Patent Publication Number: US-11662645-B2

Title: Multi-stage probabilistic signal shaping

Description:
BACKGROUND 
     Field 
     Various example embodiments relate to optical communication equipment and, more specifically but not exclusively, to signal encoders and decoders. 
     Description of the Related Art 
     This section introduces aspects that may help facilitate a better understanding of the disclosure. Accordingly, the statements of this section are to be read in this light and are not to be understood as admissions about what is in the prior art or what is not in the prior art. 
     Probabilistic signal shaping can beneficially provide energy savings often referred to as the shaping gain. In a typical implementation of probabilistic signal shaping (e.g., probabilistic constellation shaping, PCS), constellation symbols of relatively large energy are transmitted less frequently than constellation symbols of relatively small energy. For a linear communication channel, the shaping gain can theoretically approach 1.53 dB. 
     SUMMARY OF SOME SPECIFIC EMBODIMENTS 
     Disclosed herein are various embodiments of a shaping encoder capable of improving the performance of PCS in nonlinear optical channels by performing the shaping in two or more stages. In an example embodiment, a first stage employs a shaping code of a relatively short block length, which is typically beneficial for nonlinear optical channels but may cause a significant penalty in the energy efficiency. A second stage then employs a shaping code of a much larger block length, which significantly reduces or erases the penalty associated with the short block length of the first stage while providing an additional benefit of good performance in substantially linear optical channels. As a result, various embodiments may beneficially be used for both linear and nonlinear optical channels. 
     In some embodiments, a shaping encoder may employ an electronic codebook that enables encoding equivalent to that of a two-stage shaping encoder to be performed using a corresponding search-and-match operation. 
     In at least some embodiments, the disclosed shaping encoder(s) may have relatively low circuit-implementation complexity and/or relatively low cost and provide relatively high energy efficiency and relatively high shaping gain for a variety of optical channels, including but not limited to the legacy dispersion-managed fiber-optic links. 
     According to an example embodiment, provided is an apparatus, comprising: a digital encoder having, at least, first and second digital stages to produce a stream of symbols from a bitstream, the first digital stage being configured to separately encode segments of a same number of bits of the bitstream into first sequences of symbols such that each of the first sequences has a same first length and a respective total energy of the symbols therein lower than a threshold, the second digital stage being configured to encode blocks of the first sequences into second sequences of symbols, each of the blocks having a same number of first sequences therein, each of the second sequences having a same second length; and wherein a total energy of symbols of any of the second sequences divided by the second length is smaller than the threshold divided by the first length. 
     According to another example embodiment, provided is an apparatus comprising an optical data transmitter that comprises an optical front end and a digital shaping encoder, the digital shaping encoder being configured to: encode a bitstream into a stream of symbols of a constellation; and cause the optical front end to produce a modulated optical signal carrying the stream of symbols; and wherein the digital shaping encoder is configured to generate a symbol sequence for the stream in response to a segment of the bitstream, the symbol sequence including an integer number of non-overlapping subsequences of a fixed length, any one of the subsequences having a transmit energy not exceeding a first threshold energy, the symbol sequence having a transmit energy not exceeding a second threshold energy and being generated such that some symbols of higher energy have a lower probability of being generated by the digital shaping encoder than other symbols of lower energy, the first threshold energy being smaller than an average energy of an unshaped sequence of the fixed length of the symbols of the constellation, the second threshold energy being smaller than the integer number of the first threshold energies, the integer number being greater than one. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other aspects, features, and benefits of various disclosed embodiments will become more fully apparent, by way of example, from the following detailed description and the accompanying drawings, in which: 
         FIG.  1    shows a block diagram of a communication system in which various embodiments can be practiced; 
         FIG.  2    graphically illustrates certain characteristics of an example modulated optical signal that can be transmitted in the communication system of  FIG.  1    according to an embodiment; 
         FIGS.  3 A- 3 B  graphically illustrate example probability distributions that can be applied to an 8-ary Pulse-Amplitude-Modulation (8-PAM) constellation in the communication system of  FIG.  1    in some embodiments; 
         FIG.  4    shows a table of kurtosis values for several modulation formats that can be used in the communication system of  FIG.  1    in some embodiments; 
         FIG.  5    graphically illustrates an 8-PAM constellation that can be used in the communication system of  FIG.  1    according to an embodiment; 
         FIG.  6    shows a block diagram of a multistage shaping encoder that can be used in the communication system of  FIG.  1    according to an embodiment; 
         FIG.  7    shows a block diagram of a multistage shaping encoder that can be used in the communication system of  FIG.  1    according to an alternative embodiment; 
         FIG.  8    shows a block diagram of a shaping encoder that can be used in the communication system of  FIG.  1    according to yet another embodiment; 
         FIGS.  9 A- 9 D  illustrate an example codebook-construction procedure that can be used to construct a shaping codebook for the shaping encoder of  FIG.  8    according to an embodiment; 
         FIG.  10    shows a block diagram of a shaping decoder that can be used in the communication system of  FIG.  1    according to an embodiment; 
         FIG.  11    graphically illustrates example signal-to-noise ratio (SNR) improvements that can be achieved in the communication system of  FIG.  1    according to an embodiment; 
         FIG.  12    shows a block diagram of an optical transmitter that can be used in the communication system of  FIG.  1    according to an embodiment; and 
         FIG.  13    shows a block diagram of an optical receiver that can be used in the communication system of  FIG.  1    according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Herein, the generation and/or transmission of a symbol stream in which various symbols appear with different probabilities even though the various input data segments encoded onto said symbol stream have about equal probabilities is referred to as probabilistic constellation shaping (PCS). Often, preferable types of PCS generate symbol streams in which higher energy symbols are less probable than lower energy symbols. In some embodiments, forward-error-correction (FEC) encoding may be used in a manner that causes such energy shaping to be largely maintained in the corresponding FEC-encoded symbol stream. In embodiments of coherent optical fiber communication systems and optical data transmitters and receivers thereof, the PCS can advantageously be used, e.g., (i) to lower degradations related to nonlinear optical effects, which are more prominent at larger energies, (ii) to lower a required signal-to-noise ratio (SNR) for a given transmission distance, and/or (iii) to enable higher information communication rates. Various embodiments may apply the PCS to various quadrature amplitude modulation (QAM) constellations and/or other suitable symbol constellations. 
