Patent Publication Number: US-11387929-B1

Title: Systems and methods for carrier phase recovery

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 16/738,831, filed Jan. 9, 2020, which prior application claims the benefit of and priority to U.S. Provisional Patent Application Ser. No. 62/790,146, filed Jan. 9, 2019. The present application also claims the benefit of and priority to U.S. Provisional Patent Application Ser. No. 62/924,427, filed Oct. 22, 2019, and to U.S. Provisional Patent Application Ser. No. 62/934,664, filed Nov. 13, 2019. The subject matter of all of these applications are incorporated herein by reference in their entireties. 
    
    
     BACKGROUND 
     The field of the disclosure relates generally to communication networks, and more particularly, to access networks capable of digitally processing carrier signals for point-to-point (P2P) and point-to-multipoint (P2MP) communication systems. 
     Access networks, driven by ever-increasing residential data service growth rates and numbers of supported services types (e.g., business services, cellular connectivity, etc.), have been undergoing frequent technological and architectural changes. High-speed data and video service bandwidth requirements for the access paradigm are expected to grow to multi-gigabits-per-second (Gb/s) for residential offerings, and over 10-Gb/s for business markets in optical access networks of the near-future. At present, 10-Gb/s passive optical networks (PONs), such as XG-PON or IEEE 10G-EPON, are rapidly being deployed for high-bandwidth applications. 40-Gb/s PONs, based on time and wavelength division multiplexing (TWDM), have been standardized, and the IEEE 802.3ca Task Force is considering 100-Gb/s Ethernet PONs utilizing 25-Gb/s data rate per lane. However, PONs and access optical systems supporting greater than 50 Gb/s per channel have not been conventionally adopted because present direct detection optical schemes do not achieve sufficient power budgets due to their low receiver sensitivity and limited options for long-term upgrading. These direct detection challenges are particularly prevalent in the legacy fiber environment, where network operators desire the continued use of existing infrastructures to avoid costly fiber re-trenching. 
     Coherent optics technology is becoming common in the subsea, long-haul, and metro networks, but has not yet been applied to access networks due to the relatively high cost of the technology for such coherent implementations. The coherent optical technology approach is different from the direct detection approach, and enables superior receiver sensitivity that allows for an extended power budget. The high frequency selectivity of the coherent approach enables closely spaced dense or ultra-dense WDM, but without requiring narrow band optical filters. Moreover, the multi-dimensional recovered coherent signal provides additional benefits to compensate for the linear transmission impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD), and efficiently utilize the spectral resource and benefiting future network upgrades through the use of multi-level advanced modulation formats. 
     Commercial coherent optical technology was first introduced in long haul applications to overcome fiber impairments that required complex compensation techniques when using direct detection receivers. These first-generation coherent optical systems are based on a single-carrier polarization division multiplexed quadrature phase shift keying (PDM-QPSK) modulation format, and the achieved spectral efficiency (SE) is 2 bit/s/Hz greater than that of conventional 50-GHz optical grids. The system capacity according to the conventional approach is thus increased to approximately 10 Tb/s in the fiber C-band transmission window. 
     Coherent solutions have recently migrated from long haul, to metro and access networks, by leveraging the development of CMOS digital signal processing (DSP) techniques, reductions in design complexity, and decreases in the price opto-electronic components. Whereas coherent technology in long-haul optical systems utilize best-in-class discrete photonic and electronic components (e.g., the latest digital-to-analog/analog-to-digital converters (DAC/ADC) and DSP application specific integrated circuits (ASICs) based on the most recent CMOS processors), coherent pluggable modules for metro solutions have gone through C Form-factor pluggable (CFP) to CFP2 and future CFP4 via multi-source agreement (MSA) standardization for a smaller footprint, lower cost, and lower power dissipation. 
     This metro solution, however, is nevertheless considered in the field to be over-engineered, and also too expensive, large, and power-demanding to be efficiently and practically implemented in the access paradigm, which is a significantly different environment than the long haul and metro environments. The shorter transmission reach of the access network results in less distance-dependent signal degradation, and therefore requires less link equalization (e.g., fewer digital filter taps) and less processing in the DSP ASIC for impairment compensation. Such shorter-reach access applications additionally tolerate a slight reduction in optical signal-to-noise-ratio (OSNR) performance. Nevertheless, conventional DSP techniques and algorithms are unable to meet the size and cost requirements for access applications in developing and future access networks. Accordingly, it is desirable to develop DSP processing schemes for the access network paradigm that are able to reduce the DSP complexity and power consumption thereof. 
     SUMMARY 
     In an embodiment, a digital receiver is configured to process a polarization multiplexed carrier from a communication network. The polarization multiplexed carrier includes a first polarization and a second polarization. The receiver includes a first lane for transporting a first input signal of the first polarization, a second lane for transporting a second input signal of the second polarization, a dynamic phase noise estimation unit disposed within the first lane and configured to determine a phase noise estimate of the first input signal, a first carrier phase recovery portion configured to remove carrier phase noise from the first polarization based on a combination of the first input signal and a function of the determined phase noise estimate of the first input signal, and a second carrier phase recovery portion configured to remove carrier phase noise from the second polarization based on a combination of the second input signal and the function of the determined phase noise estimate of the first input signal. 
     In an embodiment, a method is provided for performing carrier phase recovery on a polarization multiplexed carrier in a digital signal processor of a coherent optics receiver. The method includes steps of dynamically estimating phase noise of a first polarization direction of the polarization multiplexed carrier to generate a single-polarization phase noise estimate from the first polarization direction, and performing phase recovery for a second polarization direction of the polarization multiplexed carrier based on the single-polarization phase noise estimate from the first polarization direction. 
    
    
     
       BRIEF DESCRIPTION 
       These and other features, aspects, and advantages of the present disclosure will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein: 
         FIG. 1  depicts a digital signal processing flow of a receiver processor. 
         FIG. 2  depicts an exemplary digital signal processing flow in an algorithmic level of the receiver processor depicted in  FIG. 1 . 
         FIG. 3  is a schematic illustration depicting a conventional carrier phase recovery process for a dual-polarization carrier by a receiver processor. 
         FIG. 4  is a schematic illustration depicting an exemplary carrier phase recovery process for a dual-polarization carrier by a receiver processor, in accordance with an embodiment. 
         FIG. 5  is a graphical illustration depicting an exemplary fixed phase rotation estimation subprocess for the carrier phase recovery process depicted in  FIG. 4 . 
         FIG. 6  is a graphical illustration depicting an alternative fixed phase rotation estimation subprocess for the carrier phase recovery process depicted in  FIG. 4 . 
         FIG. 7  is a schematic illustration depicting an alternative carrier phase recovery process, in accordance with an embodiment. 
         FIGS. 8A-B  are schematic illustrations depicting exemplary optical network architectures. 
         FIG. 9  is a schematic illustration of an exemplary test architecture for verifying experimental results implementing the receiver processing embodiments herein. 
         FIG. 10A-B  are graphical illustrations depicting experimental phase estimation result plots obtained according to the test architecture depicted in  FIG. 9 . 
         FIG. 11A-B  are graphical illustrations depicting comparative bit-error-ratio performance result plots obtained according to the test architecture depicted in  FIG. 9 . 
         FIG. 12  is a schematic illustration depicting a polarization-diversity coherent receiver. 
         FIG. 13  is a schematic illustration depicting an exemplary network communication system. 
         FIGS. 14A-C  are schematic illustrations depicting exemplary respective data architectures generated in accordance with the data unit generator depicted in  FIG. 13 . 
         FIG. 15  is a flow diagram depicting an exemplary state-of-polarization estimation technique for digital signal processing. 
         FIG. 16  is a flow diagram depicting an exemplary channel equalization process implementing the estimation technique depicted in  FIG. 15 . 
         FIG. 17  is a flow diagram depicting an alternative state-of-polarization estimation technique for digital signal processing. 
         FIGS. 18A-C  are schematic illustrations depicting alternative respective data architectures. 
         FIG. 19  is a flow diagram depicting an alternative channel equalization process implementing the estimation technique depicted in  FIG. 17 . 
         FIG. 20A  is a graphical illustration depicting a signal plot before implementation of state-of-polarization estimation and polarization recovery. 
         FIG. 20B  is a graphical illustration depicting a signal plot after implementation of state-of-polarization estimation and polarization recovery. 
         FIG. 21  is a graphical illustration depicting a comparative bit-error-ratio performance result plot obtained according to the techniques depicted in  FIGS. 15 and 17 . 
         FIG. 22A  is a schematic diagram depicting a burst-frame architecture for a conventional direct-detection passive optical network. 
         FIG. 22B  is a schematic diagram depicting an upstream recovery technique for the direct-detection burst-frame architecture depicted in  FIG. 22A . 
         FIG. 23A  is a schematic diagram depicting an exemplary burst-frame architecture for a coherent passive optical network. 
         FIG. 23B  is a schematic diagram depicting an exemplary upstream recovery technique for the coherent burst-frame architecture depicted in  FIG. 23A . 
         FIG. 24  depicts an exemplary preamble architecture for a coherent burst-mode passive optical network. 
         FIG. 25  depicts an exemplary digital signal processor for processing an upstream burst transmission implementing the preamble architecture depicted in  FIG. 24 . 
         FIG. 26  is a schematic diagram depicting an exemplary test system for upstream burst detection. 
         FIGS. 27A-D  are graphical illustrations depicting respective experimental result plots from the test system depicted in  FIG. 26 . 
         FIG. 28A  is a graphical illustration depicting a comparative plot depicting an estimated frequency offset against a target frequency offset. 
         FIG. 28B  is a graphical illustration depicting a comparative bit-error-ratio performance result plot for the subplots depicted in  FIG. 28A . 
         FIG. 29  is a graphical illustration depicting a residual frequency offset plot. 
         FIG. 30A  is a graphical illustration depicting a plot of signal mean square error before channel equalization. 
         FIG. 30B  is a graphical illustration depicting a comparative plot of signal mean square error after channel equalization. 
         FIG. 31  is a graphical illustration depicting a comparative bit-error-ratio performance result plot. 
         FIG. 32  is a graphical illustration depicting a bit-error-ratio performance plot as a function of residual chromatic dispersion. 
         FIG. 33  is a graphical illustration depicting a long-term bit-error-ratio performance result plot. 
     
    
    
