 37 may be further simplified by ignoring the magnitude a2[m] in Equation 36 and by taking into account the fact that R2[n] is a real signal centered at location

n=Ng-Δ22,
as provided in Equation 12 and illustrated inFIG. 7. It follows that IDFT{R2[n]} may be approximated as

IDFT⁢{R2⁡[n]}=exp⁡(j⁢2π⁢m⁡(Ng-Δ2)/2Ng-Δ2)[38]
for m∈{0, 1, . . . , Ng−Δ2−1}. Using the approximation in Equation 38, Equation 36 may be simplified to as follows

c2⁡[m]=∑n⁢⁢G2⁡[n]⁢exp⁡(j2⁢π⁢m⁡(n-Ng-Δ22)Ng-Δ2),[39]
for m∈{0, 1, . . . , Ng−Δ2−1}, and where Σndenotes a summation over all the frequency bins satisfying n∈{0, 1, . . . , Ng−Δ2−1}. This alternative expression for the conjugate multiplication c2[m] may be used in Equation 37 to calculate the value of Δn.

Since the expected envelope of |G2[n]| resembles a sinusoid with cycle duration of Ng−Δ2(illustrated, for example, inFIG. 7as a slowly changing signal in the frequency domain), its inverse Fourier transform is expected to be short in the time domain, and thus the summation in the Equation 37 may be further reduced to

Δn=Ng-Δ22⁢π⁢∠⁡(c2*⁡[-1]⁢c2⁡[0]+c2*⁡[0]⁢c2⁡[1]),[40]
where the superscript * denotes complex conjugation, and where c2[−1]=c2[Ng−Δ2−1] due to the periodicity of the inverse Fourier transform. In addition, Equation 39 may be implemented as 3-point IDFT with timing bins m∈{−1, 0, 1}.

Since the expected envelope of |G2[n]| is symmetrical, according to some examples the value of Δnmay be approximated by the balance point of the shape. That is, assuming a solid object having a weight of |G2[n]| at a distance n, then the center of gravity is defined as the division of the total weight distance moment by the total mass of the object, that is

∑n⁢G2⁡[n]⁢n∑n⁢G2⁡[n].
According toFIG. 7, the expected balance point of the shape with zero CFO is (Ng−Δ2)/2, and thus the value of Δnmay alternatively be expressed as

θ⁡[n]=2⁢πΔτ⁢n-(Ng-Δ2)/2Ng-Δ2[42]
where Δτis a positive real number that defines the periodicity of the rotation θ[n]. It follows that the amount of bin shift Δnmay be expressed as

The inventors have recognized that the accuracy of the estimate of Δnmay be sensitive to the value of Δτ. According to one example, good accuracy may be achieved by selecting Δτ=0.125.

The estimate of Δnin units of bin number may be converted to a CFO estimate in units of Hertz using the following Equation

CFO=κΔnN⁢FS[44]
where FSdenotes the sampling frequency in Hertz, where N denotes the size of the receiver FFT, and where κ is a positive real number that denotes a scaling factor reflecting the envelope shaping applied at the receiver C filter.

The envelope of the composite second-order Godard band components |G2[n]| represents the product of the signal envelope at the transmitter and the signal envelope at the receiver. According to some examples, the signal envelope at the transmitter is characterized by a root-raised cosine shape with a roll-off factor αTX, while the signal envelope at the receiver is characterized by a root-raised cosine shape with a roll-off factor αRX.

FIG. 11illustrates an example plot showing the impact of transmitter and receiver shaping on an expected envelope of the second-order Godard band components in the presence of a non-zero CFO. In this example, both the transmitter and the receiver use root-raised cosine shaping on G2[n] with the same roll-off factor, that is αRX=αTX. The peak location of |G2[n]| is halfway between the receiver envelope and the transmitter envelope. The distance between the receiver envelope and the transmitter envelope is created by CFO. Thus, in this example where the same root-raised cosine shaping is applied at the transmitter and the receiver, the scaling factor in Equation 44 may be selected as κ=2. According to another example, where there is no receiver shaping on G2[n], the shape of |G2[n]| is purely taken from the transmitter shaping, and the scaling factor in Equation 44 may be selected as κ=1.

