Patent Publication Number: US-7711273-B2

Title: Optical quadrature-amplitude modulation receiver

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   The subject matter of this application is related to that of U.S. patent application Ser. No. 11/204,607, filed on Aug. 15, 2005, and entitled “Coherent Phase-Shift-Keying,” which is incorporated herein by reference in its entirety. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to optical communication equipment and, more specifically, to equipment for coherent detection of optical quadrature-amplitude modulation (QAM) signals. 
   2. Description of the Related Art 
   Delivery of multimedia services (e.g., telephony, digital video, and data) that is implemented using optical phase-shift keying (PSK) signals or optical M-ary quadrature-amplitude modulation (M-QAM) signals has certain advantages, e.g., over that implemented using conventional electrical analog or digital signals. More specifically, some of the advantages are: the ability to carry various/multiple multimedia services over the same optical communication channel; the ability to maintain a selected bit-error rate (BER) with relatively low carrier-to-noise ratios; relatively high tolerance to nonlinear signal distortions; and relatively high spectral efficiency and transmission capacity. As a result, cable companies are upgrading their hybrid fiber coaxial (HFC) networks to improve/create a fully interactive, bidirectional optical network that can carry optical multimedia signals into and out of homes. It is projected that, in the near future, high definition television (HDTV) signals are likely to be delivered substantially exclusively over optical communication channels. 
   A typical coherent optical QAM receiver detects the received optical communication signal by mixing it with a local oscillator (LO) signal and processing the mixing results to determine the phase and amplitude of the communication signal in each time slot, thereby recovering the encoded data. To enable this phase and amplitude determination, the LO signal is typically phase-locked to the carrier wavelength of the communication signal using an optical phase-lock loop (PLL). More specifically, the PLL is configured to track the frequency and phase of the carrier wavelength and provide a feedback signal to the LO source, based on which feedback signal the LO source achieves and maintains the phase-lock. 
   Unfortunately, suitable coherent optical receivers are typically relatively difficult to design and/or relatively expensive to build. For example, a conventional, relatively inexpensive laser source might produce an optical signal that has a relatively large linewidth. If that laser source is used in a coherent optical receiver as an LO source, then its relatively large linewidth might produce a phase uncertainty/noise that can make the optical phase-lock between the LO and communication signals difficult to achieve and/or maintain. As a result, coherent optical receivers are often designed to have specially constructed laser sources and/or relatively complex optical PLLs, both of which can drive up the receiver cost by a substantial amount. 
   SUMMARY OF THE INVENTION 
   Problems in the prior art are addressed by various embodiments of a coherent receiver adapted to recover data encoded in a received optical quadrature-amplitude modulation (QAM) signal using an optical local oscillator (LO) signal that does not have to be phase-locked to the carrier frequency of the QAM signal and might have a relatively large linewidth. In one embodiment, a receiver of the invention has an optical detector coupled to a digital processor. The optical detector is adapted to mix the received optical QAM signal with an optical LO signal having a time-varying phase offset with respect to the carrier frequency of the QAM signal to produce two digital measures of the QAM signal. In a typical configuration, these two digital measures substantially represent the real and imaginary components, respectively, of the QAM signal in the complex plane defined with respect to the LO signal. The digital processor is adapted to: (i) determine the amplitude and phase differentials for each QAM-symbol transition based on the digital measures; (ii) adjust each phase differential for an amount of phase drift associated with the time-varying phase offset; (iii) map each QAM-symbol transition onto a constellation point of a 2D decision map using the transition&#39;s amplitude differential and adjusted phase differential; and (iv) based on the mapping results, recover the data encoded in the optical QAM signal. 
   According to one embodiment, the present invention is a receiver for an optical QAM signal, comprising: (A) an optical detector having an optical source adapted to generate an optical LO signal, said optical detector adapted to mix said optical QAM signal with said optical LO signal to produce first and second digital measures of the QAM signal, wherein the first and second digital measures correspond to different phases of the optical LO signal; and (B) a digital processor coupled to the optical detector and adapted to process the first and second digital measures in response to a time-varying phase offset between a carrier frequency of the QAM signal and the optical LO signal to recover data encoded in the optical QAM signal. 
   According to another embodiment, the present invention is a method of processing an optical QAM signal, comprising: (A) mixing said optical QAM signal with an optical LO signal to produce first and second digital measures of the QAM signal, wherein the first and second digital measures correspond to different phases of the optical LO signal; and (B) processing the first and second digital measures in response to a time-varying phase offset between a carrier frequency of the QAM signal and the optical LO signal to recover data encoded in the optical QAM signal. 
   According to yet another embodiment, the present invention is a receiver for an optical QAM signal, comprising: (A) an optical detector adapted to mix the optical QAM signal with an optical LO signal to produce first and second digital measures of the QAM signal; (B) a differentiator circuit coupled to the optical detector and adapted to determine an amplitude differential and a phase differential for a QAM-symbol transition based on the first and second digital measures; (C) an angular adjustor coupled to the differentiator circuit and adapted to adjust the determined phase differential in response to a time-varying phase offset between a carrier frequency of the QAM signal and the optical LO signal; and (D) a decoding circuit coupled to the differentiator circuit and the angular adjustor and adapted to recover data encoded in the optical QAM signal based on the determined amplitude differential and the adjusted phase differential. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other aspects, features, and benefits of the present invention will become more fully apparent from the following detailed description, the appended claims, and the accompanying drawings in which: 
       FIGS. 1A-C  graphically show three representative QAM constellations that can be used in various embodiments of the invention; 
       FIG. 2  shows a block diagram of a communication system according to one embodiment of the invention; 
       FIG. 3  shows a block diagram of a digital processor (DP) that can be used in the communication system of  FIG. 2  according to one embodiment of the invention; 
       FIGS. 4A-C  graphically illustrate the operation of the DP shown in  FIG. 3  when the communication system of  FIG. 2  is configured to transmit data using the 4-QAM constellation of  FIG. 1A ; 
       FIGS. 5A-C  graphically illustrate the operation of the DP shown in  FIG. 3  when the communication system of  FIG. 2  is configured to transmit data using the 8-QAM constellation of  FIG. 1B ; 
       FIGS. 6A-C  graphically illustrate the operation of the DP shown in  FIG. 3  when the communication system of  FIG. 2  is configured to transmit data using the 16-QAM constellation of  FIG. 1C ; and 
       FIG. 7  shows a block diagram of a DP that can be used in the communication system of  FIG. 2  according to another embodiment of the invention. 
   