     Unless otherwise indicated, the drawings provided herein are meant to illustrate features of embodiments of this disclosure. These features are believed to be applicable in a wide variety of systems including one or more embodiments of this disclosure. As such, the drawings are not meant to include all conventional features known by those of ordinary skill in the art to be required for the practice of the embodiments disclosed herein. 
     DETAILED DESCRIPTION 
     In the following specification and claims, reference will be made to a number of terms, which shall be defined to have the following meanings. 
     The singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. 
     “Optional” or “optionally” means that the subsequently described event or circumstance may or may not occur, and that the description includes instances where the event occurs and instances where it does not. 
     Approximating language, as used herein throughout the specification and claims, may be applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term or terms, such as “about,” “approximately,” and “substantially,” are not to be limited to the precise value specified. In at least some instances, the approximating language may correspond to the precision of an instrument for measuring the value. Here and throughout the specification and claims, range limitations may be combined and/or interchanged; such ranges are identified and include all the sub-ranges contained therein unless context or language indicates otherwise. 
     As used herein, the term “database” may refer to either a body of data, a relational database management system (RDBMS), or to both, and may include a collection of data including hierarchical databases, relational databases, flat file databases, object-relational databases, object oriented databases, and/or another structured collection of records or data that is stored in a computer system. 
     As used herein, the terms “processor” and “computer” and related terms, e.g., “processing device”, “computing device”, and “controller” are not limited to just those integrated circuits referred to in the art as a computer, but broadly refers to a microcontroller, a microcomputer, a programmable logic controller (PLC), an application specific integrated circuit (ASIC), and other programmable circuits, and these terms are used interchangeably herein. In the embodiments described herein, memory may include, but is not limited to, a computer-readable medium, such as a random access memory (RAM), and a computer-readable nonvolatile medium, such as flash memory. Alternatively, a floppy disk, a compact disc-read only memory (CD-ROM), a magneto-optical disk (MOD), and/or a digital versatile disc (DVD) may also be used. Also, in the embodiments described herein, additional input channels may be, but are not limited to, computer peripherals associated with an operator interface such as a mouse and a keyboard. Alternatively, other computer peripherals may also be used that may include, for example, but not be limited to, a scanner. Furthermore, in the exemplary embodiment, additional output channels may include, but not be limited to, an operator interface monitor. 
     Further, as used herein, the terms “software” and “firmware” are interchangeable, and include computer program storage in memory for execution by personal computers, workstations, clients, and servers. 
     As used herein, the term “non-transitory computer-readable media” is intended to be representative of any tangible computer-based device implemented in any method or technology for short-term and long-term storage of information, such as, computer-readable instructions, data structures, program modules and sub-modules, or other data in any device. Therefore, the methods described herein may be encoded as executable instructions embodied in a tangible, non-transitory, computer readable medium, including, without limitation, a storage device and a memory device. Such instructions, when executed by a processor, cause the processor to perform at least a portion of the methods described herein. Moreover, as used herein, the term “non-transitory computer-readable media” includes all tangible, computer-readable media, including, without limitation, non-transitory computer storage devices, including, without limitation, volatile and nonvolatile media, and removable and non-removable media such as a firmware, physical and virtual storage, CD-ROMs, DVDs, and any other digital source such as a network or the Internet, as well as yet to be developed digital means, with the sole exception being a transitory, propagating signal. 
     Furthermore, as used herein, the term “real-time” refers to at least one of the time of occurrence of the associated events, the time of measurement and collection of predetermined data, the time for a computing device (e.g., a processor) to process the data, and the time of a system response to the events and the environment. In the embodiments described herein, these activities and events occur substantially instantaneously. 
     As used herein, unless specified to the contrary, “modem termination system,” or “MTS” may refer to one or more of a cable modem termination system (CMTS), an optical network terminal (ONT), an optical line terminal (OLT), a network termination unit, a satellite termination unit, and/or other termination devices and systems. Similarly, “modem” may refer to one or more of a cable modem (CM), an optical network unit (ONU), a digital subscriber line (DSL) unit/modem, a satellite modem, etc. 
     As used herein, the term “transceiver,” unless specified otherwise, refers to a P2P coherent optics transceiver, having a coherent optics transmitting portion and a coherent optics receiving portion. In some instances, the transceiver may refer to a specific device under test (DUT) for several of the embodiments described herein. 
     As described herein, a “PON” generally refers to a passive optical network or system having components labeled according to known naming conventions of similar elements that are used in conventional PON systems. For example, an OLT may be implemented at an aggregation point, such as a headend/hub, and multiple ONUs may be disposed and operable at a plurality of end user, customer premises, or subscriber locations. Accordingly, an “uplink transmission” refers to an upstream transmission from an end user to a headend/hub, and a “downlink transmission” refers to a downstream transmission from a headend/hub to the end user, which may be presumed to be generally broadcasting continuously (unless in a power saving mode, or the like). 
     The embodiments described herein provide innovative access network architectures and processes that are useful for achieving simplified carrier phase recovery (CPR) for polarization multiplexed coherent optics in access network applications. In an exemplary embodiment, the present systems and methods leverage coherent optics technologies, and with respect to P2P or P2MP systems and communication links, to significantly improve the cable access network paradigm by reducing the cost, complexity, and power consumption from DSP on a received optical carrier. 
     In an embodiment, a CPR algorithm is implemented in three DSP steps or subprocesses for one or more single polarization signals: (1) a one-tap state-of-polarization (SoP) estimation/polarization demultiplexing step; (2) a training sequence (TS)-based frequency offset estimation (FOE)/compensation step; and (3) a digital filtering step (e.g., using two digital filters) for channel equalization. The output of the estimated carrier phase and noise from one polarization direction (e.g., X-polarization) may then be used for the signals from the other polarization direction (e.g., Y-polarization) and combined with an estimated fixed phase offset rotation between the two polarizations. 
     In another embodiment, the communication network includes a differential coded coherent system, such as polarization multiplexed differential quadrature phase shift keying (PM-DQPSK). In this embodiment, a fixed phase offset between the two polarizations is not required, thereby further reducing the total DSP complexity, which enables a significantly more hardware-efficient coherent optical system for the access network. 
     The following embodiments are described with respect to receivers operating at 100 and 200 Gb/s. However, the person of ordinary skill in the art will appreciate that such operating parameters are described by way of example, and not in a limiting sense. The principles herein are applicable to access networks, PONs, and coherent optics systems operating at different transmission speeds, and particularly as the demand for increased speed and bandwidth continues to grow. The following examples are also described with respect to exemplary fiber links of approximately 50 km. However, the person of ordinary skill in the art will further appreciate that the present techniques support links of up to 80 km, 120 km, or greater in some circumstances. 
       FIG. 1  depicts a DSP flow  100  of a receiver processor  102 . In an exemplary embodiment, receiver processor  102  is a digital coherent optical receiver and DSP flow  100  illustrates the DSP functionality for a polarization multiplexed signal with respect to a structural level  104  and an algorithmic level  106  of processor  102 . In an exemplary embodiment, the polarization multiplexed signal may be a dual-polarization (e.g., X/Y) in-phase/quadrature (I/Q) quadrature amplitude modulation (QAM, or PM-QAM) carrier signal. The structural and algorithmic functionality of the coherent optical receiver is described in further detail in co-pending U.S. patent application Ser. No. 16/370,873, filed Mar. 29, 2019, the subject matter of which is incorporated herein by reference. 
     Structural level  104  may, for example, include one or more of: a first block  108  for compensation of front-end imperfections; a second block  110  for channel impairment equalization and compensation of major channel transmission impairments; a third block  112  for timing and clock recovery; a fourth block  114  for carrier recovery; and a fifth block  116  for bit stream recovery. Algorithmic level  106  may, for example, include one or more of: a first module  118  for deskewing, normalization, and/or orthogonality correction; a second module  120  for chromatic dispersion (CD) estimation or compensation (e.g., static equalization); a third module  122  for symbol synchronization; a fourth module  124  for PMD compensation, residual CD compensation, and/or polarization demultiplexing (e.g., dynamic equalization); a fifth module  126  for estimation and/or compensation of carrier frequency offset; and a sixth module  128  for carrier phase estimation (CPE) and/or compensation. 
     In exemplary operation of DSP flow  100 , four digitized signals  130  (i.e., I and Q components for each X and Y polarization) are passed through first block  108  (i.e., in digital form, for example, after conversion by an ADC) to compensate front-end imperfections. Such front end imperfections may be compensated by one or more correction algorithms of first module  118 , which may include a deskew algorithm to correct the timing skew between the four channels resulting from the difference in both optical and electrical path lengths within the coherent receiver, normalization and orthogonality correction algorithms, and/or algorithms to compensate for differences between the respective output powers of the four channels (due to different responses of PINs and/or transimpedance amplifiers (TIAs) in the receiver), as well as quadrature imbalances resulting from a particular optical hybrid not exactly introducing a 90-degree phase shift. 
     In further operation of DSP flow  100 , major channel transmission impairments may be compensated through use of appropriate digital filters of second block  110 , which may, through second module  120 , utilize estimation and compensation algorithms to address impairments such as CD and PMD. Second module may further include algorithms for performing, based on the different time scales of the dynamics of the respective impairments, static equalization for CD compensation because of its independence of SoP and modulation format, as well as the impact on subsequent blocks of structural level  102  before the CD estimation may be needed to achieve accurate compensation. 
     At third block  112 , clock recovery for symbol synchronization may be processed within structural level  102  to track the timing information of incoming samples, for example, using third module  122 . In an embodiment, joint processing between third block  112  and fourth module  124  may be performed to achieve symbol synchronization within algorithmic level  104  after all channel impairments are equalized (e.g., as represented by respective arrows indicated in  FIG. 1 ). In at least one embodiment, a fast-adaptive equalization subprocess may be jointly performed for two polarizations within fourth module  124  through a butterfly structure and stochastic gradient algorithms, such as a constant modulus algorithm (CMA) and variants thereof. Fourth module  124  may further include one or more additional algorithms for further PMD compensation, residual CD compensation, and/or polarization demultiplexing (e.g., dynamic equalization). 
     At fourth block  114 , carrier recovery is performed in cooperation with fifth module  126 , which may include one or more algorithms to perform carrier frequency offset estimation or compensation. In an embodiment, fifth module  126  may further include algorithms configured to estimate, and then remove, the frequency offset between a source laser (not shown in  FIG. 1 ) and a local oscillator (LO), to prevent the constellation rotation at the intradyne frequency. Within sixth module  128 , algorithms may be configured such that the carrier phase noise may be estimated and removed from the modulated signal, which may further include algorithms for symbol estimation and hard or soft-decision forward error correction (FEC) for channel decoding. At fifth block  116 , the final bit streams may be recovered at both structural level  104  and algorithmic level  106 . 
     It may be noted that, for a particular digital coherent receiver, the ordering of blocks and modules for DSP flow  100  may, according to design choices at the receiver, differ from the order described above. For example, instead of, or in addition to, a feed-forward process, joint processing and feedback among different process blocks may be performed, including without limitation, clock recovery and polarization demultiplexing. 
     In some embodiments, a coherent receiver may include fewer, or additional, blocks and/or modules than those described herein. For example, an alternative algorithmic level architecture is described below with respect to  FIG. 2 . In other embodiments, similar functionality may be achieved through use of training sequences, data-aided, or blinded algorithms, as described further below with respect to  FIGS. 3-7 . 
     Coherent detection and DSP technologies have thus been key factors enabling the development of 100G coherent optical transmission systems. DSP technology has played in even more ubiquitous role, at both the transmitter and receiver, and the development of 200G coherent optical systems, and this trend is expected to continue in the development of further next-generation coherent optical systems. Although specific algorithms may be different for each block or module of the DSP, general functionality at the structural level (e.g., structural level  104 ) or functional abstractions (e.g., algorithmic level  106 ) are expected to be similar for relevant commercial products implementing such technology. 
       FIG. 2  depicts an exemplary DSP flow  200  in an algorithmic level  202  of receiver processor  102 ,  FIG. 1 . In an exemplary embodiment, algorithmic level  202  replaces algorithmic level  106 ,  FIG. 1 , within receiver processor  102 . In some embodiments, algorithmic level  202  may include one or more algorithms, modules, or subprocesses of algorithmic level  106  in a complementary fashion. 
     In the exemplary embodiment, algorithmic level  202  may, for example, include one or more of: a first module  204  for performing SoP estimation and polarization demultiplexing (e.g., 1-tap); a second module  206  for performing training sequence (TS)-based FOE and compensation; a third module  208  for performing dynamic channel equalization (e.g., two digital filters); and a fourth module  210  for performing carrier phase estimation (CPE) and compensation. 
     In exemplary operation of DSP flow  200 , first module  204  and second module  206  are all configured to functionally process all four of digitized signals  212  for the respective I/Q components of the X/Y polarizations, similar to the various respective modules of algorithmic level  106 . In the embodiment depicted in  FIG. 2  though, third module  208  may be configured to functionally process one component  212  from each polarization (e.g., YQ and XQ signals  212 , in this example). The operational functionality of first module  204 , second module  206 , and third module  208  is otherwise described in greater detail in co-pending U.S. application Ser. No. 16/412,104, filed May 15, 2019, the subject matter thereof which is incorporated by reference herein. 
     Although similar in functional operation, fourth module  210  particularly differs from sixth module  128 ,  FIG. 1 , in that whereas sixth module  128  is configured to perform carrier phase estimation and compensation on all four signals  130  (i.e., the I/Q components of both X/Y polarizations), fourth module  210  is configured such that carrier phase estimation/compensation need be performed on one of only the I/Q components of one of the two polarizations (e.g., the YQ signal  212 , in the example depicted in  FIG. 2 ). That is, DSP flow  200  represents a significantly simplified algorithmic DSP flow in the digital optical coherent receiver for the optical access network, in comparison with algorithmic level  106  of DSP flow  100 ,  FIG. 1 . Accordingly, the following DSP embodiments are described with particular focus on the innovative simplified DSP techniques of fourth module  210  that produces recovered bit streams  214  for both X and Y polarizations, but through performance of CPE on only one such polarization signal  212 . 
     According to the innovative embodiments described herein, the complexity of the DSP flow in the receiver processor is advantageously reduced such that the processor need not implement fixed CD compensation. Instead, as illustrated in the embodiment depicted in  FIG. 2 , the accumulated CD in the access network may be alternatively compensated within third module  208  (i.e., dynamic channel equalization). Moreover, the complexity of DSP flow  200  is still further reduced, in comparison with DSP flow  100  or conventional techniques, by the performance of adaptive polarization demultiplexing and PMD compensation with multiple taps in a single processing block/module, namely, first module  204 . That is, a single tap is employed for SoP tracking and polarization demultiplexing prior to channel equalization (e.g., an third module  208 ). By separating these two functional blocks/modules within DSP flow  200 , single polarization equalization may be achieved with two digital finite impulse response (FIR) filters, as opposed to conventional systems that implement a butterfly-based bank configuration with four finite impulse response (FIR) filters and crossing computation. 
     Thus, in comparison with conventional techniques, systems and methods according to the “simplified” configuration of DSP flow  200  are capable of reducing the DSP computational complexity by 50% for adaptive equalization functionality. In an exemplary embodiment of DSP flow  200 , TS-based frequency-offset estimation and compensation may be further achieved (e.g., through implementation of second module  206 ) using a training sequence having an optimized length with respect to the single-polarization signals, or with respect to the average of the dual-polarization signals. Accordingly, after frequency offset correction (e.g., second module  206 ) and channel equalization (e.g., third module  208 ) accomplished, carrier phase recovery (CPR) may then be achieved at, or by implementation of, fourth module  210 . 
       FIG. 3  is a schematic illustration depicting a conventional CPR process  300 , for a dual-polarization carrier input signal  302 , by a receiver processor (not separately shown). For each single polarization of input signal  302  (i.e.,  302 (X) and  302 (Y)), process  300  passes the original input signal  302  through a respective dynamic phase noise estimation unit  304 . An output of unit  304  is then combined, at a respective mixer  306 , with the original input signal  302  to achieve phase recovery and generate an output signal  308  for the respective single X- or Y-polarization of the dual-polarization carrier. In this example, unit  304  includes a plurality of taps  310  and a phase estimation module  312 . Phase estimation module  312  implements, for example, a Viterbi-Viterbi (VV) CPR algorithm or a blind-phase-search (BPS) algorithm to obtain the phase estimate, φ(t), for the respective single-polarization such that mixer  306  achieves phase recovery through a function e −jφ(t)  based on that phase estimate (i.e., e −jφ     x     (t)  for the X-polarization and e −jφ     y     (t)  for the Y-polarization). 
     In further operation of conventional CPR process  300 , dynamic phase noise estimation unit  304  includes L+1 taps  310  for L-tap symbols S. The symbols S are used for phase estimation of the center symbol S n+L/2 , based on, for example, a 4 th  power VV CPR or BPS algorithm. In the case where input signal  302  is a QPSK signal having four phase states, the received complex symbols of the QPSK signal are first raised to the 4 th  power to remove modulation, leaving only the phase noise present. Center symbol S n+L/2  is then added to N predecessors and successors to average the estimated phase. In conventional CPR process  300 , because the phase varies over a range of 2π, the estimated phase must be “unwrapped” to provide a continuous and unambiguous phase estimation. After the phase unwrapping, estimated phase error compensation is performed with respect to the received complex symbols. 
     Again, and as illustrated in the example depicted in  FIG. 3 , conventional CPR process  300  requires that the processing for phase estimation is performed independently for each of the X-polarization and the Y-polarization signals. These conventional techniques, therefore, require considerable processing resources for complex dual-polarization signals, which presents particular challenges to the implementation of DSP processing in the developing near-future access network paradigm. An innovative solution is challenges is described further below with respect to  FIG. 4 . 
       FIG. 4  is a schematic illustration depicting an exemplary CPR process  400  for performing carrier phase recovery and compensation on a dual-polarization carrier input signal  402  by a receiver processor (e.g., processor  102 ,  FIG. 1 ). CPR process  400  is architecturally similar, in some respects, to conventional CPR process  300 ,  FIG. 3 . CPR process  400  differs though, from conventional CPR process  300  in that CPR process  400  utilizes a single dynamic phase noise estimation unit  404  for both of the X-polarization and the Y-polarization portions (i.e., input signals  402 (X) and  402 (Y), respectively) of input signal  402 . In an exemplary embodiment, unit  404  may be similar in structure and functionality to dynamic phase noise estimation unit  304 ,  FIG. 3 , and similarly processes only a single-polarization input signal  402  (e.g., input signal  402 (X), in this example). Different from unit  304 , however, unit  404  outputs to both of two dynamic mixers  406 (X) and  406 (Y) for the two polarizations, respectively. Dynamic mixers  406 (X) and  406 (Y) may be otherwise similar to respective mixers  306 (X) and  306 (Y),  FIG. 3 . 
     CPR process further differs from conventional CPR process  300  in that CPR process  400  may include, for the other single-polarization lane (e.g., Y-polarization, in this example), a fixed phase rotation estimation unit  408  and a fixed mixer  410  configured to receive an output from unit  408 . More specifically, dynamic mixer  406 (X) combines single-polarization input signal  402 (X) with an output of single-polarization dynamic phase noise estimation unit  404  (e.g., φ x (t)-based, in this example). Thus, in the example depicted in  FIG. 4 , phase recovery for an X-polarization output signal  412 (X) is thereby achieved from dynamic mixer  406 (X) through the function e −jφ     x     (t) . 
     In contrast, dynamic mixer  406 (Y) combines single-polarization input signal  402 (Y) with the same φ x (t)-based output of the single dynamic phase noise estimation unit  404 . Since a phase recovery output  414  of mixer  406 (Y) is based on the function e −jφ     x     (t) , output  414  will exhibit rotation with respect to X-polarization output signal  412 (X). Accordingly, in the example depicted in  FIG. 4 , output  414  is passed through fixed phase rotation estimation unit  408 , and the output of unit  408  (e.g., φ y0 -based) is then combined with output  414  at fixed mixer  410  to achieve phase recovery for a Y-polarization output signal  412 (Y) through a function e −jφ     y0    relating to unit  408 . The person of ordinary skill in the art will understand, through comprehension of the present description and illustrations, that either polarization direction may be selected for processing through the single dynamic phase noise estimation unit. 
     Therefore, according to the innovative configuration of CPR process  400 , a simplified and hardware-efficient DSP flow (e.g., DSP flow  200 ,  FIG. 2 ) is accomplished. In an exemplary embodiment, CPR process  400  is accomplished in two stages: (1) phase noise estimation using only a single polarization direction; and (2) phase recovery for both polarization directions using the same single-polarization-based phase noise estimation. More particularly, phase noise estimation is performed in the first stage at only a single polarization direction, and this single-direction estimate for the first polarization signal is thus also shared with the second polarization signal to accomplish phase recovery for both polarizations in the second stage. In the exemplary embodiment, phase recovery of the second polarization signal may further utilize fixed phase rotation estimation and recovery through implementation of data-aided or blind estimation processes. 
     Thus, according to the present systems and methods, DSP processing for a dual-polarization carrier signal be effectively accomplished through performance of only one dynamic phase noise estimation processing stage for both polarizations of the dual-polarization signal. Dynamic phase noise estimation processing is time varying, with high computational complexity. The innovative configuration depicted in  FIG. 4  advantageously reduces this computational burden and complexity by approximately half. Whereas the particular example for CPR process  400  described herein does include an additional fixed phase rotation estimation that is not performed in conventional CPR process  300 ,  FIG. 3 , this fixed phase rotation estimation is considered, in comparison with dynamic phase noise estimation, to be a one-time process having a relatively negligible computation complexity. An exemplary technique for performing fixed phase rotation estimation is described below with respect to  FIG. 5 . 
       FIG. 5  is a graphical illustration depicting an exemplary fixed phase rotation estimation subprocess  500  for CPR process  400 ,  FIG. 4 . In an exemplary embodiment, subprocess  500  may be implemented at fixed phase rotation estimation unit  408  for the second polarization direction that is not subject to dynamic phase noise estimation (i.e., through unit  404 ). Thus, according to the exemplary embodiment depicted in  FIG. 4 , because X-polarization input signal  402 (X) and Y-polarization  402 (Y) originate from the same carrier, subprocess  500  is able to advantageously leverage the relationship between these two signal portions to utilize a single dynamic estimation from only one signal polarization to achieve CPR for both signal polarizations. That is, even though the phase of the respective individual signal polarizations may change substantially in relation to one another (e.g., from multiple DSP stages on the individual polarization lanes), fixed phase estimation subprocess  500  may utilize one or more training sequences  502  in the second polarization signal (i.e., signal  402 (Y), in this example) to achieve a TS-based estimate for phase recovery of the second polarization signal portion. 
     More particularly, and as illustrated in the example depicted in  FIG. 5 , a training sequence, Ts, is inserted into second input signal polarization  402 (Y) to coincide with first input signal polarization  402 (X). Thus, a given Ts  502  may be represented according to [Ts 1 , Ts 2 , . . . Ts N ], where N represents the training length in the Y-polarization direction. In this manner, a received signal, Rs, at the X-polarization is [Rs 1 , Rs 2 , . . . Rs N ], and may represent input signal  402 (X), which may have been subject to FOE and channel equalization (e.g., from second module  206  and third module  208 , respectively,  FIG. 2 ), and after dynamic phase noise estimation (e.g., by dynamic phase noise estimation unit  404 ,  FIG. 4 ). Using these values, subprocess  500  is able to determine the fixed phase rotation φ y0  according to:
 