In practice, where different shaping is applied at the transmitter and the receiver, that is αTX≠αRX, an optimum value for the scaling factor may be 1<κ<2. A mismatch in the shaping of the two envelopes may be addressed by forcing the receiver envelope to have a flat shape rather than a root-raised cosine shape. That is, by initializing the C filter to have a flat response (i.e., no shaping, such that the response has a uniform magnitude over frequency), |G2[n]| is expected to have almost the same shape as the transmitter shaping window. Under these conditions, it is possible to use a scaling factor of κ=1 in Equation 44, regardless of the transmitter shaping. In practice, a “flat” C filter generally does not have a perfectly flat passband, but instead has a small amount of roll-off toward the edges. According to some examples, this roll-off may be compensated for by the scaling factor κ.

In practice, when the CFO is large, one tail of |G2[n]| may land outside of the Ng−Δ2frequency bins. In such cases, the shape of |G2[n]| may not be symmetrical, resulting in bias on the estimate of the CFO. This bias may be avoided by removing values of |G2[n]| having magnitudes lower than a predefined threshold δ, that is, by pre-trimming the tails on both sides of |G2[n]| to maintain the symmetrical shape. Accordingly, truncated composite second-order Godard band components Ĝ2[n] may be expressed as

Selecting a higher value for the threshold δ may reduce the bias in the estimate of Δn, while selecting a lower value for the threshold δ may increase the SNR of the estimate of Δnby taking more samples of |G2[n]| into account. According to one example, the threshold δ may be defined by the following expression:

Taking into account this technique for eliminating bias in the estimation of the CFO, Equation 43 may be modified to the following:

FIG. 12illustrates an example method1200for blind estimation of residual CD and CFO in accordance with some examples of the technology disclosed herein. The method1200may be performed at a receiver device, such as the receiver device104or200. In general, the method1200may be implemented using circuitry configured to perform the various steps of the method1200. The circuitry may comprise various combinations of processors (including DSPs), computer-readable media storing computer-executable instructions or code, ASICs, and the like.

At1202, filtered signals are generated by applying filter coefficients to a digital representation of a received communications signal. The communications signal may comprise a degraded version of a communications signal generated by a transmitter device. For example, the communications signal may comprise a degraded version of the optical signal204transmitted by a transmitter device, where the optical signal204conveys a stream of symbols transmitted at an angular symbol frequency

ωs=2⁢πT,
where T denotes the symbol period. The digital representation of the communication signal may comprise, for example, the signals314,316described with respect toFIG. 3. The filter coefficients may comprise the coefficients319applied by the C filter318. The filtered signals may comprise the signals320,322. As described previously, the filtered signals may have a substantially raised cosine shape in the frequency domain which is characterized by a roll-off factor α, where a is a real number satisfying 0<α≤1. According to some examples, the substantially raised cosine shape may result from applying root-raised cosine shapes at both the transmitter and the receiver. According to other examples, the substantially raised cosine shape may result from applying a root-raised cosine shape at the transmitter, and applying a substantially flat shape at the receiver. These conditions may be used temporarily during receiver start-up, and once CD and/or CFO have been estimated according to the method1200, the receiver may be configured to apply a root-raised cosine shape similar to the one applied at the transmitter, thereby minimizing ISI during subsequent operation of the receiver.

At1204, Godard band components are extracted or selected from the filtered signals calculated at1202. For example, the Godard band extraction operation344may be applied to the signals320,322, thereby resulting in the signals346,348. The Godard band components may comprise those components defined over frequency bins that correspond to the range of angular frequencies

At1206, first-order Godard band components are calculated from a phase derivative of the Godard band components extracted at1204. For example, the first-order Godard band calculation350is applied to the signals346,348, thereby resulting in the signals352. The phase derivative performed at1206may use a first differential distance Δ1=Ng, where

Ng=(L-1)⁢NL,
and where L denotes the up-sampling factor of the signal.

At1208, second-order Godard band components are calculated from a phase derivative of the first-order Godard band components calculated at1206. For example, the second-order Godard band calculation354is applied to the signals352, thereby resulting in the signals356. The phase derivative performed at1208may use a second differential distance Δ2, where Δ2is a positive integer.