   DETAILED DESCRIPTION 
     FIGS. 1A-C  graphically show three representative two-layered QAM constellations that can be used in various embodiments of the invention. More specifically,  FIG. 1A  graphically shows a symbol set and transition diagram for a two-layered 4-QAM constellation;  FIG. 1B  graphically shows a symbol set and transition diagram for a two-layered 8-QAM constellation; and  FIG. 1C  graphically shows a symbol set and transition diagram for a two-layered 16-QAM constellation. 
   Referring to  FIG. 1A , symbol set A 4  of the two-layered 4-QAM constellation has four symbols labeled ( 0 ) through ( 3 ) that are described by Eq. (1):
 
 A   4 =±1,±2 j   (1)
 
where symbols ( 0 ) and ( 2 ) lie on the real (Re) axis of the complex plane, and symbols ( 1 ) and ( 3 ) lie on the imaginary (Im) axis of the complex plane. Symbols ( 0 ) and ( 2 ) represent the first layer of the 4-QAM constellation, and symbols ( 1 ) and ( 3 ) represent the second layer of that constellation. Using the constellation of  FIG. 1A , data are encoded in a differential manner by assigning a particular two-bit value to each transition between the constellation symbols. The arrows in  FIG. 1A  illustratively show four possible transitions that involve symbol ( 0 ) as a start state, with the assigned binary values indicated next to the respective arrows. For example, the ( 0 )→( 1 ) transition is assigned a binary value of 00. Similarly, the ( 0 )→( 2 ) and ( 0 )→( 3 ) transitions are assigned binary values of 10 and 11, respectively. Finally, the ( 0 )→( 0 ) transition is assigned a binary value of 01. A transition diagram for transitions that originate at any one of symbols ( 1 ), ( 2 ), and ( 3 ) can be obtained from the shown transition diagram using the following principles: (i) the binary value assigned to a transition between two constellation symbols is invariant with respect to the transition direction and (ii) transitions characterized by the same absolute value of the amplitude increment and the same absolute value of the phase increment are assigned the same binary value.
 
   Referring to  FIG. 1B , symbol set A 8  of the two-layered 8-QAM constellation has eight symbols labeled ( 0 ) through ( 7 ) that are described by Eq. (2):
 
 A   8   =e   jkπ/2 ,2 e   j(k+0.5)π/2 ; k=0, . . . , 3  (2)
 
The even symbols of symbol set A 8  represent the first layer of the 8-QAM constellation, lie at the complex-plane-axis intersections with a unit circle, and are separated from one another by 90 degrees. The odd symbols of symbol set A 8  represent the second layer of the 8-QAM constellation, lie on a circle having a radius of two at the points that are equidistant from the complex-plane axes, and are also separated from one another by 90 degrees.
 
   Using the 8-QAM constellation of  FIG. 1B , data are also encoded in a differential manner by assigning a particular binary value to each transition between the symbols. However, one difference between the 4-QAM and 8-QAM constellations is that, in the latter, each binary value is a three-bit value, as opposed to a two-bit value in the former. The arrows in  FIG. 1B  illustratively show eight possible transitions that involve symbol ( 0 ) as a start state, with the assigned binary values indicated next to the respective arrows. A transition diagram for transitions that originate at any one of the remaining symbols can be obtained from the shown transition diagram using the above-specified two principles, i.e., (i) the binary value assigned to a transition between two constellation symbols is invariant with respect to the transition direction and (ii) transitions characterized by the same absolute value of the amplitude increment and the same absolute value of the phase increment are assigned the same binary value. 
   Referring to  FIG. 1C , symbol set A 16  of the two-layered 16-QAM constellation has sixteen symbols labeled ( 0 ) through ( 9 ) and (A) through (F) that are described by Eq. (3):
 
 A   16   =e   jkπ/4 ,2 e   j(k+0.5)π/4 ; k=0, . . . , 7  (3)
 