φ y     0   =avg(angle( Rs/Ts ))  (Eq. 1)
 
       FIG. 6  is a graphical illustration depicting an alternative fixed phase rotation estimation subprocess  600  for CPR process  400 ,  FIG. 4 . In an exemplary embodiment, subprocess  600  may be implemented as an alternative to the implementation of TS-based fixed phase rotation estimation subprocess  500 ,  FIG. 5 , within or in conjunction with, an alternative embodiment of fixed phase rotation estimation unit  408 ′,  FIG. 4 . In the exemplary embodiment, subprocess  600  represents a blind phase estimation processing technique useful to determine an estimate of the fixed phase rotation φ y0 . That is, similar to the innovative technique described above with respect to  FIG. 5 , the fixed phase rotation φ y0  is still determined to achieve phase recovery for the second of the two single-polarization signals after dynamic phase noise estimation for only the first of the single-polarization input signals. 
     As depicted in the example illustrated in  FIG. 6 , a plurality of received symbols  602  (i.e., 1-N received symbols  602 ) are fed into and processed by an algorithm of a phase noise estimation unit  604 , which in turn generates the fixed phase rotation estimation φ y0 . That is, through this alternative subprocessing technique, the same fixed phase rotation estimation (i.e., φ y0 ) is obtained according to this blind phase estimation approach subprocess  600 , as is obtained through implementation of training sequence-based subprocess  500 . Both techniques fully support the simplified DSP flow approach described above with respect to  FIGS. 2 and 4 . 
     Although the blind phase estimation approach described with respect to  FIG. 6  is similar, in some respects, to conventional blind phase recovery methods (e.g., BPS, or even VV). However, according to the innovative and simplified approach of subprocess  600 , the sliding window that is necessary to the conventional approach, is no longer needed according to the blind phase estimation approach of subprocess  600 . Indeed, according to subprocess  600  only a one-time phase estimation of the N symbols (R 1 -R N ) is performed, and then the fixed phase rotation estimate φ y0  may be obtained by averaging these N symbols. 
       FIG. 7  is a schematic illustration depicting an alternative CPR process  700 . CPR process  700  is similar in many respects to CPR process  400 ,  FIG. 4 , and performs carrier phase recovery and compensation on the respective polarizations of a dual-polarization carrier input signal  702  (i.e.,  702 (X) and  702 (Y)) by a receiver processor (e.g., processor  102 ,  FIG. 1 ). In the exemplary embodiments depicted in  FIG. 7 , CPR process  700  represents the implementation of the innovative and simplified algorithmic embodiments described above, but in this example, applied to a dual-polarized signal according to a differential modulation format, such as a DQPSK signal. 
     Similar to CPR process  400 , CPR process  700  also implements only a single dynamic phase noise estimation unit  704 , which may be similar in structure and function to dynamic phase noise estimation unit  404 ,  FIG. 4 . Also similar to unit  404 , the phase noise estimation φ x (t) output from unit  704  is based only on a single polarization (i.e., the X-polarization signal, in this example) but shared with respective mixers  706  for both polarizations, that is, mixer  706 (X) for the X-polarization and mixer  706 (Y) for the Y-polarization. The phase recovery from both mixers  706 (X),  706 (Y) thus also utilizes the same function e −jφ     x     (t)  corresponding to the phase noise estimation value from unit  704 . 
     However, for the exemplary embodiment depicted in  FIG. 7 , because dual-polarization carrier input signal  702  is a DQPSK signal, to implement the simplified DSP flow techniques described above, CPR process  700  implements only the dynamic phase noise estimation stage of processing, and may avoid the need for fixed phase rotation recovery for the polarization signals. Accordingly, in this example, CPR process  700  may employ an individual differential decoding unit  708  at the output of each mixer  706 , respectively, to obtain the relevant output polarization signal  710 . Accordingly, the person of ordinary skill the art can see that the complexity of DSP processing may be even further substantially reduced in the case of input carriers utilizing differential modulation formats. 
       FIGS. 8A-B  are schematic illustrations depicting exemplary optical network architectures  800 ,  802 , respectively. More particularly, optical network architecture  800  illustrates an exemplary implementation of the present DSP embodiments within a P2P configuration, and optical network architecture  802  illustrates an exemplary implementation of the present DSP embodiments within a P2MP configuration. 
     In an embodiment, P2P optical network architecture  800  includes a first transceiver  804  in operable communication with a second transceiver  806  over an optical communication transport medium  808 . First transceiver  804  includes a first transmitter  810  and a first receiver  812 , and second transceiver  806  includes a second receiver  814  and a second transmitter  816 . In the exemplary embodiment, first receiver  812  includes a first DSP unit  818 , and/or second receiver  814  includes a second DSP unit  820 . In this exemplary P2P configuration, both of first and second receivers  812 ,  814  may be configured to operate as continuous mode coherent optical receivers, and either or both of first and second DSP units  818 ,  820  are configured to implement the reduced-complexity DSP flow techniques described above. 
     In contrast, P2MP optical network architecture  802  includes an upstream hub transceiver  822  (e.g., at a headend) in operable communication with a plurality (i.e., 1-k) of downstream transceivers  824  over an optical communication transport medium  826 . Hub transceiver  822  includes a downstream transmitter  828  and an upstream receiver  830 . In this exemplary P2MP configuration of architecture  802 , each of downstream transceivers  824  may therefore include a respective downstream receiver  832  and an upstream transmitter  834 . In an exemplary embodiment, one or more of downstream receivers  832  includes a respective downstream DSP unit  836 , and upstream receiver  830  includes an upstream DSP unit  838 . In an exemplary embodiment, some or all of upstream DSP unit  838  and downstream DSP units  836  are configured to implement the reduced-complexity DSP flow techniques described above. In an embodiment, downstream (DS) transmissions from downstream transmitter  828  to downstream receivers  832  may be sent as continuous mode coherent optical transmissions, and upstream (US) transmissions from respective upstream transmitters  834  to upstream receiver  830  may represent burst mode coherent optical transmissions. 
       FIG. 9  is a schematic illustration of an exemplary test architecture  900  for verifying experimental results implementing the receiver processing embodiments herein. More particularly, test architecture  900  was implemented in a real-world experimental setup to verify the proof of concept for the CPE and DSP flow systems and methods, as well as the several algorithmic blocks modules thereof, described above. 
     Test architecture  900  simulated a real-world operation of a coherent optics communication network, and included transmitter end  902  operably coupled to a receiver end  904  by an optical communication medium  906  (e.g., a 50-km single mode fiber (SMF), in this case). Transmitter end  902  included an arbitrary waveform generator (AWG)  908  (e.g., including an 80 GSa/s DAC), which generated of 25 GBaud polarization multiplexed QPSK and 16QAM signals  910 . Signals  910  were modulated using an I/Q modulator  912  coupled with a laser source  914  (e.g., a laser diode, 100 kHz), and then amplified by amplifier  916  (e.g., a booster erbium-doped fiber amplifier (EDFA) for transmission over the 50-km SMF of medium  906 . 
     At the receiver end  904 , the power of the transmitted signal was measured after a variable optical attenuator (VOA)  918  deployed along medium  906  at an input of receiver end for coherent detection. The received signal was then amplified by a pre-EDFA  920 , input to an integrated coherent receiver (ICR)  922  in operable communication with a local oscillator (LO) source  924 , sampled by a digital sampling oscillator (DSO)  926  (e.g., also 80 GSa/s), and processed by a Matlab-capable computer (PC)  928 . That is, in the actual experimental setup of test architecture  900 , the several reduced-complexity algorithms, described above, for the receiver were implemented to demodulate the transmitted signal through a Matlab offline process employed by PC  928 . In practical applications, such functionality may be performed within the coherent receiver itself, or by a DSP unit thereof. Results obtained from the experimental setup of test architecture  900  are described further below with respect to  FIGS. 10A-11B . 
       FIG. 10A-B  are graphical illustrations depicting experimental phase estimation measurement plots  1000 ,  1002 , respectively, obtained according to test architecture  900 ,  FIG. 9 . More particularly, plots  1000 ,  1002  illustrate phase estimation results for both polarizations of a multi-symbol dual-polarization signal, as well as the comparative differences between the conventional approach (e.g.,  FIG. 3 ) and the reduced-complexity/simplified CPR systems and methods described herein (e.g.,  FIGS. 2, 4-7 ). 
     For example, plot  1000  illustrates the estimated phase-versus-symbol results according to the conventional technique that requires independent estimation of dynamic phase noise for each of the X- and Y-polarizations individually. As shown in plot  1000 , an X-polarization phase subplot  1004  has the same phase evolution, but with a fixed phase offset, as a Y-polarization phase subplot  1006 . That is, since the independent phase noise from fiber nonlinearity (e.g., from medium  906 ,  FIG. 9 ) is considered to be relatively rather small at the transmission distances associated with the access paradigm, the respective phase noise in the two polarizations exhibits effectively the same behavior, except the fixed phase rotation. 
     In contrast, plot  1002  illustrates the estimated phase-versus-symbol results according to the CPR processing techniques described herein for the simplified DSP flow of a receiver processor. More particularly, a first subplot  1008  (solid line) illustrates the residual phase for one polarization using a dynamic phase estimation result, and a second subplot  1010  (dotted line) illustrates the results obtained using fixed phase rotation for the other polarization. As can be seen from the graphical illustration depicted in  FIG. 10B , first and second subplots  1008 ,  1010  substantially align with one another, thereby demonstrating the particular effectiveness of embodiments according to the present systems and methods. 
       FIG. 11A-B  are graphical illustrations depicting comparative BER performance result plots  1100 ,  1102 , respectively, obtained according to test architecture  900 ,  FIG. 9 . More particularly, plot  1100 ,  FIG. 11A  illustrates a comparative BER-versus-symbol length overlay of a first subplot  1104  utilizing a training sequence-based fixed phase rotation estimation (e.g.,  FIG. 5 ) against a second subplot  1106  utilizing a BPS-based fixed phase rotation estimation (e.g.,  FIG. 6 ). As can be seen from the graphical illustration depicted in  FIG. 11A , first and second subplots  1104 ,  1106  substantially align with one another. That is, BER performance is similar using either of the TS or blind estimation algorithms (in the condition of fixed receiver power at −38.3 dBm, for the experimental results of this example). As can also be seen from plot  1100 , a converged result  1108  for fixed phase rotation estimation is obtained for a training sequence or average window size of 64 symbols. 
     In contrast, plot  1102  illustrates BER-versus-received optical power comparative overlays  1110 ,  1112  for a 100G QPSK signal and a 200G 16QAM signal, respectively. More particularly, comparative overlay  1110  superimposes a first subplot  1114  depicting the BER performance of the QPSK signal according to conventional CPR techniques (i.e., where both polarizations are independently subject to dynamic phase noise estimation) with a second subplot  1116  depicting the BER performance of the same QPSK signal according to the simplified CPR techniques described herein. Similarly, comparative overlay  1112  superimposes a third subplot  1118  depicting the BER performance of the 16QAM signal according to the conventional CPR techniques with a fourth subplot  1120  depicting the BER performance of the same 16QAM signal according to the present simplified CPR techniques. As can be seen from the optical power sensitivity comparisons of plot  1102 , the innovative reduced-complexity DSP flow techniques of the present embodiments may be effectively implemented for different modulation formats with no significant or observable performance degradation therefrom. 
     The systems and methods described herein are therefore of particular advantageous use for the access network paradigm, for example, in the cable environment or other telecommunication applications, and may be implemented with respect to 4G, 5G, and 6G networks and related applications, as well as fronthaul, backhaul, and midhaul deployments, and also for both short- and long-haul architectures. 
     Exemplary embodiments of DSP systems and methods for digital and/or optical communication networks are described above in detail. The systems and methods of this disclosure though, are not limited to only the specific embodiments described herein, but rather, the components and/or steps of their implementation may be utilized independently and separately from other components and/or steps described herein. 
     Data-Aided SoP Estimation and Channel Equalization for Coherent Access Networks 
     As described above mobile Internet, 5G technology, cloud networking, and video streaming services are presently driving the growth of bandwidth requirements in optical access networks. As a P2MP system, PON technologies have been one of the dominant architectures to meet such high capacity demand for the end users. A high-speed PON, based on a single wavelength having a time-division multiplexing (TDM) mechanism, has been an attractive solution in the field due to its capability of reducing the number of required optical components and associated costs, while also saving wavelength resources. However, the limited sensitivity of such systems has become a critical challenge to support high-speed PONs with high power budgets using direct detection technologies. One such direct detection PON, for example, has a PR 30 power budget, transmits at greater than 50 Gbps per wavelength, and at a distance greater than 20 km. 
     Coherent detection offers a solution that both enables high-speed data transmission with advanced modulation formats, and also enhances the link power budget due to the increased sensitivity of the coherent receivers. However, implementation of digital coherent technologies into the optical access network paradigm creates new challenges arising from the differences between the access network and present long haul coherent technologies. 
     A first challenge arises from the fact that coherent detection in long haul transmissions requires powerful DSP at the receiver-side of the network to compensate for the channel linear and nonlinear distortions. In the access network (e.g., PON, P2P Ethernet), however, the transmission distance is generally limited, that is, over considerably shorter distances than in the long haul paradigm. In the access network, many distortions such as CD, PMD, and fiber nonlinearity, may be relatively small, and may often be ignored with little penalty. The long haul coherent detection techniques are considered too costly, in terms of hardware and power budget, to simply drop into the access network as is. Accordingly, it is necessary to fundamentally redesign the computation complexity of DSP functionality in the access network to reduce both the cost and power consumption in access network applications. 
     A second challenge arises with respect to upstream burst-mode digital coherent detection at the coherent digital receiver. In the long haul transmission paradigm, coherent detection operates in continuous mode, which may, for signal processing, tolerate convergence over significantly longer time durations, or larger latencies. One of the critical DSP functions at the coherent digital receiver though, is polarization recovery. The embodiments described above estimate the polarization and apply channel equalization to compensate for polarization dependent effects using such techniques as (i) blind estimation algorithms (e.g., 2×2 multi-input/multi-output (MIMO)-based adaptive equalization), (ii) CMA, and/or other known techniques, such as a multi-modulus algorithm (MMA). However, since these algorithms are based on the error signal feedback to update the filter coefficients, each such algorithm requires considerable convergence time. Furthermore, the blind algorithms are prone to sub-optimum convergence and instability, including the possibility of wrong convergence. Therefore, such techniques are not suitable for burst-mode coherent detection in the access network paradigm, particularly in the case of short burst frames. 
     Accordingly, the present embodiments offer an innovated technique for providing a data-aided method for performing both state-of-polarization (SoP) estimation, as well as, simplified channel equalization. In some embodiments, the present systems and methods are based on a specially designed data unit, in combination with a redesigned corresponding DSP, which separates the polarizations directly in a feedforward manner. 
     The unique data unit of the present embodiment is uniquely configured to generate special frame structuring for burst-mode signal frames. In some embodiments, by basing the proposed innovation on the feedforward estimation, the convergence time may be greatly reduced, which is a particular advantage in a burst-mode coherent communication system. 
     The present systems and methods achieve still further advantages over conventional techniques through the innovative implementation of data-aided features. That is, although the estimation techniques of the present embodiments may be data-aided, the estimation does not depend on the specific bit information carried by the data itself. Instead, according to the present systems and methods, the estimation may be advantageously based on the relationship between two detected polarization-diversity signals. Accordingly, a data unit that is specially designed according to these principles is not limited to SoP estimation only; such a data unit apparatus is further useful for carrying net bit information, as well as other DSP functions beyond estimation. 
     The following embodiments are therefore of particular use with channel equalization algorithms, while achieving equalization results with significantly reduced complexity and shorter convergence time. In some embodiments, the present techniques may reduce convergence time implementing a conventional 2×2 channel equalization structure, for example, by initializing the filter taps for the four adaptive equalizers of the 2×2 structure, or by pre-separating the respective polarizations before equalization. An exemplary embodiment of this principle, implemented with respect to a 2×2 channel equalization structure, is described further below with respect to  FIG. 15 . 
     In an alternative embodiment, a simplified channel equalization structure may utilize only two adaptive equalizers. In this example, after SoP estimation, the two adaptive equalizers may be independently utilized for channel equalization for each polarization of a polarization multiplexed signal, thereby reducing the computation complexity by approximately half, in comparison with non-data-aided techniques. For a single polarization signal, this simplified structure may be even further simplified, since only one adaptive equalizer would be needed according to this technique. An exemplary embodiment of this principle, implemented with respect to a simplified channel equalization structure, is described further below with respect to  FIG. 17 . 
       FIG. 12  is a schematic illustration depicting a polarization-diversity coherent receiver  1200 . In the exemplary embodiment depicted in  FIG. 12 , coherent receiver  1200  is implemented to demonstrate an operational principle of detecting two polarizations with cross coupling. More particularly, in the exemplary embodiment, coherent receiver  1200  includes a first polarization beam splitter (PBS)  1202  configured to receive an input polarization multiplexed signal  1204  (e.g., from an optical transport medium, not shown in  FIG. 12 ), and a second PBS  1206  configured to receive a local oscillator (LO) signal from an LO source  1208 . In some embodiments, second PBS  1206  may be a conventional splitter. 
     Coherent receiver further includes a first 90 degree optical hybrid  1210  and a second 90 degree optical hybrid  1212 . In this example, first 90 degree optical hybrid  1210  is configured to receive as inputs an X-polarization signal component from first PBS  1202  and the LO signal from second PBS  1204 . Similarly, second 90 degree optical hybrid  1212  is configured to receive as inputs a Y-polarization signal component from first PBS  1202  and the LO signal from second PBS  1204 . Each 90 degree optical hybrid  1210 ,  1212  is further configured to output separate I and Q components  1214  for its respective polarization signal component (i.e., XI and XQ, or YI and YQ, in this example). These components are described for purposes of illustration, and are not intended to be limiting. The person of ordinary skill in the art will understand, for example, that coherent receiver  1200  may include additional components  1216 , such as photodetectors (PDs), amplifiers or transimpedance amplifiers (TIAs), ADCs, and/or additional components conventionally utilized in coherent optical receivers, without departing from the scope herein. 
     As illustrated in  FIG. 12 , polarization-diversity coherent receiver  1200  operates to detect signal  1204  on two polarizations. However, when signal  1204  is received over fiber transmission, the respective polarizations of signal  1204  may no longer be aligned to LO  1208 . That is, along fiber transmission, the signal polarization may become randomly rotated due to birefringence from the fiber. Other polarization effects, such as PMD, may also affect the phase difference between the two polarizations, and the post-transmission SoP of signal  1204  may drift with time as a result of environmental conditions on the installed fibers/cables. Accordingly, in the exemplary embodiment, polarization-diversity coherent detection is achieved by separately detecting the two selected orthogonal polarizations (i.e., X/Y) of the received input signal  1204 , thereby enabling coherent receiver  1200  to implement SoP estimation and recovery in the digital domain. The following embodiments are described with respect to particular specially-designed data sequences and/or frame structures for dual-polarization signals, that is, polarization division multiplexed signals. These examples are provided by way of illustration, and not in a limiting sense. The principles described herein will be understood to be applicable to other types of polarized multiplexed signals and multiplexed signals having multiple signal subcomponents. 
       FIG. 13  is a schematic illustration depicting an exemplary network communication system  1300 . In an exemplary embodiment, system  1300  includes a transmitter-side  1302  and a receiver-side  1304 , and transmitter-side  1302  includes a data unit generator  1306  configured for frame structuring and design of data units communicated to a corresponding digital signal processor (DSP)  1308  at receiver-side  1304 . In the exemplary embodiment, data unit generator  1306  is configured to generate a data unit having a specially-designed frame structure for each of two polarizations, and then modulate the generated data unit onto the optical carrier corresponding to each polarization for transmission, for example, as a dual-polarization multiplexed signal (e.g., signal  1204 ,  FIG. 12 ) to receiver-side  1304  over an optical fiber medium  1310 . 
     At receiver-side  1304 , DSP  1308  is configured to have knowledge of the frame structure(s) of the data unit(s) generated by data unit generator  1306 , and applies DSP functions corresponding to the known data units after the respective signal components are coherently detected (e.g., at ICR  922 ,  FIG. 9 , optical hybrids  1210 / 1212 ,  FIG. 12 ). In the exemplary embodiment, DSP  1308  is logically disposed after conversion by an ADC (not shown in  FIG. 13 ), and operates in the digital domain to estimate and recover SoP with channel equalization, in coordination with data unit generator  1306  and/or knowledge of the special frame structures of the data units generated thereby. 
     In the exemplary embodiment, SoP estimation is therefore based on the data unit frame structures generated by specially-designed data unit generator  1306 . In this respect, the exemplary embodiment depicted in  FIG. 13  may be considered “data-aided,” due to the insertion of the data units on to the respective polarization signal frames. Nevertheless, because the corresponding SoP estimation performed by DSP  1308  need not be based on the actual bit information carried by the data of the respective signal frames, and is instead based on the relation between the two polarization-diversity detected signals, the generated data units may also be utilized for information carrying, that is, in addition to their implementation for SoP estimation and recovery. In some embodiments, may alternatively or additionally be used to coordinate with other corresponding DSP functions at DSP  1308 , such as those described above. Exemplary data unit frame structures are described further below with respect to  FIGS. 14A-C . 
       FIGS. 14A-C  are schematic illustrations depicting exemplary respective data architectures  1400 A-C generated in accordance with data unit  1306 ,  FIG. 13 . In the exemplary embodiment, each respective data architecture  1400  includes at least one specially-designed data unit  1402  placed with respect to a dual-polarization signal frame  1404  (e.g., the data payload) in the time domain. In the example depicted in  FIGS. 14A-C , dual-polarization signal frame  1404  is illustrated with respect to orthogonal X- and Y-polarizations, with data carried by their respective optical signals represented as Data X and Data Y (or Data Xi and Data Yi, in the case of multiple signal frames within the same data architecture  1400 ). Accordingly, data unit  1402  correspondingly structured to include an X-data component and a Y-data component represented herein as Ux and Uy (or Uxi and Uyi). 
     More particularly,  FIG. 14A  illustrates an implementation example of data unit  1402 A including N-symbol length data only on the X-data component for the X-polarization, and zeros on the Y-data component for the Y-polarization; that is, the frame structure of data unit  1402 A is configured such that Ux includes N-symbol length data in the time slot of data unit  1402 A, and Uy includes zeros (no data) within that same time slot. Similarly,  FIG. 14B  illustrates the counter-implementation example of data unit  1402 B including zeros on the X-data component Ux, and N-symbol length data on only the Y-data component Uy within the same time slot. 
       FIG. 14C , on the other hand, illustrates an implementation example of data unit  1402 C having a hybrid frame structure spanning at least two time slots within a single data unit. That is, data unit  1402 C includes N-symbol length data on both polarizations, but in different time slots within a single data unit. More particularly, within the first time slot of data unit  1402 C, Ux includes N-symbol length data, and Uy includes zeros, whereas in the second time slot of data unit  1402 C, Ux includes zeros, and Uy includes N-symbol length data. It will be understood by persons of ordinary skill in the art that the disposition of data within the two time slots of data unit  1402 C may be reversed, as long as one polarization of data unit  1402 C is structured no with zeros while the other polarization is structured to include data within a particular time slot. 
     The operating principles of data architectures  1400 A-C may otherwise be considered similar to one another with respect to SoP estimation and recovery. However, the hybrid implementation example illustrated in  FIG. 14C  may be considered to provide more balanced data loading between the X- and Y-polarizations. In an exemplary embodiment, implementation of data architecture  1400 C may further provide a more robust performance due to the enablement of corresponding DSP calculations on both of the two polarizations. An analysis of such DSP calculations and related processing is described further below. 
     In general, for an access network having relatively limited transmission distance, the polarization dependent loss and fiber nonreality may be ignored. Based on this principle, the Jones matrix of the fiber channel after signal transmission can be expressed, as a unitary matrix, according to: 
     