At1210, composite second-order Godard band components are calculated from an average over a plurality of time intervals of the second-order Godard band components generated at1208. For example, the summation operation358is applied to the signals356, thereby resulting in the signal360.

At1212, an estimate of residual CD is calculated using the composite second-order Godard band components calculated at1210. For example, the residual CD calculation362is performed on the signal360, thereby resulting in the signal364. As described previously, the residual CD calculation362may include calculating third-order Godard band components from a third phase derivative of the composite second-order Godard band components at a third differential distance, calculating an estimate of a third-order phase term using the third-order Godard band components, and using the estimate of the third-order phase term to calculate an estimate of the second-order phase term. Alternatively, the residual CD calculation362may include using the composite second-order Godard band to directly calculate an estimate of the third-order phase term, and using that estimate to calculate the estimate of the second-order phase term. For example, the estimate of residual CD may be calculated using a timing bin corresponding to a peak magnitude of an inverse Fourier transform of the composite second-order Godard band components.

At1214, updated filter coefficients are calculated using the residual CD estimate calculated at1212. For example, the first coefficient calculation operation366is applied to the signal364, thereby generating the signal319. The method1200may then return to step 1202, with the newly updated filter coefficients being applied to a newly received communications signal.

At1216, an estimate of CFO is calculated using the composite second-order Godard band components calculated at1210. For example, the CFO calculation368is performed on the signal360, thereby resulting in the signal370. As described previously with respect to Equations 33-44, the estimate of the CFO may be calculated using the magnitudes of the composite second-order Godard band components. According to one example, the CFO calculation368may include determining a frequency (represented, for example, by a number of frequency bins Δn) that corresponds to the peak value of |G2[n]|, as expressed in Equation 33, and using that frequency to estimate the CFO. According to another example, the CFO calculation368may include determining a frequency (represented, for example, by a number of frequency bins Δn) that corresponds to a center of gravity of |G2[n]|, as expressed in Equation 41, and using that frequency to estimate the CFO. According to another example, the CFO calculation368may include calculating a frequency shift (represented, for example, by a number of frequency bins Δn) of an expected shape of G2[n] that maximizes a correlation between G2[n] and the expected shape of G2[n], as expressed in Equation 34, and using that frequency shift to estimate the CFO. According to another example, the CFO calculation368may include calculating an autocorrelation of an inverse Fourier transform of G2[n], as expressed in Equation 37 or 40, and using the autocorrelation to estimate the CFO. According to some examples, CFO calculation368may be performed using only the composite second-order Godard band components having magnitudes equal to or greater than a predefined threshold, such as the threshold δ defined in Equation 46. In other words, the CFO calculation368may exclude the use of any composite second-order Godard band components having magnitudes lower than the predefined threshold (see Equation 45 in combination with Equation 47).

At1218, carrier recovery is performed using the estimate of the CFO calculated at1216. For example, the carrier recovery operation330is applied to the signals326,328based on the signal370generated by the CFO calculation368.

According to some examples, the calculations at1212,1214may be performed in parallel to performing the calculations at1216,1218. According to other examples, the calculations at1212,1214may be performed before or after performing the calculations at1216,1218. According to other examples, the calculations at1212,1214may be performed without performing the calculations at1216,1218. According to other examples, the calculations at1216,1218may be performed without performing the calculations at1212,1214.

According to some examples, the phase derivative performed at1208is initially performed using a first value of the differential distance Δ2, and then the method1200is repeated using a second higher value of the differential distance Δ2.

Although the calculations have been described as being performed in the frequency domain, they may alternatively be performed in the time domain. The algorithms for estimation of residual CD and CFO may be fully applied in ASIC hardware, or may be partly accomplished by a low-speed firmware processor and/or field-programmable gate array (FPGA) which assists the ASIC in the residual CD and CFO calculations.

According to the method1200, it may be possible to obtain blind estimates of residual CD and/or CFO, meaning estimates that are independent of clock phase offset, and optionally independent of DGD, SOP rotation, and PDL, where the blind estimates also do not require any prior knowledge of transmitter signal patterns. Thus, the method1200may be performed during the first stages of start-up of a receiver device, prior to the clock recovery circuit being activated.