The 16-QAM constellation of  FIG. 1C  is analogous to the 4-QAM and 8-QAM constellations of  FIGS. 1A-B  in that its symbols lie on two circles, the first circle having a radius of one and the second circle having a radius of two. In particular, the even symbols of symbol set A 16  (i.e., symbols ( 0 ), ( 2 ), ( 4 ), ( 6 ), ( 8 ), (A), (C), and (E)) represent the first layer of the 16-QAM constellation, lie on the first circle, and are separated from one another by 45 degrees. Similarly, the odd symbols of symbol set A 16  (i.e., symbols ( 1 ), ( 3 ), ( 5 ), ( 7 ), ( 9 ), (B), (D), and (F)) represent the second layer of the 16-QAM constellation, lie on the second circle, and are also separated from one another by 45 degrees. Note, however, that the even symbols are oriented with respect to the complex-plane axes such that four of them lie at the intersection points of the first circle with the complex-plane axes, while the odd symbols are oriented such that there are no odd symbols at the four intersection points of the second circle with the complex-plane axes and each of those four intersection points is equidistant from the respective two adjacent odd symbols.
 
   Similar to the 4-QAM and 8-QAM constellations of  FIGS. 1A-B , the 16-QAM constellation of  FIG. 1C  is used to encode data in a differential manner by assigning a particular binary value to each transition between the symbols. However, in the 16-QAM constellation, each binary value is a four-bit value. The arrows in  FIG. 1C  illustratively show sixteen possible transitions that involve symbol ( 0 ) as a start state, with the assigned binary values indicated next to the respective arrows. A transition diagram for transitions that originate at any one of the remaining symbols can be obtained from the shown transition diagram using the same two principles as those specified above for the 4-QAM and 8-QAM constellations of  FIGS. 1A-B . 
     FIG. 2  shows a block diagram of a communication system  200  according to one embodiment of the invention. System  200  has a transmitter  210  and a receiver  230  coupled via an optical communication link  220 . Transmitter  210  has an optical source (e.g., a laser)  212  coupled to an optical modulator (OM)  214 , which is controlled by a driver  216 . Driver  216  receives a binary input sequence X(n), transforms it into a sequence of constellation symbols, e.g., using a selected one of the QAM constellations shown in  FIGS. 1A-C , and generates a control signal that is applied to OM  214  to produce an optical QAM signal  218  that carries that symbol sequence. 
   After propagating through link  220 , signal  218  is received at receiver  230  as signal  228 , which is then split into first and second copies in a splitter  236   a . A local oscillator (LO) signal  234 , which is produced at receiver  230  by an optical source (e.g., a laser)  232 , is similarly split into first and second copies in a splitter  236   b . The first copy of signal  228  and the first copy of signal  234  are then applied to an optical mixer  240   a . The second copy of signal  228  and a phase-shifted copy of signal  234  are similarly applied to an optical mixer  240   b , with the phase-shifted copy of signal  234  obtained from the second copy of signal  234  (produced by splitter  234   b ) by passing that copy through an optical phase shifter (OPS)  238 . In a typical configuration, OPS  238  is configured to introduce a π/2 (i.e., 90-degree) phase shift. It is desirable for the phase shift introduced by OPS  238  to fall between 45 and 135 degrees, and it is preferred that said phase shift is between 75 and 105 degrees. 
   Each of optical mixers  240   a - b  is designed to combine its input signals to produce two interference signals, each having an intensity that is: (i) proportional to the intensities of the input signals and (ii) related to an instant phase offset between those input signals. More specifically, the interference signals produced by optical mixer  240  are such that the intensity difference between these interference signals is proportional to cos(Δφ), where Δφ is the instant phase offset. A pair of balanced photodetectors  242  coupled to a respective one of differential amplifiers  244   a - b  continuously measures the intensity difference for the interference signals produced by the respective one of optical mixers  240   a - b  and applies the measurement results to a respective one of synchronized analog-to-digital converters (ADCs)  246   a - b . Using these measurement results, each of ADCs  246   a - b  produces a respective one of digital signals  248   a - b , both of which are applied to a digital processor (DP)  250 . 
   Note that the above-described signal processing implemented in receiver  230  substantially causes digital signal  248   a  to be proportional to I 228  cos(Δθ), where I 228  is the instant intensity of signal  228  and Δθ is the instant phase offset between signals  228  and  234 . Note also that, if OPS  238  introduces a π/2 phase shift, then the signal processing implemented in receiver  230  causes digital signal  248   b  to be substantially proportional to I 228  sin(Δθ). Thus, the signal processing implemented in receiver  230  substantially provides, in the form of digital signals  248   a - b , instant measures of the real and imaginary components, respectively, of signal  228  in the complex plane defined with respect to LO signal  234 . 
   The absence of a phase-lock between the carrier frequency (wavelength) of signal  228  and LO signal  234  generally manifests itself in that different instances of the same symbol carried by signal  228  fall onto different portions of the complex plane defined with respect to LO signal  234 . More specifically, if a sufficiently large number of instances of the same symbol are received and mapped onto the complex plane, those instances form a substantially continuous circular band having a radius corresponding to the distance between the center of coordinates and the symbol position in the QAM constellation. For example, repetitive transmission of symbol ( 0 ) of the 4-QAM constellation (see  FIG. 1A ) will produce a circular band having a radius of one. Similarly, repetitive transmission of symbol ( 2 ) of the 4-QAM constellation will produce a circular band having a radius of one, which circular band will overlap with the circular band corresponding to symbol ( 0 ). Repetitive transmission of symbols ( 1 ) and ( 3 ) of the 4-QAM constellation (see  FIG. 1A ) will produce two overlapping circular bands having a radius of two. One consequence of this band overlapping is that the direct mapping, using digital signals  248   a - b , of the received symbols onto the complex plane defined with respect to (not phase-locked) LO signal  234  does not enable appropriate symbol recognition and/or data extraction. As described in more detail below, DP  250  processes digital signals  248   a - b  such that, even in the absence of a phase-lock between signals  228  and  234 , symbol transitions in communication signal  228  are ascertained to enable substantial reconstruction of the original binary sequence X(n). 
   In certain embodiments of receiver  230 , DP  250  is optionally configured to generate a control signal  252  and apply that control signal to optical source  232  for the purpose of loosely controlling the frequency (ω LO ) of that optical source. In one embodiment, based on control signal  252 , optical source  232  is configured to adjust its frequency, e.g., such that |ω LO −ω S |≦Δω 0 , where ω S  is the carrier frequency of communication signal  228 , and Δω 0  is a selected maximum frequency mismatch value. Keeping the frequency mismatch between signals  228  and  234  within certain bounds might be advantageous because, in the presence of a relatively large frequency mismatch, the magnitudes of the signals generated by photodetectors  242  become relatively low. As such, control signal  252  can help to maintain optimal performance of photodetectors  242 . Note, however, that the feedback loop that provides control signal  252  is not designed to phase-lock optical source  232  to the carrier frequency of communication signal  228  (as would be the case in a PLL). 
     FIG. 3  shows a block diagram of a digital processor (DP)  350  that can be used as DP  250  ( FIG. 2 ) according to one embodiment of the invention. DP  350  is configured to receive digital input signals  348   a - b  that are analogous to, e.g., digital signals  248   a - b  (see  FIG. 2 ), respectively. In one configuration, signals  348   a - b  represent the real and imaginary parts, respectively, of communication signal  228  and can be expressed as follows:
   y   348 ( t )=α{ E   B ( t ) e   j(Φ     W     +Δωt)   +N ( t )}  (4) 
where: y 348 (t) is the complex value of signal  348  at time t; α is a conversion coefficient;
 