       
         
           
             
               
                 
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                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, θ represents the overall polarization rotation effect, whereas ϕ 1  and ϕ 2  represent the phase caused by PMD after fiber transmission. To solve this equation with three variables, the problem to be simplified by expanding Eq. 2 into: 
     
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ϕ 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       j 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             1 
                                           
                                           + 
                                           
                                             ϕ 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             ϕ 
                                             1 
                                           
                                           + 
                                           
                                             ϕ 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                           
                             
                               [ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ϕ 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ϕ 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ] 
                             
                             = 
                             
                               
                                 
                                   [ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               γ 
                                             
                                           
                                         
                                       
                                       
                                         
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             - 
                                             sin 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             e 
                                             
                                               
                                                 - 
                                                 j 
                                               
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               γ 
                                             
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 [ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           e 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ϕ 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               = 
                               
                                 
                                   J 
                                   1 
                                 
                                 ⁢ 
                                 
                                   J 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, γ=ϕ 1 +ϕ 2 , and it may thus be seen that J now has two parts, J 1  and J 2 , with the first part J 1  having only two variables (θ and δ), which may be solved more easily considering that the second part J 2  has no contribution to the polarization crosstalk (i.e., power transfer between the X- and Y-polarizations). Accordingly, this second part J 2  only adds a phase difference between the X- and Y-polarizations, which may be solved in the phase recovery process. Thus, only the first part of the equation requires a solution. 
     It is known that, for a unitary matrix, its inverse H is the conjugate transpose of that unitary matrix. Accordingly, the equation need be solved only to obtain: 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 γ 
                               
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 γ 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     The systems and methods of the present embodiments thus provide an advantageously simplified technique to solve this equation. For example, assuming that the received signal E R  after polarization diversity detection may be represented as: 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       R 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               E 
                               x 
                             
                           
                         
                         
                           
                             
                               E 
                               y 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     then the recovery signal E T  may be represented according to: 
     
       
         
           
             
               
                 
                   
                     E 
                     T 
                   
                   = 
                   
                     
                       H 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 E 
                                 x 
                               
                             
                           
                           
                             
                               
                                 E 
                                 y 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     γ 
                                   
                                 
                                 ⁢ 
                                 
                                   E 
                                   x 
                                 
                               
                               - 
                               
                                 sin 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   E 
                                   y 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   E 
                                   x 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     γ 
                                   
                                 
                                 ⁢ 
                                 
                                   E 
                                   y 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, E x  and E y  represent the revised signals on the two respective polarizations. As described above, the second part J 2  has no contribution on the power transfer between the X- and Y-polarizations. Accordingly, the innovative data unit frame structure of the present embodiments enables a simplified solution to the equation, due to the unique property of the present data unit that one polarization is null with zeros. For example, in the case where the Y-polarization of the transmitted data unit is null with zeros (e.g., data unit  1402 A), the following is true:
 
sin θ E   x +cos θ e   jγ   E   y =0  (Eq. 7)
 
     Under this principle, the equation may then be solved according to: 
     
       
         
           
             
               
                 
                   
                     
                       sin 
                       ⁢ 
                       θ 
                     
                     = 
                     
                       
                         
                           
                              
                             
                               
                                 E 
                                 y 
                               
                               / 
                               
                                 E 
                                 x 
                               
                             
                              
                           
                           2 
                         
                         
                           1 
                           + 
                           
                             
                                
                               
                                 
                                   E 
                                   y 
                                 
                                 / 
                                 
                                   E 
                                   x 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           + 
                           
                             
                                
                               
                                 
                                   E 
                                   y 
                                 
                                 / 
                                 
                                   E 
                                   x 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     γ 
                     = 
                     
                       angle 
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             
                               E 
                               x 
                             
                             
                               E 
                               y 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     Accordingly, the inverse matrix H for SoP estimation, polarization recovery, and demultiplexing may be easily obtained. A similar algorithmic process may be implemented in the case where the X-polarization of the transmitted data unit is null with zeros (e.g., data unit  1402 B). That is, in the case where the X-polarization of the transmitted data unit is null with zeros, the following is true: 
     
       
         
           
             
               
                 
                   
                     
                       sin 
                       ⁢ 
                       θ 
                     
                     = 
                     
                       
                         
                           
                              
                             
                               
                                 E 
                                 x 
                               
                               / 
                               
                                 E 
                                 y 
                               
                             
                              
                           
                           2 
                         
                         
                           1 
                           + 
                           
                             
                                
                               
                                 
                                   E 
                                   x 
                                 
                                 / 
                                 
                                   E 
                                   y 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           + 
                           
                             
                                
                               
                                 
                                   E 
                                   x 
                                 
                                 / 
                                 
                                   E 
                                   y 
                                 
                               
                                
                             
                             2 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     γ 
                     = 
                     
                       - 
                       
                         angle 
                         ⁡ 
                         
                           ( 
                           
                             
                               E 
                               y 
                             
                             
                               E 
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, the inverse matrix H may be solved through implementation of any data units  1402 A-C. Namely, as long as one of the respective data components of the polarizations is null with zero, the equation may be solved according to either Eq. 8 or Eq. 9, depending on the configuration of the particular data unit  1402  that is used. For example, in the case of hybrid data unit  1402 C, the equation may be solved using both Eq. 8 and Eq. 9, but in different respective time slots. 
     In an exemplary embodiment, in practical use, each data unit  1402  may contain N symbols to improve the estimation accuracy. Therefore, for either of data  1402 A and  1402 B, the respective results from Eq. 8 and Eq. 9 may be averaged by the N symbols. In the case where data unit  1402 C is implemented, since both of Eq. 8 and Eq. 9 are separately calculated for the inverse matrix H, the matrix is counted twice as many times. In this case, the results from both equations may also be separately averaged. Alternatively, the results of the first equation may be averaged prior to calculation of the second equation, namely, between the X- and Y-polarizations, to further improve accuracy. In at least one embodiment, Eq. 8 may be implemented for the lower two elements of the inverse matrix H, and Eq. 9 may be implemented for the upper two elements of the inverse matrix H. 
     In an exemplary embodiment, the respective implementation examples depicted in  FIGS. 14A-C  may be further implemented with a channel equalization. In this case, the inverse Jones Matrix H may be a 2×2, 1-tap filter, which can be used for polarization recovery and demultiplexing independently before any channel equalization is performed, as described further below with respect to  FIG. 15 . Alternatively, the inverse matrix H may be used for the initialization of a 2×2 multi-tap adaptive equalizer, as described further below with respect to  FIG. 17 . 
       FIG. 15  is a flow diagram depicting an exemplary SoP estimation technique  1500  for digital signal processing. In the exemplary embodiment depicted in  FIG. 15 , technique  1500  illustrates the DSP flow of an SoP estimation unit  1502  implemented together with a 2×2 multi-tap equalization schemes using adaptive equalizers  1504  for initialization. More specifically, each of the four adaptive equalizers  1504 ( 1 ),  1504 ( 2 ),  1504 ( 3 ),  1504 ( 4 ) are utilized in the exemplary embodiment for channel equalization. In some embodiments each of adaptive equalizers  1504  is further enabled to compensate for both intra-polarization linear channel distortions (e.g., inter-symbol-interference (ISI) from transmitter and receiver bandwidth limitations, residual CD) and inter-polarization cross-talk. 
     In exemplary operation of technique  1500 , both of revised polarization signals E x  and E y  are input to SoP estimation unit  1502 , whereas only revised polarization signal E x  is input to adaptive equalizers  1504 ( 1 ) and  1504 ( 2 ), and only revised polarization signal E y  is input to adaptive equalizers  1504 ( 3 ) and  1504 ( 4 ). Outputs from adaptive equalizers  1504 ( 1 ) and  1504 ( 2 ) are summed to generate an output X-polarization signal X out , and outputs from adaptive equalizers  1504 ( 3 ) and  1504 ( 4 ) are summed to generate an output Y-polarization signal Y out . Both of output signals X out  and Y out  may then be fed back into an error function unit  1506 . In an exemplary embodiment, the respective filter coefficients of adaptive equalizers  1504  (i.e., [d xx , d xy ; d yx , d yy ]) may then be updated (e.g., using CMA, MMA, or LMS algorithms, described above) based on the error signal feedback processed by error function unit  1506 . 
     As discussed above, using all-blind adaptive equalization (e.g.,  FIGS. 3, 6 ) the initialization of the filter coefficients of adaptive equalizers  1504  may be non-optimized with a default starting point. For example, assuming no polarization rotation, and setting the values of the center tap coefficients d xx , d xy , d yx , d yy  as [1 0 0 1], a gradual update to these filter coefficients may cost and unreasonable amount of time when significant polarization rotation occurs. Accordingly, considering that the most significant determinant of the convergence time for adaptive equalizers  1504  is the initialization, the present embodiments advantageously avoid this cost by implementing the SoP estimation (i.e., SoP estimation unit  1502 ) together with the four multi-tap adaptive equalizers  1504  for initialization, as depicted in  FIG. 15 . Furthermore, since four adaptive equalizers  1504  are utilized for adaptive equalization, the polarization change may be tracked, including the inter-polarization time of d xy  and d yx . That is, for each frame, no further estimation is needed after initialization. Accordingly, the respective frame structures described above with respect to  FIGS. 14  A-C may be implemented with technique  1500 , with only the respective data unit  1402  heading data architecture  1400  in time before the data payload of signal frame  1404 . 
     Therefore, according to technique  1500 , using the detected signals from each of two polarizations (i.e., E x  and E y ), the SoP estimation may be initially performed based on the first data unit  1402  in the frame head of the respective data architecture  1400 . That is, SoP estimation unit  1502  may be programmed with algorithms or computer-executed instructions to perform the calculations described above with respect to Eq. 8 and 9, and thereby obtain the inverse matrix H. 
     In an embodiment, technique  1500  further includes a multiplication unit  1508 , a channel response storage unit  1510 , and a normalization unit  1512 . In exemplary operation of this embodiment, the convergence time may be further reduced by multiplying, using multiplication unit  1508 , the inverse matrix H output from SoP estimation unit  1502  with an initial normalized channel response stored in channel response storage unit  1510  to initialize the four adaptive equalizers  1504 . That is, the normalized channel response D=[D xx , D xy ; D yx  D yy ] may be initially set, at the very beginning of system operation according to technique  1500 , with the initial pre-stored channel response such that the center taps of D xx  and D yy  are 1, and all other elements of D are set to zero. 
     For example, in the case of a 5-tap channel response, the initial pre-stored normalized channel response D in channel response storage unit  1510  may be set according to: 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               D 
                               xx 
                             
                           
                           
                             
                               D 
                               xy 
                             
                           
                         
                         
                           
                             
                               D 
                               
                                 y 
                                 ⁢ 
                                 x 
                               
                             
                           
                           
                             
                               D 
                               yy 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             00100 
                           
                           
                             
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                             
                           
                         
                         
                           
                             
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                               ⁢ 
                               0 
                             
                           
                           
                             00100 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     The initialization for the four adaptive equalizers  1504  may then be set according to: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             d 
                             xx 
                           
                         
                         
                           
                             d 
                             
                               x 
                               ⁢ 
                               y 
                             
                           
                         
                       
                       
                         
                           
                             d 
                             
                               y 
                               ⁢ 
                               x 
                             
                           
                         
                         
                           
                             d 
                             yy 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     H 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               D 
                               xx 
                             
                           
                           
                             
                               D 
                               
                                 x 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         
                           
                             
                               D 
                               
                                 y 
                                 ⁢ 
                                 x 
                               
                             
                           
                           
                             
                               D 
                               yy 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, after this initialization procedure, the SoP estimation is complete. In further operation according to technique  1500 , respective adaptive equalizers  1504  may then proceed with continuous updating taps to track the channel response and polarization changes. As described above, the respective filter coefficients of adaptive equalizers  1504  (i.e., [d xx , d xy ; d yx , d yy ]) may then be updated based on the error signal feedback from error function unit  1506  (e.g., CMA, MMA, LMS algorithms). In some embodiments, training sequences may be implemented when updating the filter coefficients to achieve faster convergence. 
     In further exemplary operation of technique  1500 , when channel equalization is completed, the tap values of adaptive equalizers  1504  (i.e., [d xx , d xy ; d yx , d yy ]) may be fed to normalization unit  1512 , normalized as an updated channel response D, and stored in channel response storage unit  1510 . The value of updated channel response D may then be utilized to equalize the next sequential frame. In the exemplary embodiment, any or all of the respective components illustrated with respect to technique  1500  may be contained within a DSP of a receiver (e.g., DSP  1308  of receiver-side  1304 ,  FIG. 13 ), and/or executed as a software module by a processor thereof. 
     According to the exemplary embodiment depicted in  FIG. 15 , technique  1500  is of particular use for, and fully compatible with, upstream burst detection in a PON, where each ONU of the PON may have its own stored channel response D (e.g., and a memory thereof), that is, a particular channel response D i  for the respective ONU i . Thus, according to technique  1500 , the channel response D i  may be utilized in the initialization process for next burst frame coming from that ONU i , whereas different ONUs may have different pre-stored coefficients. An exemplary implementation process of technique  1500  is described below with respect to  FIG. 16 . 
       FIG. 16  is a flow diagram depicting an exemplary channel equalization process  1600  implementing estimation technique  1500 ,  FIG. 15 . In the exemplary embodiment, process  1600  begins at step  1602 , in which SoP estimation is performed (e.g., by SoP estimation unit  1502 ) using a data unit (e.g., data unit  1402 A,  1402 B, or  1402 C,  FIG. 14 ) in the frame header of a received data frame (e.g., respective data architecture  1400 A,  1400 B, or  1400 C,  FIG. 14 ) of an input data signal (e.g., E x  and/or E y ). In an exemplary embodiment of step  1602 , the SoP estimation utilizes non-zero data in the data unit to calculate the inverse matrix H. 
     Step  1604  is a decision step. If, in step  1604 , process  1600  determines that the received data frame is the first frame in the received signal sequence, process  1600  proceeds to step  1606 . In step  1606 , the calculated inverse matrix H is multiplied (e.g., by multiplication unit  1508 ) by an initial pre-stored normalized channel response D (e.g., in channel response storage unit  1510 ), and the multiplied result thereof is fed the respective taps of adaptive equalizers (e.g., adaptive equalizers  1504 ) for adaptive channel equalization. 
     In step  1608 , the adaptive filters perform adaptive channel equalization on the multiplied inverse matrix H. In step  1610 , process  1600  determines that channel equalization has been completed, outputs the respective polarization output signal (e.g., X out  and/or Y out ), and feeds an updated normalized channel response D to the channel response storage unit (e.g., from adaptive filters  1504  by way of normalization unit  1512 ). In step  1612 , the channel response storage unit stores the updated normalized channel response D. In an exemplary embodiment of step  1612 , the updated normalized channel response D is stored within a table contained within the channel response storage unit. 
     Referring back to step  1604 , if process  1600  alternatively determines that the received data frame is not the first frame in the signal sequence, process  1600  instead proceeds to step  1614 . In step  1614 , the calculated inverse matrix H is multiplied (e.g., by multiplication unit  1508 ) by a stored updated normalized channel response D (e.g., from step  1612 ) read from the channel response storage unit, and the multiplied result thereof is fed the respective taps of adaptive equalizers (e.g., adaptive equalizers  1504 ) for adaptive channel equalization. Process  1600  then proceeds from step  1614  to step  1608 , and process  1600  may then be repeated for each successive received signal frame. 
     In an alternative embodiment, SoP estimation, polarization recovery, and demultiplexing may be implemented independently from adaptive channel equalization. An exemplary technique for such independent SoP estimation is described further below with respect to  FIG. 17 . 
       FIG. 17  is a flow diagram depicting an alternative SoP estimation technique  1700  for digital signal processing. In the exemplary embodiment, components of processing technique  1700  are implemented within, and/or using software-based processing algorithms of, a DSP of a coherent receiver (e.g., DSP  1308 ,  FIG. 13 ). In the embodiment depicted in  FIG. 17 , technique  1700  is similar in some respects to technique  1500 ,  FIG. 15 , and includes an SoP estimation unit  1702 , two adaptive equalizers  1704  (as opposed to the four adaptive equalizers  1504  utilized according to technique  1500 ), an error function unit  1706 , and a channel response storage unit  1708 . 
     Although individual elements of technique  1700  are thus similar to analogous elements of technique  1500 , the operating principle of technique  1700  is different from that of technique  1500 . For example, different from technique  1500 , technique  1700  utilizes a 1-tap inverse matrix H unit  1710  to achieve instant polarization recovery, and before channel equalization by adaptive equalizers  1504 ( 1 ),  1504 ( 2 ). Accordingly, because unit  1710  is able to recover both of the X- and Y-polarization signals prior to equalization, only two adaptive equalizers  1704 ( 1 ) and  1704 ( 2 ) (e.g., d xx  and d yy , respectively), are needed at the receiver, thereby significantly reducing the computation complexity, in comparison with technique  1500 , by approximately half. 
     However, since technique  1700  does not include inter-polarization equalizers (e.g., d xy  and d yx  in example 1), the two adaptive equalizers  1704 ( 1 ) and  1704 ( 2 )/d xx  and d yy  will not be able to track slower polarization changes. For example, in the case of short burst frames, there may be little or no change in the polarization, and therefore the inability to track slower polarization changes would be considered to result in a very small penalty on the performance. However, in the case of longer bursts or continuous mode operation, technique  1700  may be further configured to periodically check the polarization state and apply SoP estimation and recovery as needed. Nevertheless, even with this additional periodic checking, receiver DSP systems and methods according to technique  1700  represent significantly simplified channel equalization schemes, and with respect to both hardware costs and the processing resource burdens thereof. 
     In exemplary operation of technique  1700 , both of revised polarization signals E x  and E y  are input to SoP estimation unit  1702 , and both also to 1-tap inverse matrix H unit  1710 . The outputs from adaptive equalizer  1704 ( 1 ) becomes the output X-polarization signal X out , and the outputs from adaptive equalizer  1704 ( 2 ) becomes the output Y-polarization signal Y out . Both of output signals X out  and Y out  may again be fed back into error function unit  1706 , similar to the analogous operation in technique  1500 . 
     In further exemplary operation of technique  1700 , using the detected signals from two polarizations (e.g., E x  and E y ), the SoP estimation is initially performed based on the data unit received in the frame head (e.g., data unit  1402 A,  1402 B,  1402 C,  FIGS. 14A-C , described further below with respect to  FIGS. 18A-C ). That is, SoP estimation unit  1702  is programmed to process the detected polarization signals according to either or both of Eq. 8 and Eq. 9, above, to solve for the inverse matrix H. Instant polarization recovery unit  1710  then performs polarization recovery for the received signals E x  and E y  using the inverse matrix H produced by SoP estimation unit  1702  to output two polarization-recovered signals for the respective X- and Y-polarizations, which may then be separately fed to respective adaptive equalizers  1704 . In the example illustrated in  FIG. 17 , the recovered X-polarization signal is input to first adaptive equalizer  1704 ( 1 ), and the recovered Y-polarization signal is input to second adaptive equalizer  1704 ( 2 ), both of which may then apply channel equalization to the respective input recovered polarization signal. 
     As discussed above, in the example depicted in  FIG. 17 , only two adaptive equalizers  1704  are implemented, and are not expected to include tracking capability for polarization state changes according to this configuration. Nevertheless, a receiver DSP that implements technique  1700  may easily configured such that SoP estimation unit  1702  is enabled to perform SoP estimation a plurality of times for a single frame, depending on the length of the particular frame. Technique  1700  is thus further different from technique  1500 ,  FIG. 15 , in that technique  1500  would be expected implement SoP estimation (e.g., by SoP estimation unit  1502 ) only once for each received frame/data unit. In contrast, technique  1700  may advantageously perform SoP estimation the plurality of times between received data units. 
     In further exemplary operation of technique  1700 , after SoP estimation is performed, adaptive equalizers  1704  may be initialized using a pre-stored channel response D=[D xx ; D yy ] stored in a memory of channel response storage unit  1708 . Similar to the exemplary embodiment described with respect to  FIG. 15 , channel response storage unit  1708  may also set with a pre-stored default channel response value to reduce the convergence time. Thus, at the very beginning of system operation according to technique  1700 , the pre-stored default channel response may be set with center tap of D xx  and D yy  having a value of 1, and all other elements of D set to zero. 
     Accordingly, again considering the case of a 5-tap channel response, the initial pre-stored channel response D in channel response storage unit  1708  may be set according to: 
     
       
         
           
             
               
                 