                 E   B     ⁡     (   t   )       =       ∑   n     ⁢         A   B     ⁡     (   n   )       ⁢     p   ⁡     (     t   -   nT     )             ,         
where A B (n) is the respective constellation symbol (e.g., from a selected one of constellations A 4 , A 8 , and A 16  of  FIGS. 1A-C ) in the n-th time slot, p(t) is the waveform envelope associated with each constellation symbol, and T is the symbol period (time-slot duration); Φ W =Φ S +Φ LO , where Φ S  is the linewidth-related phase noise in the communication signal (e.g., signal  228 ) and Φ LO  is the linewidth-related phase noise in the LO signal (e.g., signal  234 ); Δω=ω LO −ω S , and N(t) is the additive complex Gaussian noise.
 
   Signals  348   a - b  are applied to an amplitude differentiator  310  and an angular differentiator  320 . Amplitude differentiator  310  has an amplitude extractor  312  that is configured to compute, for each symbol period, the amplitude of signal  348 , e.g., by (i) presenting the received value of y 348  (see Eq. (4)) in the form given by Eq. (5):
 
 y   348 ( t )= r ( t ) e   jψ(t)   (5)
 
where r(t) is the signal amplitude and ψ(t) is the signal phase, and (ii) extracting the value of r(t). The extracted value of r(t) is then applied to a delay element (Z −1 )  314  and an adder  316 . Delay element  314  delays the value of r(t) by one symbol period T, multiplies the delayed value by −1, and applies the result to adder  316 . Adder  316  then sums the current value r(t) and the negative delayed value r(t−T), thereby computing an amplitude differential, dr(n)=r(n)−r(n−1), for each symbol transition.
 
   Angular differentiator  320  has a phase extractor  322  that is configured to compute, for each symbol period, the phase of signal  348 , e.g., by (i) presenting the received value of y 348  in the form given by Eq. (5) and (ii) extracting the value of ψ(t). The extracted value of ψ(t) is then applied to a delay element (Z −1 )  324  and an adder  326 . Delay element  324  delays the value of ψ(t) by one symbol period T, multiplies the delayed value by −1, and applies the result to adder  326 . Adder  326  then sums the current value ψ(t) and the negative delayed value ψ(t−T), thereby computing a phase differential, dψ(n)=ψ(n)−ψ(n−1), for each symbol transition. 
   The output produced by angular differentiator  320  is applied to an angular adjustor  330 . Angular adjustor  330  has a frequency offset estimator (FOE)  332  and a phase adjustor  334 . FOE  332  is configured to compute and track the value of Δω. The speed at which FOE  332  computes and updates the value of Δω is determined by the frequency offset drift rate, dΔω/dt. More specifically, FOE  332  is configured to accumulate a statistically sufficient number (determined by the frequency offset drift rate) of phase differentials dψ(n) and compute the value of Δω under the assumption that: (i) for a sufficiently long pseudo-random bit sequence, the sum of dψ(n) is substantially zero and (ii) the mean of the linewidth-related phase noise Φ W  is also zero. As such, FOE  332  determines the cumulative phase differential over an appropriately long time interval and then computes the value of Δω by simply dividing the determined cumulative phase differential by the duration of that time interval. In alternative embodiments, FOE  332  can be configured to use other suitable averaging methods to compute the value of Δω. 
   When DP  350  is initially brought online, FOE  332  is normally able to produce a first estimate of Δω after a certain induction period, during which the FOE accumulates the phase-differential statistics. After that initial induction period, FOE  332  can be configured to update the value of Δω as often as each symbol period using, e.g., a known sliding-window averaging method, in which a fixed number of most-recent phase differentials is used to calculate the cumulative phase differential value that goes into the frequency offset calculation. Based on the determined value of Δω, FOE  332  can optionally produce a control signal  352  that can be used as control signal  252  in receiver  230  (see  FIG. 2 ). FOE  332  also supplies the determined value of Δω to phase adjustor  334 . 
   Similar to FOE  332 , phase adjustor  334  is configured to receive the sequence of phase differentials dψ(n) produced by angular differentiator  320 . Since normally Δω≠0, each phase differential contains a frequency offset component that can be expressed as ΔωT. Based on the value of Δω received from FOE  332 , phase adjustor  334  is configured to remove this frequency offset component from each phase differential to produce, for each symbol transition, a frequency-offset-adjusted phase differential. As shown in more detail below, the frequency-offset-adjusted phase differential, together with the corresponding amplitude differential, can be used to recover the encoded data. 
   The frequency-offset-adjusted phase differential and the amplitude differential produced by angular adjustor  330  and amplitude differentiator  310 , respectively, are applied to a QAM mapper  340 , which is configured to perform at least two functions. The first function of QAM mapper  340  is to form a two-dimensional (2D) decision map. The second function of QAM mapper  340  is to use the formed 2D decision map to sort symbol transitions in the received communication signal for data decoding in a decoder  342 . 
   Referring to the first function of QAM mapper  340 , the 2D decision map is constructed after the receiver (e.g., receiver  230 ) having DP  350  has received a sufficiently large number of symbols. QAM mapper  340  is configured to fold each frequency-offset-adjusted phase differential into a 2π range and plot each symbol transition as a point on a 2D plane, in which the point&#39;s x-coordinate is the folded phase differential value and the y-coordinate is the absolute value of the amplitude differential. As further illustrated below, the plotted symbol-transition points normally form clusters, the number of which corresponds to the number of different binary values assigned to various symbol transitions in the corresponding QAM constellation. For example, for the 4-QAM, 8-QAM, and 16-QAM constellations of  FIGS. 1A-C , the plotted symbol-transition points form 4, 8, and 16 clusters, respectively. For each cluster, QAM mapper  340  determines a mean amplitude-differential value and a mean folded-phase-differential value, which mean values provide the coordinates of the respective constellation point on the 2D decision map. As such, the decision map for the 4-QAM, 8-QAM, and 16-QAM constellations of  FIGS. 1A-C  has 4, 8, and 16 constellation points, respectively. These constellation points of the 2D decision map are used to sort the symbol transitions of the communication signal and decode the data. 
   In one configuration, when DP  350  is initially brought online, QAM mapper  340  is typically configured to plot a relatively large number (e.g., 50) of symbol-transition points to obtain the initial coordinates of the constellation points on the 2D decision map. After the initial coordinates has been determined, QAM mapper  340  can be configured to continuously update the 2D decision map by recalculating the mean amplitude and folded-phase differential values corresponding to each cluster based, e.g., on the most recently received 100 symbol-transition points. As such, at any time after the initial start-up period, QAM mapper  340  maintains a current 2D decision map that is used for decoding information carried by the received communication signal. 
   In an alternative configuration, the initial coordinates of the constellation points on the 2D decision map are set in advance of the above-described calculation, e.g., based on the 2D decision map stored in the memory from the previous runs. These coordinates are then adjusted based on the data statistics collected on the fly during the current run substantially as described above. As a result, the receiver has a workable 2D decision map as soon as it is brought online, which 2D decision map is gradually improved to reflect the current channel/noise conditions. 
   Referring now to the second function of QAM mapper  340 , the QAM mapper is configured to sort the received symbol-transition points by mapping each symbol-transition point onto one of the constellation points of the 2D decision map. In one representative configuration, QAM mapper  340  calculates the Euclidian distance between a received symbol-transition point and each of the constellation points. QAM mapper  340  then compares the calculated Euclidian distances to determine the shortest Euclidian distance. Finally, QAM mapper  340  assigns to the symbol-transition point a logical value associated with the constellation point corresponding to the shortest distance. If the symbol-transition point has the same (shortest) distance from two or more constellation points, then that symbol-transition point can be discarded as an error. Alternatively or in addition, QAM mapper  340  can be configured to use other suitable mapping methods known to those skilled in the art to map the received symbol-transition points onto the constellation points of the 2D decision map. 
   By performing the above-described two functions, QAM mapper  340  substantially transforms the received streams of phase and amplitude differentials into a stream of logical values representing the constellation points of the 2D decision map. This stream of logical values is applied to decoder  342 , which is configured to recover from that stream the original bit sequence X(n) (see also  FIG. 2 ). More specifically, decoder  342  converts each logical value received from QAM mapper  340  into a corresponding binary value assigned to a set of symbol transitions of the QAM constellation (e.g., one of the constellations shown in  FIGS. 1A-C ) having the requisite amplitude and phase increments. Note that the differential nature of the encoding algorithm makes it unnecessary to determine the exact QAM symbols carried by the communication signal because the encoded data can unequivocally be recovered by correctly ascertaining only the respective amplitude and phase increments for each QAM-symbol pair, and not the exact QAM-symbol pair that produced those increments. 