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     After this initialization, each adaptive equalizer  1704  may then start with continuously updating taps to track the channel response and polarization changes. The corresponding filter coefficients [d xx ; d yy ] of adaptive equalizers  1704  may then be updated based on the error signal feedback from error function unit  1706  (e.g., which may use algorithms such as CMA, MMA, LMS etc.). Similar to technique  1500 , technique  1700  may further utilize training sequences when updating the filter coefficients to achieve faster convergence. 
     Once channel equalization is completed, the tap values [d xx ; d yy ] of adaptive equalizers  1704  may be stored channel response storage unit  1708  as an updated channel response D. This stored value for the updated channel response D may then be used for equalization processing of the next frame in the signal sequence. Different from technique  1500 , technique  1700  does not include a normalization unit or normalization processing, since SoP estimation is performed before channel equalization according to technique  1700 . 
     According to the exemplary embodiment depicted in  FIG. 17 , technique  1700  is also of particular use for, and fully compatible with, upstream burst detection in a PON, where each ONU of the PON may have its own stored channel response D (e.g., and a memory thereof), that is, a particular channel response D i  for the respective ONU i . Thus, according to technique  1700 , the channel response D i  may be utilized in the initialization process for next burst frame coming from that ONU i , whereas different ONUs may have different pre-stored coefficients. 
     According to technique  1700 , the respective data architectures and data units thereof may be similar to those described above with respect to  FIG. 15 . However, whereas the operating principles may be the same, the particular data architectures used with respect to technique  1700  may differ somewhat from those used with respect to technique  1500 , as described below with respect to  FIGS. 18A-C . Additionally, an exemplary implementation process of technique  1700  is described further below with respect to  FIG. 19 . 
       FIGS. 18A-C  are schematic illustrations depicting alternative respective data architectures  1800 A-C. Data architectures  1800 A-C are similar in principle to data architectures  1400 A-C,  FIGS. 14A-C , but illustrate alternative designs and/or dispositions of data units  1802  with respect to a sequential series of dual-polarization signal frames  1804  containing data payloads. Similar to data architectures  1400 A-C, data units  1802  are structured to include an X-data component and a Y-data component (i.e., Uxi and Uyi), and signal frames  1804  include both X-polarization data and Y-polarization data (i.e., Data Xi and Data Yi).  FIGS. 18A-C  thus depict three respective exemplary implementation examples of data unit frame structures having periodic frame units  1802  generated within the respective frame architecture  1800  for the simplified channel equalization processing described above with respect to technique  1700 ,  FIG. 17 . 
     More particularly,  FIG. 18A  illustrates an implementation example of periodic data units  1802 Ai including N-symbol length data only on the X-data component Ux, and zeros on the Y-data component Uy. In this example, the same frame structure is employed for each data unit  1802 A in the sequential series of data architecture  1800 A, such that a first data unit  1802 A( 1 ) that precedes a first signal frame  1804 A( 1 ) has the same frame structure as a second data unit  1802 A( 2 ) preceding a second signal frame  1804 A( 2 ) and a third data unit  1802 A( 3 ) preceding a third signal frame  1804 A( 3 ), etc. Data architecture  1800 A is therefore substantially similar to data architecture  1400 A,  FIG. 14A , but applied to a sequence of signal frames  1804 Ai. 
     The frame structure of data architecture  1800 A is depicted in  FIG. 18A , by way of example and not in a limiting sense, as implementing a frame structure having X-data Uxi on the X-polarization and zeros on the Y-polarization. Persons of ordinary skill in the art will understand though, that the operating principle of data architecture  1800 A will be substantially the same if implemented using frame structure having zeros on the X-polarization and Y-data Uyi on the Y-polarization, which is therefore similar in principle to data architecture  1400 B,  FIG. 14B , but applied to a sequence of signal frames  1804 Ai. 
       FIG. 18B  illustrates an implementation example of data architecture  1800 B having periodic data units  1802 Bi that are individually similar to data units  1802 A,  FIG. 18A  (i.e., N-symbol length data on one polarization, zeros on the other polarization), but is different than data architecture  1800 A when the plurality of data units  1802 Bi are considered in the aggregate, seen over the entire series of signal frames  1804 Bi included within data architecture  1800 B. That is, although each data unit  1800 B N-symbol length data on one polarization and zeros on the other polarization, the polarization that includes the N-symbol length data alternates for each sequential data unit  1802 B in the series. Thus, in the example depicted in  FIG. 18B , the frame structure for a first data unit  1802 B( 1 ) preceding a first signal frame  1804 B( 1 ) includes data Ux on the X-polarization and zeros on the Y-polarization, but a second data unit  1802 B( 2 ) preceding a second signal frame  1804 B( 2 ), which is next in the sequence, reverses the frame structure to include zeros on the X-polarization and data Uy on the Y-polarization. The frame structure of a third data unit  1802 B( 3 ) then alternates again to a frame structure substantially similar to that of first data unit  1802 B( 1 ), and the subsequent sequence of frames is processed accordingly. 
       FIG. 18C  illustrates an implementation example of data architecture  1800 C utilizing data units  1802 Ci having a hybrid frame structure substantially similar to that of data units  1402 C,  FIG. 14C , but applied to the sequence of signal frames  1804 Ci. That is, each of data units  1802 Ci similarly include at least two time slots within a single data unit, with N-symbol length data on one polarization/zeros on the other polarization in the first time slot, and alternating in the second time slot of the same data unit  1802 C. Because the respective U-data and zeros thus alternate within a single data unit, the same overall frame structure may remain the same for each periodic data unit  1802 C( 1 ),  1802 C( 2 ),  1802 C( 3 ), etc. 
     Accordingly, a general operational principle of the present systems and methods fundamental is to periodically leave one polarization of periodic data units  1802  null with zeros, while placing data on the other polarization. According to the exemplary embodiments depicted in  FIGS. 18A-C , SoP estimation may be easily and rapidly performed for sequential frames in a data stream. Thus, data units  1802 Ai and  1802 Ci operate according to the principles described above, with the implementation of data units  1802 Ci providing more balanced data loading between the X- and Y-polarizations, as well as a more robust performance due to the calculation on both of the two polarizations. Implementation of data units  1802 Bi, on the other hand, strike a balance between the different approaches that implement data units  1802 Ai or  1802 Ci. That is, use of data units  1802 Bi may achieve the more balanced data loading between polarizations, similar to that achieved through use of data units  1802 Ci, but implementing only a single time slot within each periodic data unit. 
       FIG. 19  is a flow diagram depicting an alternative channel equalization process  1900  implementing the estimation technique  1700 ,  FIG. 17 . In the exemplary embodiment, process  1900  begins at step  1902 , in which SoP estimation is performed (e.g., by SoP estimation unit  1702 ) using a data unit (e.g., data unit  1802 A,  1802 B, or  1802 C,  FIG. 18 ) in the frame header of a received data frame (e.g., respective data architecture  1800 A,  1800 B, or  1800 C,  FIG. 18 ) of an input data signal (e.g., E x  and/or E y ). In an exemplary embodiment of step  1902 , the SoP estimation utilizes non-zero data in the data unit to calculate the inverse matrix H, and then uses the calculated inverse matrix H (e.g., in instant polarization recovery unit  1710 ) to recovered outputs for each respective polarization. 
     Step  1904  is a decision step. If, in step  1904 , process  1900  determines that the received data frame is the first frame in the received signal sequence, process  1900  proceeds to step  1906 . In step  1906 , an initial pre-stored channel response D (e.g., from channel response storage unit  1708 ) is applied to each of the recovered polarization outputs (e.g., at adaptive equalizers  1704 ). In step  1908 , the adaptive filters perform adaptive channel equalization on the recovered polarization outputs according to the channel response D (e.g., obtained from channel response storage unit  1708 ). 
     Step  1910  is also a decision step, in which process  1900  determines whether a last SoP estimation cycle has been performed on the data sequence. If, in step  1910 , process  1900  determines that the SoP estimate is not for the last estimation cycle, process  1900  proceeds to step  1912 . In step  1912 , process  1900  performs an additional SoP estimation and recovery operation (e.g., using SoP estimation unit  1702  and instant polarization recovery unit  1710 ) using the periodic data units in the frame and the calculated inverse matrix H, after which, process  1900  returns to step  1908  for additional adaptive channel equalization. If, however, in step  1910 , process  1900  determines that the last SoP estimation cycle has been completed, process  1900  outputs the respective polarization output signal (e.g., X out  and/or Y out ), and then proceeds to step  1914 , in which an updated channel response D is provided to the channel response storage unit (e.g., from adaptive filters  1704 ). In an exemplary embodiment of step  1914 , the updated channel response D is stored within a table contained within the channel response storage unit. 
     Referring back to step  1904 , if process  1900  alternatively determines that the received data frame is not the first frame in the signal sequence, process  1900  instead proceeds to step  1916 . In step  1916 , the stored updated channel response D (e.g., from step  1914 ) is read from the channel response storage unit, and then applied to each of the recovered polarization outputs by the adaptive equalizers. Process  1900  then proceeds from step  1916  to step  1908 , and process  1900  may then be repeated for each successive received signal frame or subsequent SoP cycle. 
     In accordance with the DSP systems and methods described above, performance of the respective SoP estimation techniques was tested in an experimental simulation set up. For the simulation performance testing, 25 GBaud dual-polarization 16QAM training symbols were used. Experimental results of the simulated performance testing are described further below with respect to  FIGS. 20A-B  and  21 . 
       FIG. 20A  is a graphical illustration depicting a signal plot  2000  before implementation of SoP estimation and polarization recovery.  FIG. 20B  is a graphical illustration depicting a signal plot  2002  after implementation of SoP estimation and polarization recovery. A comparison of the respective results of signal plots  2000  and  2002  indicates that the polarizations of the respective PDM-16QAM signals, seen before SoP estimation and polarization recovery in plot  2000 , were correctly separated implementing the innovative techniques described above, as indicated in plot  2002 , scene after SoP estimation and polarization recovery. 
       FIG. 21  is a graphical illustration depicting a comparative BER performance result plot  2100  obtained according to techniques  1500 ,  FIGS. 15, and 1700 ,  FIG. 17 . More particularly, plot  2100  superimposes the BER-versus-received optical power results, scene after SoP estimation and polarization recovery, for both of a two-equalizer DSP implementation and a four-equalizer DSP implementation (i.e., FIR equalizers, for this performance test). More particularly, the two-equalizer implementation is representative of the simplified channel equalization technique illustrated in  FIG. 17 , and the four-equalizer implementation is representative of the more complex channel equalization technique depicted in  FIG. 15 . As may be seen from plot  2100 , the penalty resulting from the selection of one technique over the other is substantially negligible. Nevertheless, implementation of DSP equalization according to the simplified channel equalization technique of  FIG. 17  will reduce the computation complexity by half. 
     According to the systems and methods described above, an innovative data-aided technique is provided for SoP estimation and simplified channel equalization. These techniques advantageously utilize a specially designed data unit at the transmitter-side, which, in cooperation with complementary DSP at the receiver-side, efficiently and correctly separate the polarizations of a dual-polarization directly in a feedforward manner. The innovative data units of the present embodiments therefore include a frame structure that is particularly useful for burst-mode signal frames. Additionally, because the present systems and methods are based on feedforward estimation, convergence time may be greatly reduced, which is a unique advantage, in comparison with conventional techniques, to burst-mode coherent communication systems. 
     Additionally, because the DSP estimation techniques described herein are not based on the bit information carried by the data, but instead based on the relation between polarization-diversity detected signals, the innovative data units of the present embodiments may also be used for carrying bit information, if desired, or other DSP functions. The present techniques are thus also fully compatible with the utilization of channel equalization algorithms having reduced complexity and reduced convergence time, and may be implemented utilizing a conventional 2×2 channel equalization architecture to reduce he convergence time, whether by initializing the filter taps of the four adaptive equalizers therein, or by pre-separating the polarizations of the dual-polarization signal. 
     Alternatively, the present techniques further provide significantly simplified channel equalization using two adaptive equalizers instead of the four equalizers of the conventional 2×2 channel equalization architecture. That is, after SoP estimation, two independent adaptive equalizers perform channel equalization on each respective polarization of a polarization multiplexed signal, thereby reduces the computation complexity by 50% when compared with non-data-aided methods. Additionally, for single polarization signals, these techniques may be even further simplified to utilize only one adaptive equalizer. Efficient Preamble Design and DSP in Coherent-PON Upstream Burst-Mode Detection 
     As described above, the advance of high-speed optical access networks has been propelled by new business and application drivers, such as 5G, mobile x-haul, cloud networking, and high-bandwidth 4K/8K video streaming services. As a result of this advance, the bandwidth requirements in the optical access network have grown significantly in proportion. PON technologies have been a dominant solution to meet such high-capacity demand from end users, by offering relatively low-cost P2MP services. 
     Accordingly, the industry expects to upgrade the access network to 25/50-Gb/s, and even 100-Gb/s, PON technologies in the near future. The IEEE 802.3ca Task Force has recently, for example, released a 25/50G NG-EPON specification based on wavelength multiplexing of 25 Gb/s per single channel, and ITU-T/FSAN has launched new projects to standardize higher speed PONs, such as 50G single-wavelength TDM-PONs. However, both of these recent PON standardization proposals are based on intensity modulation and direct detection (IM/DD) in physical layer, and not, for example, based on coherent detection. 
     Single-wavelength high-speed TDM-PON systems are nevertheless of great interest in the industry, in comparison with system mechanisms for bonding multiple wavelengths, because the single-wavelength solution not only reduces the number of required optical components and the associated costs thereof, but also saves wavelength resources. Furthermore, 100G PON proposals using wavelength multiplexing and IM/DD of four 25 Gb/s, or two 50 Gb/s, channels are considered in the industry to be too challenged by their limited power budget and complicated wavelength resource management techniques. For example, a 100G PON based on O-band IM/DD has been recently proposed downstream transmission, however, this proposal requires a prohibitively large launch power at the OLT-side. A correspondingly high launch power is therefore considered out of reach at the ONU-side, that is, for upstream transmission. Moreover, appropriate transmission wavelength windows are difficult to obtain in the O-band in consideration of coexistence with legacy PON services. Therefore, the limited sensitivity of the 100G TDM-PON is considered too great of a challenge to increasing the data rate on a single wavelength to meet the PR-30 (&gt;29-dB link loss) power budget using direct detection in the O-band. 
     The present embodiments overcome these challenges by providing a 100-Gb/s, single wavelength, coherent detection TDM-PON. Coherent PONs, for example, provide higher sensitivity, and due to continuing DSP advancements, coherent PONs enable significantly higher access capacity and longer coverage reach. Coherent technology though, remains costly. Recent efforts to reduce the cost and complexity of coherent optics in the access network include semi-coherent systems using heterodyning, amplitude modulation, and Alamouti-coding based polarization-independent detection. However, these efforts to simplify the complexity have resulted in trade-offs that have penalized the sensitivity of the network, increased the device bandwidth requirements, and required non-standard coherent transceiver architectures. For example, where a single Mach-Zehnder modulator has been substituted for the dual-polarization I/Q modulator at the transmitter-side, this reduction to the complexity of the transmitter as required a corresponding increase to the complexity of the receiver, which in this example requires twice the bandwidth in comparison with an analogous receiver in a full-coherent QPSK system. This example of a semi-coherent system also still suffers the sensitivity penalty trade-off. 
     Because coherent optics in a fully-coherent system is at present the only practically-available, commercially-developed, and mass-deployed optical coherent communication technology in the field, the present embodiments build on this existing mature platform, in consideration of recent developments in opto-electronic integration and CMOS technology, as well as the existing market size in the access network, to achieve 100G coherent PON in a full-coherent system. The present embodiments are further fully compatible with related techniques that reduce and optimize the costs, complexity, and power consumption of the access network. 
     To realize these advantageous results, the following embodiments provide systems and methods for robustly achieving upstream burst mode coherent detection. That is, as discussed above, upstream transmission in the TDM-PON is burst-mode, which is different from the operation of the downstream transmission, where signals are continuously broadcast to all end users. In an exemplary embodiment, a centralized OLT receives signals, burst-by-burst, from different user-side ONUs. The different respective incoming upstream bursts signals are typically received by the OLT at different respective signal powers, carrier phases, times or clocks, and/or SoPs. 
     The present embodiments thus realize significant improvements to signal recovery and processing of the upstream burst signals at the OLT. For efficient recovery and processing of upstream transmissions at the OLT, the OLT must be able to respond rapidly to recover the burst signals from the various ONUs within a short time duration, and then be able to reset itself for the next incoming upstream burst. In comparison with burst-mode signal recovery techniques used by direct-detection PONs, signal recovery in the coherent PON is considerably more challenging due to the greater complexity of coherent optical signals, which are modulated and multiplexed on phase, polarization, and amplitude. 
     The present embodiments still further overcome the unsuitability of conventional continuous-mode coherent detection and DSP used in P2P links, which are typically based on blind or feedback-type equalization techniques, and thus required too long an acquisition time to accomplish signal recovery for burst-mode detection. The present systems and methods additionally effectively address the additional challenges arising from burst-mode DSP, such as: (i) other non-DSP subsystems are required to operate at sufficient similar high speed to detect the short optical bursts; and (ii) frequency-offset estimation must be similarly sufficiently fast, and also able to withstand a large offset range due to possible laser wavelength drift. 
     Some recent proposals attempt to address these additional challenges through techniques such as: (i) designed preambles and fast DSPs to achieve fast polarization separation, which fit pilot sequences into the burst-mode detection of a 100G PDM-QPSK coherent TDM-PON; (ii) a real-time 20-Gb/s single-polarization QPSK coherent burst-mode detection using 1.0-MHz clock frequency difference; and (iii) fast I/Q imbalance compensation for 100G PDM-QPSK burst-mode detection with using an 826-ns preamble. However, none of these recent proposals provide practical details on the overall preamble design, the related burst-mode signal processing performance, or importantly, how to reduce and optimize the preamble length for the 100G coherent TDM-PON. Although a burst-mode DSP architecture for coherent PONs has also been proposed, this previous architectural proposal utilizes pre-calculated tap coefficients for adaptive equalization during the ONU discovery process to enable preamble lengths, and is thus unsuitable for easy integration with different architectural configurations. 
     The following embodiments therefore further solve these additional problems, by providing systems and methods for reliable and efficient preamble design, with corresponding burst-mode DSP, for coherent upstream burst-mode detection in a 100G coherent TDM-PON. The present systems and methods still further provide detailed descriptions of the preamble design configuration and associated principles, as well as related key DSP functions, such as frame synchronization, SoP estimation, and FOE. 
     The following description additionally provides detailed analyses that demonstrate the utility of the present systems and methods, including experimental results showing improvements with respect to frequency-offset and fiber CD, as well as verification of the efficiency and overall performance using the present designed preamble techniques under different test conditions. According to the present systems and methods, the preamble length may be advantageously reduced by sharing the preamble unit among multiple DSP functions, and a robust performance in large frequency-offset and residual fiber dispersion is confirmed. 
       FIG. 22A  is a schematic diagram depicting a burst-frame architecture  2200  for a conventional direct-detection PON.  FIG. 22B  is a schematic diagram depicting an upstream recovery technique  2202  for direct-detection burst-frame architecture  2200 ,  FIG. 22A . In the embodiments depicted in  FIGS. 22A-B , burst frame architecture  2200  and recovery technique  2202  respectively represent burst frame structures and upstream burst-mode signal recovery functions for a conventional direct-detection TDM-PON according to the IEEE 802.3ca NG-EPON example. 
     As illustrated in  FIG. 22A , upstream burst-frame architecture  2200  begins with three synchronization patterns (SPs)  2204 , which are not under the FEC protection. More specifically, a first SP  2204 ( 1 ) (SP 1 ) is used for receiver (Rx) settling with the function of automatic gain control, a second SP  2204 ( 2 ) (SP 2 ) is used for the function of burst clock and data recovery (BCDR), and a third SP  2204 ( 3 ) (SP 3 ) is used for frame synchronization with the state-of-burst delimiter (SBD) function to indicate the start of the burst after synchronization. Upstream burst-frame architecture  2200  further includes a payload portion  2206  following SPs  2204 , and an end-of-burst delimiter (EBD) portion  2208  following payload portion  2206 . 
     As illustrated in  FIG. 22B , corresponding burst-mode signal recovery functions of conventional technique  2202  include an automatic gain control section  2210  and a burst-mode signal processor  2212 . In operation, automatic gain control section  2210  is configured to first perform automatic gain control with SP 1 , for example, using a burst-mode transimpedance amplifier (BM-TIA, not separately shown in  FIG. 22B ). After the BM-TIA achieves steady-state, the burst-mode signal processor in the receiver of the OLT begins processing the steady-state signal from automatic gain control section  2210  to acquire phase lock on the incoming data stream at a clock and data recovery portion  2214 . In the conventional direct-detection PON, no channel equalization is generally required. Once processor  2212  of the receiver successfully locks the clock of burst signal, the data will also be recovered with SP 2 . Once recovered, a frame synchronization unit of processor  2212  then processes payload portion  2206  after the SBD of SP 3  indicates the start-of-burst. Finally, end of the upstream burst is indicated upon detection of EBD portion  2208 . 
     A comparison of this conventional direct-detection frame architecture and processing functionality may be seen with respect to the exemplary coherent architecture and functionality described further below with respect to  FIGS. 23A-B . For example although some of the burst-mode detection principles of the conventional direct-detection PON may be applicable to a coherent PON, coherent upstream burst-mode detection is known to be significantly more challenging due to the increased complexity of coherent optical signals, which are modulated and multiplexed on phase, polarization, and amplitude. As described further below, the DSP of a coherent OLT receiver is required to process different clocks, different carrier frequency-offsets, different carrier phases, random SoPs, and different channel responses from different bursts. 
       FIG. 23A  is a schematic diagram depicting an exemplary burst-frame architecture  2300  for a coherent passive optical network.  FIG. 23B  is a schematic diagram depicting an exemplary upstream recovery technique  2302  for coherent burst-frame architecture  2300 ,  FIG. 23A . The exemplary embodiments depicted in  FIGS. 23A-B  respectively depict exemplary burst-frame structures and upstream burst-mode signal recovery functions for a coherent PON. 
     As illustrated in  FIG. 23A , different from the three-SP structure of upstream burst-frame architecture  2200 ,  FIG. 22A , upstream burst-frame architecture  2300  includes four different SPs  2304  for coherent detection of upstream burst PDM signals, which generally require polarization separation and channel equalization for DSP, as described in greater detail with respect to the embodiments above. According to the exemplary frame structure of burst-frame architecture  2300 , the overall preamble is advantageously designed for coherent burst synchronization and channel equalization. 
     More specifically, a first SP  2304 ( 1 ) (SP 1 ) is used for receiver (Rx) settling with the function of automatic gain control (e.g., similar to first SP  2204 ( 1 ),  FIG. 22A ), a second SP  2304 ( 2 ) (SP 2 ) is designed for digital clock recovery (Clock RCY), a third SP  2304 ( 3 ) (SP 3 ) may be optimized for channel synchronization (CH SYNC) with multiple functions, and a fourth SP  2304 ( 4 ) (SP 4 ) is used for channel adaptive equalization (CH EQ). Upstream burst-frame architecture  2300  further includes a payload portion  2306  following SPs  2304 , and an EBD portion  2308  following payload portion  2306 . 
     As illustrated in  FIG. 23B , corresponding burst-mode signal recovery functions of coherent recovery technique  2302  include an automatic gain control section  2310  and a burst-mode signal processor  2312 , similar to conventional technique  2202 ,  FIG. 22B . In operation, automatic gain control section  2310  is configured to first perform automatic gain control with first SP  2304 ( 1 ). At the time of this application there is no commercially-available BM-TIA for coherent upstream burst-mode detection. Accordingly, in an embodiment, automatic gain control section  2310  may perform optical automatic gain control using a semiconductor optical amplifier (SOA) or an erbium doped fiber amplifier (EDFA). Nevertheless, the present inventors contemplate that, once a linear coherent BM-TIA is demonstrated and available, such a coherent BM-TIA may be integrated with automatic gain control section  2310  to perform optical automatic gain control without departing from the scope of the embodiments herein. 
     In further operation of coherent recovery technique  2302 , burst-mode signal processor  2312  is configured to perform burst-mode digital signal processing functions after automatic gain control section  2310 , and based on the preamble design of second, third, and fourth SPs  2304 ( 2 - 4 ). In the exemplary embodiment, all functions of burst-mode signal processor  2312  may be implemented digitally, acting as a DSP, and which may follow an ADC (not separately shown in  FIG. 23B ). Burst-mode signal processor  2312  may further include one or more of an optional CD compensation unit  2314 , a clock recovery unit  2316 , a channel synchronization unit  2318 , a channel equalization unit  2320 , and a payload processing unit  2322 . 
     After the receiver achieves steady-state from automatic gain control section  2310 , digital clock recovery may be implemented by clock recovery unit  2316 , with second SP  2304 ( 2 ), to acquire frequency and phase lock to the clock of an incoming burst structured according to burst-frame architecture  2300 . After digital clock recovery, channel synchronization unit  2318  may perform channel synchronization, with third SP  2304 ( 3 ), and which may employ additional multiple sub-functions, including one or more of accurate frame synchronization, carrier frequency-offset estimation, and SoP estimation for polarization separation and recovery. Channel equalization unit  2320  may then apply, with fourth SP  2304 ( 4 ), channel response estimation for adaptive channel equalizations. Using the relevant respective information obtained from SPs  2304 ( 2 - 4 ) of the preamble, a payload demodulation process implemented by payload processing unit  2322  may be greatly simplified, along with a significant reduction of the convergence time. 
     Persons of ordinary skill in the art will appreciate that the particular order of burst-mode DSP functions/functional units of burst-mode signal processor  2312  are illustrated in  FIG. 23B  for illustration purposes, and are not intended to be limiting. The functional order may, for example, differ according to the particular algorithms implemented, as shown by the different implementation techniques described above with respect to  FIGS. 15 and 17 . Additionally, because many DSP algorithms that based on training sequences typically require accurate starting positions, frame synchronization is a first sub-function implemented in the channel synchronization process performed by channel synchronization unit  2318 , in the case where training sequences are employed. In some embodiments, two or more of SPs may be combined into a single SP using the same sequence pattern. Nevertheless, all of the corresponding functions described herein may still be applied to incoming bursts. 
     Additional robust and efficient preamble architectures, having data-assisted burst-mode DSPs in coherent upstream burst-mode detection after Rx-settling, are described further below. 
       FIG. 24  depicts an exemplary preamble architecture  2400  for a coherent burst-mode passive optical network. In the embodiment depicted in  FIG. 24 , preamble architecture  2400  represents an innovative high-efficiency upstream frame structure for preambles preceding payload sections  2402  of respective polarization signals  2404  of an upstream burst transmission polarization multiplexed signal in a 100G coherent PON. 
     In an exemplary embodiment, preamble architecture  2400  includes a first preamble processing SP  2406  (SP-A), a second preamble processing SP  2408  (SP-B), and a third preamble processing SP  2410  (SP-C). In this example, first, second, and third preamble processing SPs  2406 ,  2408 ,  2410  are respectively analogous to SP  2304 ( 2 ), SP  2304 ( 3 ), SP  2304 ( 4 ),  FIG. 23A . That is, SP-A is analogous to SP 2 , SP-B is analogous to SP 3 , and SP-C is analogous to SP 4 . As illustrated in  FIG. 24 , second preamble processing SP  2408  is configured for each respective polarization signal  2404  to have a 2×N conjugate symmetric symbol length over each of a first time slot  2412  and a second time slot  2414  within SP-B, for a total length of 4N symbols (described further below). As illustrated, each time slot  2412 ,  2414  includes 2N conjugate symmetric symbols in one polarization, and 2N zeros on the other polarization in that time slot with the symbol/zero polarization relationship alternating in the other time slot. In this respect, second preamble processing SP  2408  may be seen to include a frame structure similar to data unit  1402 C,  FIG. 14C . 
     It may be noted here that the exemplary preamble architecture  2400  depicted in  FIG. 24  does not exclude the additional inclusion of a preceding control SP analogous to SP 1  of  FIG. 23A , namely, a preamble control SP used for Rx-settling and automatic gain control. However, as illustrated above with respect to  FIGS. 22A and 23A , because the control SP 1  is not utilized for DSP, such a control SP is not further illustrated in  FIG. 24  to simplify the illustration. That is, the design of preamble processing SPs  2406 ,  2408 ,  2410  is drawn toward the corresponding burst-mode DSP functions of the receiver that utilize the relevant preamble processing SPs. 
     Therefore, in the exemplary embodiment, in practical application of the techniques described herein, a preamble control SP 1  is included in the preamble before preamble architecture  2400 , and therefore the length (in time) of the entire preamble will be the sum of the respective lengths of SP 1 , SP-A, SP-B, and SP-C. As described further below with respect to  FIG. 25 , the overall burst-mode DSP flow at the corresponding receiver may be designed according to a feed-forward configuration to reduce the processing latency and shorten the burst preamble length of preamble architecture  2400 . 
       FIG. 25  depicts an exemplary DSP  2500  for processing an upstream burst transmission implementing preamble architecture  2400 ,  FIG. 24 . In the example depicted in  FIG. 25 , DSP  2500  represents a data-aided burst-mode DSP for a 100G coherent PON. In an exemplary embodiment, DSP  2500  includes one or more of a frame detection and normalization unit  2502 , a CD compensation unit  2504 , a burst clock recovery unit  2506 , a frame synchronization unit  2508 , a burst SoP estimation and polarization demultiplexing unit  2510 , a preamble-based FOE unit  2512 , a channel estimation unit  2514 , a payload signal processing unit  2516 , and a phase recovery unit  2518 . As with processor  2312  of coherent recovery technique  2302 ,  FIG. 23B , some of the respective functional units of DSP  2500  may be optional and/or disposed in a different functional order, without departing from the scope of the embodiments herein. In some embodiments, additional processing units or functionality may also be included beyond the DSP functions shown in  FIG. 25 . In the exemplary embodiment, DSP  2500  is disposed after Rx-settling has occurred (e.g., after an automatic gain control section of that particular receiver). 
     In exemplary operation of DSP  2500 , after normalization and non-data-aided CD compensation is performed by frame detection and normalization unit  2502  and CD compensation unit  2504 , respectively, five particular data-aided DSP functions are performed based on the three preamble processing SPs  2406 ,  2408 ,  2410  (SP-A, SP-B, SP-C). In this example, first preamble processing SP  2406 /SP-A is used by burst clock recovery unit  2506  for burst-clock-recovery based on DC-balanced, state QPSK symbols that are distributed nearly equally. In an exemplary embodiment, a fast square-timing-recovery algorithm may additionally be applied based on the received symbols (not separately shown) within SP-A. In this case, because the square-timing-recovery algorithm is not training based, there would be no need for accurate frame synchronization, which may eliminate the need for the separate frame synchronization unit  2508 . In at least one embodiment, the particular pattern used for SP-A may also be used to achieve burst-mode automatic gain control. For example, SP-A may include a symbol portion corresponding to an additional preamble control SP 1 . In this case, the overall length of SP-A would be increased to include the additional SP  1  symbol portion. 
     In further operation of DSP  2500 , second preamble processing SP  2408 /SP-B is of particular importance, and which may be specially designed to perform one or more of three key data-aided DSP functions: (1) frame synchronization (e.g., by frame synchronization unit  2508 ); (2) SoP estimation (e.g., by burst SoP estimation and polarization demultiplexing unit  2510 ); and (3) FOE (e.g., by preamble-based FOE unit  2512 ). Accordingly, by utilizing a single preamble SP to perform all three of these DSP functions, the overall preamble length is advantageously reduced by sharing the same preamble SP (i.e., SP-B). Of these three key functions, it may be desirable to implement accurate frame synchronization first, in the case where the other key functions may be based on a training sequence that requires perfect frame synchronization. Thus, where training sequences may be implemented, frame synchronization unit  2508  may be logically placed prior to burst SoP estimation and polarization demultiplexing unit  2510  and preamble-based FOE unit  2512 . In this example, it is therefore assumed that the relevant frame synchronization algorithm is tolerant of carrier frequency offset. 
     In an exemplary embodiment, the sub-architecture of second preamble processing SP  2408  is advantageously designed to include 4N symbols within SP-B, including 2N conjugate symmetric symbols and 2N zeros on each respective polarization, as described above. As illustrated in  FIG. 24 , the dual-polarization SP-B may be transmitted according to the pattern [S X , 0; 0, S Y ]. In this example, S X =[s x1 , . . . , s xN , s xN *, . . . , s x1 *], and S Y =[s y1 , . . . , s yN , s yN *, . . . , s y1 *]. In this manner, essentially all of the 4N symbols in SP-B of the preamble may be staggered and transmitted with 2N symbols for each polarization. Additionally, without inter-polarization crosstalk between the X- and Y-polarizations, accurate frame synchronization may, for example, be realized using a sliding window with normalized auto-correlation processing on each polarization according to: 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       
                         x 
                         , 
                         y 
                       