     FIGS. 4A-C  graphically illustrate the operation of DP  350  when communication system  200  is configured to transmit data using the 4-QAM constellation of  FIG. 1A . More specifically, the data shown in  FIGS. 4A-C  correspond to a system configuration, in which the data transmission rate is 10 GBaud/s, the frequency offset (Δω) is 1 GHz, the optical-source linewidth is 100 MHz, the signal-to-noise ratio (SNR) is 15.3 dB, and the bit-error rate (BER) is 1.5×10 −5 . 
     FIG. 4A  graphically shows the symbol-point distribution corresponding to signals  248   a - b  ( FIG. 2 ) or  348   a - b  ( FIG. 3 ). As already explained above, the presence of a non-zero frequency offset between optical sources  212  and  232  causes the symbol points applied to DP  250  or  350  to form two circular bands, each representing a respective layer of the QAM constellation. 
     FIG. 4B  graphically illustrates the data corresponding to the data of  FIG. 4A  after the phase drift associated with the non-zero frequency offset and the accumulated linewidth-induced phase noise are removed. As seen in  FIG. 4B , removal of the phase drift and linewidth-induced phase noise differentiates the circular bands of  FIG. 4A  into four clusters. 
     FIG. 4C  illustrates the process of constructing a 2D decision map implemented in QAM mapper  340 . More specifically, the data of  FIG. 4A  are processed by amplitude differentiator  310 , angular differentiator  320 , and angular adjustor  330  and the processing results are applied to QAM mapper  340 , where these processing results are transformed by plotting each symbol-transition point on a 2D plane using the point&#39;s folded phase differential value as the abscissa and its absolute amplitude-differential value as the ordinate. Note that four symbol-transition-point clusters are evident in  FIG. 4C . The center of mass of each such cluster represents the respective constellation point of the 2D decision map. After the coordinates of the constellation points on the 2D decision map have been determined, QAM mapper  340  maps each subsequently received symbol-transition point onto one of those constellation points and outputs the corresponding logical value to decoder  342 , which then recovers the encoded data. 
     FIGS. 5A-C  graphically illustrate the operation of DP  350  when communication system  200  is configured to transmit data using the 8-QAM constellation of  FIG. 1B . More specifically, the data shown in  FIGS. 5A-C  correspond to a system configuration, in which the data transmission rate is 10 GBaud/s, Δω=1 GHz, the optical-source linewidth is 50 MHz, SNR=20.1 dB, and BER=2×10 −5 . 
     FIG. 5A  graphically shows the symbol-point distribution corresponding to signals  248   a - b  ( FIG. 2 ) or  348   a - b  ( FIG. 3 ). Similar to  FIG. 4A , a non-zero frequency offset between optical sources  212  and  232  manifests itself in the formation of two circular bands representing the layers of the 8-QAM constellation. 
     FIG. 5B  graphically illustrates the data corresponding to the data of  FIG. 5A  after the phase drift associated with the non-zero frequency offset and the accumulated linewidth-induced phase noise are removed. As seen in  FIG. 5B , removal of the phase drift and linewidth-induced phase noise differentiates the circular bands of  FIG. 5A  into eight clusters. 
     FIG. 5C  illustrates the process of constructing a 2D decision map implemented in QAM mapper  340 . More specifically, the data of  FIG. 5A  are processed by amplitude differentiator  310 , angular differentiator  320 , and angular adjustor  330  and the processing results are applied to QAM mapper  340 , where these processing results are transformed by plotting each symbol-transition point on a 2D plane using the point&#39;s folded phase differential value as the abscissa and its absolute amplitude-differential value as the ordinate. Note that, after taking into account the wrap-around effect for two borderline clusters, eight symbol-transition-point clusters are evident in  FIG. 5C . The center of mass of each such cluster represents the respective constellation point of the 2D decision map. After the coordinates of the constellation points on the 2D decision map have been determined, QAM mapper  340  maps each subsequently received symbol-transition point onto one of those constellation points and outputs the corresponding logical value to decoder  342 , which then recovers the encoded data. 
     FIGS. 6A-C  graphically illustrate the operation of DP  350  when communication system  200  is configured to transmit data using the 16-QAM constellation of  FIG. 1C . More specifically, the data shown in  FIGS. 6A-C  correspond to a system configuration, in which the data transmission rate is 10 GBaud/s, Δω=1 GHz, the optical-source linewidth is 10 MHz, SNR=22 dB, and BER=8.3×10 −5 . 
     FIG. 6A  graphically shows the symbol-point distribution corresponding to signals  248   a - b  ( FIG. 2 ) or  348   a - b  ( FIG. 3 ). Similar to  FIGS. 4A and 5A , a non-zero frequency offset between optical sources  212  and  232  manifests itself in the formation of two circular bands representing the layers of the 16-QAM constellation. 
     FIG. 6B  graphically illustrates the data corresponding to the data of  FIG. 6A  after the phase drift associated with the non-zero frequency offset and the accumulated linewidth-induced phase noise are removed. As seen in  FIG. 6B , removal of the phase drift and linewidth-induced phase noise differentiates the circular bands of  FIG. 6A  into sixteen clusters. 
     FIG. 6C  illustrates the process of constructing a 2D decision map implemented in QAM mapper  340 . More specifically, the data of  FIG. 6A  are processed by amplitude differentiator  310 , angular differentiator  320 , and angular adjustor  330  and the processing results are applied to QAM mapper  340 , where these processing results are transformed by plotting each symbol-transition point on a 2D plane using the point&#39;s folded phase differential value as the abscissa and its absolute amplitude-differential value as the ordinate. Note that, after taking into account the wrap-around effect for two borderline clusters, sixteen symbol-transition-point clusters are evident in  FIG. 6C . The center of mass of each such cluster represents the respective constellation point of the 2D decision map. After the coordinates of the constellation points on the 2D decision map have been determined, QAM mapper  340  maps each subsequently received symbol-transition point onto one of those constellation points and outputs the corresponding logical value to decoder  342 , which then recovers the encoded data. 
     FIG. 7  shows a block diagram of a DP  750  that can be used as DP  250  according to another embodiment of the invention. In the foregoing description of DP  350 , it is implicit that OPS  238  ( FIG. 2 ) introduces a phase shift that is substantially 90 degrees, without significant (e.g., exceeding 15 degrees) deviations from that value. However, in certain embodiments of system  200 , OPS  238  might have imperfections resulting in a deviation of the introduced phase shift from the desired 90 degrees. Such deviation is often referred to as the conjugate phase misalignment (CPM). If the value of CPM is relatively large, then the processing implemented in DP  350  might produce erroneous results. In contrast, the use of DP  750  in system  200  in place of DP  250  enables receiver  230  to recover the transmitted data even in the presence of relatively large CPM. DP  750  can generally be used for processing communication signals having any amount of CPM, including those having substantially no CPM. 
   DP  750  incorporates an instance of DP  350 , which is preceded in DP  750  by a correction block  710 . DP  750  is configured to receive digital input signals  708   a - b  that are analogous to, e.g., digital signals  248   a - b  (see  FIG. 2 ), respectively, produced in receiver  230  in the presence of CPM. Signals  708   a - b  are applied to correction block  710 , which determines the amount of CPM in those signals and transforms them into signals  748   a - b  that would substantially be produced in receiver  230  in the absence of CPM. Signals  748   a - b  are then applied to DP  350 , where these signals are processed substantially as described above to recover the original bit sequence X(n). 
   To illustrate the data processing implemented in correction block  710 , let us suppose that, in the absence of CPM, signals  708   a - b  are described by Eqs. (6):
 
 y   708a ( n )= ReA   B ( n ) e   j(Φ     W     +ΔωnT) ≡ρ i ( n )  (6a)
 
 y   708b ( n )= ImA   B ( n ) e   j(Φ     W     +ΔωnT) ≡ρ q ( n )  (6b)
 
where y 708a (n) and y 708b (n) are the values of signals  708   a - b , respectively, in the n-th time slot; A B (n) is the respective constellation symbol; Φ W =Φ S +Φ LO , where Φ S  is the linewidth-related phase noise in the communication signal (e.g., signal  228 ) and Φ LO  is the linewidth-related phase noise in the LO signal (e.g., signal  234 ); Δω=ω LO −ω S , and T is the symbol period. In the presence of CPM, signal  708   b  is distorted to become:
 
 y   708b ( n )′=ρ i ( n )sin φ+ρ q ( n )cos φ≡ρ′ q ( n )  (7)
 
where φ is the value of CPM. Making the substitution given by Eqs. (8),
 
 u ≡√{square root over (2)}(ρ i +ρ′ q )  (8a)
 
 v ≡√{square root over (2)}(ρ′ q −ρ i )  (8b)
 
one finds that the following is true:
 
                       u   2         ρ   2     ⁡     (     1   +     sin   ⁢           ⁢   ϕ       )         +       v   2         ρ   2     ⁡     (     1   -     sin   ⁢           ⁢   ϕ       )           =   1           (   9   )               
where ρ 2 =ρ i   2 +ρ q   2 . Since Eq. (9) is an equation of an ellipse, it implies that one effect of CPM on data processing is that the CPM transforms the generally circular QAM layer bands shown, e.g., in  FIGS. 4A ,  5 A, and  6 A, into generally elliptical bands having their ellipse axes oriented at 45 degrees with respect to the coordinate axes of the complex plane.
 