                     
                     ⁡ 
                     
                       ( 
                       m 
                       ) 
                     
                   
                   = 
                   
                     
                       abs 
                       ⁡ 
                       
                         [ 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 r 
                                 
                                   x 
                                   , 
                                   y 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   m 
                                   + 
                                   k 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   x 
                                   , 
                                   y 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   m 
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     N 
                                   
                                   + 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     / 
                     
                       P 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     Here, C x  and Cy represent respective normalized auto-correlation functions on each polarization, P N  represents a normalization signal power factor, and r x  and r y  represent the received signals from X- and Y-polarizations, respectively. Because of the conjugate symmetric symbol distribution across the two time slots SP-B, it may therefore be demonstrated that the correlation results of Eq. 14 is tolerant of FOE. Accordingly, referring to the transmitted signals by T(mt s ), the received signals r(mt s ) may be expressed according to:
 
 r ( mt   s )= T ( mt   s )exp( j 2πΔ f ( mt   s )+φ)  (Eq. 15)
 
     Here, φ represents the carrier phase, and Δf represents the frequency-offset between the burst signal and the LO in the OLT. Assuming m 0  two symbolize the first symbol of the designed 2N conjugate symmetric symbols, the following is true:
 
 T   S ( k+ 1)= T   S (2 N−k )*= S   k , 0≤ k≤N   (Eq. 16)
 
     Therefore, when synchronized, the normalized auto-correlation peak may be expressed according to: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     17 
                   
                   ) 
                 
               
             
           
         
       
     
     According to this advantageous processing configuration, both the frequency-offset and the signal phase may be seen to have no impact on these processing results, while the normalized auto-correlation peak is nevertheless tolerant of carrier frequency offset errors. 
     Polarization, however, is known to randomly rotate after fiber transmission. Accordingly, to improve the tolerance to such polarization rotations, a combining scheme may be further implemented according to:
 
 C ( m )= W   x ( m ) C   x ( m )+ W   y ( m ) C   y ( m ),  (Eq. 18)
 
     where C(m) represents the combined function for peak searching, and W x  and W y  are defined to represent the respective power ratio of each polarization. 
     For example, W x  and W y  may be expressed according to: 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         m 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           P 
                           x 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                       
                         
                           
                             P 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             m 
                             ) 
                           
                         
                         + 
                         
                           
                             P 
                             y 
                           
                           ⁡ 
                           
                             ( 
                             m 
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         W 
                         y 
                       
                       ⁡ 
                       
                         ( 
                         m 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           P 
                           y 
                         
                         ⁡ 
                         
                           ( 
                           m 
                           ) 
                         
                       
                       
                         
                           
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                             x 
                           
                           ⁡ 
                           
                             ( 
                             m 
                             ) 
                           
                         
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                             y 
                           
                           ⁡ 
                           
                             ( 
                             m 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
     In this manner, an exact location of the SP-B symbols may be found from the received signal, and the synchronization algorithm discussed above is shown to be robust to carrier frequency-offset and polarization rotations. 
     In an exemplary embodiment, the same SP-B portion of the preamble may also be used for SoP estimation. For example, assuming that the received SP-B symbols may be expressed according to [r x1 , r x2 ; r y1 , r y2) ], the SoP may be instantly estimated after frame synchronization from the received SP-B symbols. 
     Thus, considering the single polarization case described above, the inverse Jones Matrix H may be estimated according to: 
     
       
         
           
             
               
                 
                   
                     H 
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   α 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     γ 
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       α 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     α 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   α 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     γ 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     20 
                   
                   ) 
                 
               
             
           
         
       
     
     where α 2  and γ 2  may be calculated based on the received signals according to: 
     
       
         
           
             
               
                 
                   
                     
                       α 
                       1 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               r 
                               
                                 x 
                                 ⁢ 
                                 1 
                               
                             
                             / 
                             
                               r 
                               
                                 y 
                                 ⁢ 
                                 1 
                               
                             
                           
                            
                         
                         2 
                       
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                                
                               
                                 
                                   r 
                                   
                                     x 
                                     ⁢ 
                                     1 
                                   
                                 
                                 / 
                                 
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     In a similar manner, α 2  and γ 2  may be obtained using the second half of the symbols within SP-B (e.g., within second time slot  2414 ,  FIG. 24 ). Polarization separation may thus be effectively realized based on the inverse of Jones Matrix H. After separating the two polarizations, frequency-offset may be estimated based on the same training symbols contained in SP-B. 
     In an exemplary embodiment, to achieve fast and accurate FOE, a maximum likelihood (ML) criteria FOE algorithm may be modified, by considering the different polarizations, and implemented to estimate the carrier frequency-offset according to:
 