   In one embodiment, correction block  710  has a CPM calculator  712  and a signal corrector  714 . CPM calculator  712  is configured to determine the value of φ and supply the determined value to signal corrector  714 . For example, in one embodiment, CPM calculator  712  accumulates a relatively large number of symbol points that are sufficient to form the above-mentioned elliptical bands representing the QAM constellation layers. CPM calculator  712  then fits each of those bands using, e.g., Eq. (9), and determines the value of φ from the fit results. Using the value of φ determined by CPM calculator  712 , signal corrector  714  then transforms signals  708   a - b  into signals  748   a - b , respectively. For example, in one embodiment, signal corrector  714  determines the value of ρ i (n) using signal  708   a  and outputs that value as signal  748   a . Signal corrector  714  also determines the value of ρ q (n) and outputs that value as signal  748   b , with the value of ρ q (n) determined using: (i) signal  708   b , (ii) the value of φ received from CPM calculator  712 , (iii) the determined value ρ i (n), and (iv) Eq. (7). 
   In yet another embodiment, correction block  710  can be modified to correct an aggregate crosstalk between the I (in-phase) and Q (quadrature-phase) signal components as well as certain other signal imperfections, e.g., originating at the transmitter and/or in the communication link. For example, let us suppose that signals  708   a - b  are distorted as follows:
 
 y   708 ( n )′= y   708a ( n )+ jy   708b ( n )=(ρ i −δ i )+ j [β(ρ i  sin φ+ρ q  cos φ)−δ q ]  (10)
 
where δ i  and δ q  are the offsets of the undistorted I (i.e., ρ i ) and Q (i.e., ρ q ) signal components, respectively; β is the relative scaling distortion of the I and Q signal components; and φ is the aggregate crosstalk value, which is equivalent to CPM when the offsets are zero and β=1 (see also Eq. (7)). The data described by Eq. (10) generally form an elliptical band that can be described by Eqs. (11):
 
   
     
       
         
           
             
               
                 
                   
                     ay 
                     
                       708 
                       ⁢ 
                       a 
                     
                     2 
                   
                   + 
                   
                     
                       by 
                       
                         708 
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                         a 
                       
                     
                     ⁢ 
                     
                       y 
                       
                         708 
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                         b 
                       
                     
                   
                   + 
                   
                     cy 
                     
                       708 
                       ⁢ 
                       b 
                     
                     2 
                   
                   + 
                   
                     dy 
                     
                       708 
                       ⁢ 
                       a 
                     
                   
                   + 
                   
                     ey 
                     
                       708 
                       ⁢ 
                       b 
                     
                   
                   + 
                   f 
                 
                 = 
                 0 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 where 
                 ⁢ 
                 
                   : 
                 
               
             
             
               
                   
               
             
           
           
             
               
                 β 
                 = 
                 
                   1 
                   / 
                   
                     c 
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
           
             
               
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ϕ 
                 
                 = 
                 
                   
                     - 
                     b 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     β 
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                     2 
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   c 
                 
                 ) 
               
             
           
           
             
               
                 
                   δ 
                   q 
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       e 
                     
                     - 
                     bd 
                   
                   
                     
                       4 
                       ⁢ 
                       c 
                     
                     - 
                     
                       b 
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   d 
                 
                 ) 
               
             
           
           
             
               
                 
                   δ 
                   i 
                 
                 = 
                 
                   d 
                   - 
                   
                     
                       δ 
                       q 
                     
                     ⁢ 
                     b 
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   e 
                 
                 ) 
               
             
           
         
       
     
   
   Correction block  710  is configured to accumulate a relatively large number of symbol points that are sufficient to form the elliptical band(s) described by Eqs. (11). Correction block  710  then fits each of those bands using Eq. (11a), and determines parameter values a, b, c, d, and e from the fit results. Using these parameter values, correction block  710  then calculates the values of δ i , δ q , β, and φ using Eqs. (11b-11e). Finally, correction block  710  uses these calculated values to remove the above-specified distortions and produce the undistorted I (ρ i ) and Q (ρ q ) signal components as signals  748   a - b , respectively. In one configuration, the processing described by Eqs. (10-11) can be performed off-line, e.g., during a calibration procedure. Alternatively or in addition, correction block  710  can be configured to send the estimated distortion parameters back to transmitter  210  and OPS  238  (see  FIG. 2 ) to fine tune their respective configurations to reduce the amounts of signal distortion. 
   While this invention has been described with reference to illustrative embodiments, this description is not intended to be construed in a limiting sense. For example, different QAM constellations having different numbers of layers and/or constellation symbols can similarly be used in other configurations of communication system  200 . Various modifications of the described embodiments, as well as other embodiments of the invention, which are apparent to persons skilled in the art to which the invention pertains are deemed to lie within the principle and scope of the invention as expressed in the following claims. 
   Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments necessarily mutually exclusive of other embodiments. The same applies to the term “implementation.” 
   Embodiments of the present invention may be implemented as circuit-based processes, including possible implementation on a single integrated circuit. As would be apparent to one skilled in the art, various functions of circuit elements may also be implemented as processing steps in a software program. Such software may be employed in, for example, a programmable digital signal processor, micro-controller, or general-purpose computer. 
   Unless explicitly stated otherwise, each numerical value and range should be interpreted as being approximate as if the word “about” or “approximately” preceded the value of the value or range. 
   It will be further understood that various changes in the details, materials, and arrangements of the parts which have been described and illustrated in order to explain the nature of this invention may be made by those skilled in the art without departing from the scope of the invention as expressed in the following claims. 
   It should be understood that the steps of the exemplary methods set forth herein are not necessarily required to be performed in the order described, and the order of the steps of such methods should be understood to be merely exemplary. Likewise, additional steps may be included in such methods, and certain steps may be omitted or combined, in methods consistent with various embodiments of the present invention.