Δ f =avg(Δ f   x   ,Δf   y ),  (Eq. 22)
 
     where Δf x  and Δf y  represent the estimated frequency-offset in the X- and Y-polarizations based on the 2N non-zero symbols described above. 
     In the exemplary embodiment, third preamble processing SP  2410 /SP-C may be advantageously designed to include training QPSK symbols for channel estimation (e.g., by channel estimation unit  2514 ), and which may be based on a CMA algorithm for DSP. It is noted that this implementation of CMA in DSP is different than the conventional CMA implementation in a continuous-mode DSP, where the CMA is blind, without any information regarding the SoP. Here we apply the inverse of Jones Matrix H to reduce the convergence time of CMA. In contrast, the CMA implementation for SP-C enables all relevant information to be obtained from the preamble, and then applied to the following payload processing performed by payload signal processing unit  2516 , thereby greatly simplifying the payload demodulation process while also significantly reducing the convergence time in comparison with the conventional blind CMA techniques that are devoid of any SoP information of SOP, as confirmed by the experimental demonstration results described further below. In at least one embodiment, a feed-forward phase recovery algorithm is implemented by phase recovery unit  2518  as a final step in the signal recovery process before BER measurements. 
       FIG. 26  is a schematic diagram depicting an exemplary test system  2600  for upstream burst detection. The embodiment depicted in  FIG. 26 , system  2600  represents an experimental setup to demonstrate coherent upstream burst detection in a 100-Gb/s/TDM coherent PON, which exhibited a detected power waveform  2602  of a burst frame transmitted therein including a preamble design according to preamble architecture  2400 ,  FIG. 24 . 
     For the experimental demonstration set up depicted in  FIG. 26 , system  2600  included an ONU-side  2604  transmitting upstream to an OLT-side  2606  over a 50 km single mode fiber (SMF)  2608 . At ONU-side  2604 , two synchronized ONUs  2610  were separately run to generate respective 25-GBaud PDM-QPSK burst frames utilizing a preamble frame structure according to preamble architecture  2400 ,  FIG. 24 . For this setup, the burst frames were generated by respective 80-GSa/s arbitrary waveform generators (AWGs)  2612 , and then fed into respective dual-polarization I/Q modulators  2614 . For this setup, each dual-polarization I/Q modulator  2614  included four drivers for optical signal modulation. Each ONU  2610  further included a respective tunable DFB laser  2616 , each tuned to a 1550-nm wavelength with a linewidth of approximately 1 MHz as the laser source of that ONU  2610 . 
     For demonstration purposes, after modulation by dual-polarization I/Q modulator  2614 ( 1 ), the burst signals generated from first ONU  2610 ( 1 ) are combined with a dummy signal from second ONU  2610 ( 2 ) using a 3-dB optical coupler (OC)  2618 , and the respective burst frames from the two ONUs  2610  were staggered to avoid collision. Using an automatic bias-control and synchronization  2620  between the two AWGs  2612 , the burst signal from one ONU  2610  was coupled only with the null signal from the other ONU  2610 . 
     The combined burst signals from ONU-side  26   044  then transmitted over 50-km SMF  2608 , and received optical power to OLT-side  2606  was controlled by a variable optical attenuator (VOA)  2622  for BER testing. At OLT-side  2606 , a burst-mode EDFA  2624  was used for signal pre-amplification. The pre-amplified signal was then mixed with LO  2626  in an integrated coherent receiver (ICR)  2628  for coherent detection. In this setup, LO  2626  included a tunable external-cavity-laser (ECL) at 1550-nm and a linewidth&lt;100 kHz. After coherent detection by ICR  2628 , the received signals were sampled by an 80-GSa/s digital sampling oscilloscope (DSO)  2630  and then processed using an offline burst-mode  2632  conforming to the exemplary configuration of DSP  2500 ,  FIG. 25 . For this setup, DSO  2630  was free-running, and there was no synchronization deployed between DSO  2630  and AWGs  2612 . 
     The symbol lengths of the respective preamble processing SPs, used in the burst frames generated by ONUs  2610 , are listed below in Table 1, which features a summary of the respective preamble SP types, lengths, and functions. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Preamble 
                 Length (Symbols) 
                 Functions 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 SP-A 
                 1024 
                 Burst Clock Recovery 
               
               
                   
                 SP-B 
                 512 
                 Frame Sync, Pol DeMux, FOE 
               
               
                   
                 SP-C 
                 256 
                 Channel Estimation 
               
               
                   
                   
               
            
           
         
       
     
     As may be seen from Table 1, SP-A, SP-B, and SP-C have symbol length of 1024, 512, and 256 symbols, respectively. Accordingly, each burst frame may be calculated to contain a total preamble length of 71.68 ns (i.e., 1792 symbols), a payload length of 3.072 μs, an end of burst (EOB) length of 30.72 ns. For this setup, a guard interval (GI), having a length of 102.4 ns, was included to separate the bursts. 
       FIGS. 27A-D  are graphical illustrations depicting respective experimental result plots  2700 ,  2702 ,  2704 ,  2706  from test system  2600 ,  FIG. 26 . Because SP-B is configured to have the most notable processing impact of the present preamble embodiments, plots  2700 ,  2702 , and  2704  were generated to demonstrate the testing performance of SP-B with respect to frame synchronization, SoP estimation, and FOE, respectively. 
     More particularly, plot  2700  of  FIG. 27A  depicts the experimental results of the normalized auto-correlation output for peak search. Plot  2700  therefore demonstrates the frame synchronization result based on the combined auto-correlation result of C(m). It may be seen from plot  2700  that two peaks  2708  appear at the respective X- and Y-polarizations, and peak very sharply at the synchronization point locations. These sync-peak locations therefore represent the start of non-zero SP-B symbols in the received signals. To quantify this performance of the frame synchronization, peak-to-maximum-noise ratio (PMNR) metric  2710  is here defined to indicate the quality of sync-peaks in comparison with noise peaks. 
     Plot  2702  of  FIG. 27B  depicts experimental results of PMNR against the length of SP-B non-zero symbols. As may be seen from plot  2702 , each polarization had 256 non-zero symbols in SP-B, that is, 512 symbols total within SP-B that included 256 zeros, provided over 10-dB PMNR, showing very high-quality peaks. 
     Plot  2704  of  FIG. 27C  depicts experimental results of PMNR against frequency offset. As described above, the present frame synchronization techniques are tolerant of FOEs, which is demonstrably verified by the experimental results shown in plot  2704 . Plot  2704  further confirms that a PMNR of over 10 dB may be achieved even with a 25 GHz offset range (−12.5 to 12.5 GHz, in this example). 
     Plot  2706  of  FIG. 27D , on the other hand, depicts the experimental results of PMNR against different polarization rotations namely, the X-polarization, the Y-polarization, and a combination thereof. Plot  2706  thus demonstrates the performance of frame synchronization under different polarization rotation states. According to plot  2706 , it may further be seen that the PMNR from each individual polarization is polarization-dependent, and changes according to the polarization rotation, as shown by individual polarization PMNR subplots  2712 . In contrast, as shown by combined PMNR subplot  2714 , the combined PMNR is substantially polarization-independent thereby verifying the mathematical principles described above with respect to Eq. 18 that indicate the advantageous tolerance of polarization rotation. 
       FIG. 28A  is a graphical illustration depicting a comparative plot  2800  depicting an estimated frequency offset against a target frequency offset. More particularly, comparative plot  2800  illustrates the FOE performance of the present preamble-based DSP FOE (e.g., of preamble-based FOE unit  2512  using second preamble processing SP  2408 /SP-B), over the 25 GHz estimation range, shown in a first subplot  2802  of plot  2800 . For comparison, a second subplot  2804  of plot  2800  is superimposed to show the performance of a feed-forward blind FOE DSP technique, which, in this example, was based on a long data sequence. A comparison of first subplot  2802  with second subplot  2804  demonstrates that the present preamble-based FOE technique, which utilizes the innovative SP-B architecture described above, outperforms the blind long-data-based technique. Additionally, it is notable that the present preamble-based FOE technique realized a larger estimation range (e.g., from −12.5 to 12.5 GHz), and the blind FOE technique used 2500 symbols, whereas the present preamble-based FOE technique used only 512 symbols within SP-B (i.e., 256 non-zero symbols on each polarization). 
       FIG. 28B  is a graphical illustration depicting a comparative BER performance result plot  2806  for subplots  2802 ,  2804 ,  FIG. 28A . More particularly, comparative BER performance result plot  2800  includes a first subplot  2808  demonstrating the BER performance of the present preamble-based FOE over the 25 GHz range, and a second subplot  2010  demonstrating the BER performance of the blind long-data-based FOE over the same range. From first subplot  2808 , it may be seen that the BER penalty is almost negligible for the present preamble-based FOE when the frequency offset is less than ±10 GHz. However, as illustrated by second subplot  2810 , due to the phase ambiguity, the feed-forward blind FOE method fails when the frequency-offset is greater than ±2.5 GHz. 
       FIG. 29  is a graphical illustration depicting a residual frequency offset plot  2900 . More particularly, plot  2900  illustrates the performance of residual FOE against the length of non-zero training symbols in SP-B. The results illustrated in plot  2900  thus confirm that 256 non-zero symbols for each polarization in SP-B is sufficient to accurately achieve a residual offset of &lt;2-MHz. In an exemplary embodiment, this residual offset value may be subsequent processed in carrier phase recovery function/functional unit of the DSP (e.g., phase recovery unit  2518 ,  FIG. 25 ). 
       FIG. 30A  is a graphical illustration depicting a plot  3000  of signal mean square error (MSE) before channel equalization.  FIG. 30B  is a graphical illustration depicting a comparative plot  3002  of signal MSE after channel equalization. More particularly, plot  3000  illustrates the results of SoP estimation by plotting the MSE against the length of non-zero symbols in SP-B, and comparative plot  3002  illustrates the signal MSE against the training length of symbols in SP-C. These SoP estimation results of comparative plot further demonstrate the impact on the required symbol length in SP-C used for channel estimation (e.g., by channel estimation unit  2514 ,  FIG. 25 ). 
     Plot  3000  further shows the results from testing the required SP-B symbols for SoP estimation before the channel equalization. It may be seen from plot  3000  that 256 non-zero symbols on each polarization in SP-B (512 symbols in total) is sufficient to minimize the impact from MSE. Comparative plot further shows the results from testing the impact on adaptive channel equalization (i) without using SP-B for SoP estimation, as illustrated in a first subplot  3004 , and (ii) with SP-B, as illustrated in a second subplot  3006 . First subplot  3004  thus illustrates how, without SP-B as described herein, the CMA process for channel equalization requires a considerably long convergence time due to the random polarization rotation. In contrast, as illustrated in second subplot  3006 , use of SP-B for SoP estimation, drastically reduces the minimum convergence time (i.e., indicating the length of SP-C training symbols) from 2560 symbols without SP-B to only 256 symbols with SP-B. Accordingly, by greatly reducing the channel response estimation time in this manner, the overall preamble length is also similarly reduced. 
       FIG. 31  is a graphical illustration depicting a comparative BER performance result plot  3100 . More particularly, plot  3100  depicts the BER performance against received optical power for an ECL-based continuous 100G PDM-QPSK signal, a back-to-back (B2B) DFB-based burst signal, and the DFB-based burst signal transmitted over 50 km SMF  2608 ,  FIG. 26 . After 50-km fiber transmission, plot  3100  demonstrates that a required optical power value  3102 , at an average BER of 1×10 −3 , is shown to be −39 dBm. Plot  3100  therefore further includes, for illustration purposes, a first constellation  3104  of the 50 km DFB-based burst signal and a second constellation  3106  of the B2B DFB-based burst signal, with both of first and second constellations  3104 ,  3106  taken at −39 dBm. 
     For further comparison, the plotted results of the ECL-based continuous 100G PDM-QPSK signals demonstrate the consistent performance over the different signal types. Furthermore, due to the high receiver sensitivity offered by the coherent detection technology employed in the test setup of  FIG. 26 , system  2600  was able to implement a pre-FEC BER threshold of 1×10 −3 , as opposed to the 1×10 −2  pre-FEC threshold of non-coherent PON systems. That is, lower coding and decoding complexities are expected from such simpler FEC coding schemes. Plot  3100  further shows that, when compared with the ECL-based continuous signals, there is less than a 0.3-dB penalty for the DFB laser-based burst signals after burst-mode coherent detection. 
     Plot  3100  still further illustrates the results from testing a dynamic range  3108  of the coherent receiver. That is, without having changed the receiver setup in OLT-side  2606  (i.e., the same BM-EDFA  2624  and ICR  2628  were kept), a dynamic range  3108  of approximately 20 dB is exhibited for the received power of the 100G coherent PON upstream burst signals. For system  2600 , dynamic range  3108  only depicts the test results using BM-EDFA  2624 . Nevertheless, the present inventors contemplate that an effective dynamic range will also be achieved using an SOA instead of an EDFA. 
       FIG. 32  is a graphical illustration depicting a BER performance plot  3200  as a function of residual CD. More particularly, plot  3200  demonstrates the test results of the overall BER performance under different residual CD values. Plot  3200  thus confirms that there is no overt BER penalty experienced when the residual dispersion is within 87.5 ps/nm (i.e., −43.75 to +43.75 ps/nm, in this test setup), and only is a small BER penalty is experienced for residual dispersion within 437.5 ps/nm (i.e., −218.75 to +218.75 ps/nm, in this test setup). For this test, the received optical power was maintained at −38.5 dBm during the entirety of the test from which the results are illustrated in plot  3200 . 
       FIG. 33  is a graphical illustration depicting a long-term bit-error-ratio performance result plot  3300 . For plot  3300 , the BER was tested continuously for a duration of approximately 6 hours. Plot  3300  thus effectively demonstrates testing of BER performance of the present systems and methods for long-term operation. For this testing implementation, a margin of 1 dB was reserved, and thus the received optical power was maintained at −38 dBm. As shown in plot  3300 , measured BER values  3302 , over the 6-hour test duration, all stayed below an FEC threshold value  3304  of BER at 1×10 −3 , thereby further confirming the long-term stability of the present preamble architectures and corresponding burst-mode DSP for upstream burst-mode coherent detection. For the testing considerations that produced the results shown in plot  3300 , the output power from the ONU was −2 dBm. Therefore, plot  3300  still further demonstrates the achievement of a 36-dB power budget (including a 1-dB margin). 
     According the embodiments described above, an innovative preamble architectural design is provided, as well as a corresponding burst-mode DSP solution, enabling significantly improved coherent upstream burst-mode detection in a 100G TDM coherent-PON. The above embodiments further demonstrate that these advantageous architectural and DSP function systems and methods are experimentally verified to be both reliable and efficient over a variety of different relevant test scenarios and test conditions. 
     The unique preamble architectural configuration described herein provides still further advantages over conventional techniques by enabling individual portions of the new preamble structure to be shared by multiple DSP functions, or functional units, thereby greatly reducing the overall preamble length. The experimental results described above further confirmed a robust performance of the present embodiments over a large frequency-offset, residual fiber dispersion, and long running times. As a proof-of-concept, a relevant testing system setup achieved effective coherent upstream burst-mode detection of a 100 Gb/s PDM-QPSK signal, with 36-dB power budget, and after 50-km SMF transmission using the present preamble architectures having a length of 71.68 ns at the transmission-side, with corresponding burst-mode DSP at the receiver-side. The present systems and methods still further demonstrated approximately 20 dB of received power dynamic range for burst signal detection in a 100-Gb/s/TDM coherent-PON. 
     Although specific features of various embodiments of the disclosure may be shown in some drawings and not in others, this convention is for convenience purposes and ease of description only. In accordance with the principles of the disclosure, a particular feature shown in a drawing may be referenced and/or claimed in combination with features of the other drawings. 
     Some embodiments involve the use of one or more electronic or computing devices. Such devices typically include a processor or controller, such as a general purpose central processing unit (CPU), a graphics processing unit (GPU), a microcontroller, a reduced instruction set computer (RISC) processor, an application specific integrated circuit (ASIC), a programmable logic circuit (PLC), a field programmable gate array (FPGA), a digital signal processing (DSP) device, and/or any other circuit or processor capable of executing the functions described herein. The processes described herein may be encoded as executable instructions embodied in a computer readable medium, including, without limitation, a storage device and/or a memory device. Such instructions, when executed by a processor, cause the processor to perform at least a portion of the methods described herein. The above examples are exemplary only, and thus are not intended to limit in any way the definition and/or meaning of the term “processor.” 
     This written description uses examples to disclose the embodiments, including the best mode, and also to enable any person skilled in the art to practice the embodiments, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the disclosure is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal language of the claims.