     An important benefit of probabilistic signal shaping is that the amount of shaping (e.g., specific characteristics of the corresponding shaping code) can be selected to optimize a desired set of performance characteristics of any given communication channel. For example, depending on specified performance requirements, probabilistic signal shaping can be adjusted to achieve an optimal (e.g., the highest) spectral efficiency or an optimal (e.g., the highest) net bit-rate for any given transmission distance. 
       FIG.  1    shows a block diagram of a communication system  100  in which various embodiments can be practiced. System  100  comprises an optical data transmitter  104  and an optical data receiver  108  that are coupled to one another by way of a communication link  106 . In an example embodiment, communication link  106  can be implemented using one or more spans of optical fiber or fiber-optic cable. 
     System  100  carries out PCS using (i) an electronic encoder  110  appropriately interfaced with an electrical-to-optical (E/O) converter (also sometimes referred-to as the optical transmitter front end)  140  at transmitter  104 , and (ii) an optical-to-electrical (O/E) converter (also sometimes referred-to as the optical receiver front end)  150  appropriately interfaced with an electronic decoder  160  at receiver  108 . 
     In an example embodiment, one or both of electronic encoder  110  and electronic decoder  160  can be implemented using a respective digital signal processor (DSP) or a portion thereof. 
     Electronic encoder  110  operates to generate one or more electrical radio-frequency (RF) signals  138  in response to receiving input data  102 . In response to electrical RF signal(s)  138 , the optical modulator(s) of the E/O converter  140  then generate(s) a corresponding modulated optical signal  142  suitable for transmission over link  106  and having encoded thereon a segment of the stream of input data  102 . In an example embodiment, E/O converter  140  comprises an optical modulator that can be implemented as known in the pertinent art, e.g., using: (i) a laser configured to generate an optical carrier wave; (ii) one or more modulator configured to generate modulated optical signal  142  by modulating the optical carrier wave generated by the laser; and (iii) one or more driver circuits, e.g., analog electrical circuits, configured to electrically drive the modulator(s) using electrical RF signal(s)  138 , thereby causing the E/O converter  140  to generate modulated optical signal  142 . Depending on the embodiment, the optical modulator(s) used in E/O converter  140  can be implemented using one or more optical IQ modulators, Mach-Zehnder modulators, amplitude modulators, phase modulators, and/or intensity modulators. The optical modulators may also be nested to generate a modulated optical carrier whose orthogonal polarization components are separately modulated. In some embodiments, E/O converter  140  may employ directly modulated lasers, e.g., laser diodes configured to generate modulated optical signals in response to modulated electrical currents that drive the diodes. 
     Communication link  106  typically imparts noise and other linear and/or nonlinear signal impairments onto signal  142  and delivers a resulting impaired (e.g., noisier) signal  142 ′ to O/E converter  150  of receiver  108 . O/E converter  150  operates to convert optical signal  142 ′ into one or more corresponding electrical RF signals  152 . Electronic decoder  160  then uses decoding processing to recover data  102  from the electrical RF signal(s)  152 . 
     In some embodiments, O/E converter  150  comprises an optical demodulator that can be configured, as known in the pertinent art, for coherent (e.g., intradyne or homodyne) detection of signal  142 ′. In such embodiments, O/E converter  150  may include: (i) an optical local-oscillator (LO) source; (ii) an optical hybrid configured to optically mix signal  142 ′ and the LO signal generated by the optical LO source; and (iii) one or more photodetectors (e.g., photodiodes) configured to convert optical interference signals generated by the optical hybrid into the corresponding components of electrical RF signal(s)  152 . 
     In some other embodiments, O/E converter  150  comprises an optical demodulator that can be configured for direct (e.g., square law, intensity, optical power) detection of signal  142 ′. In such embodiments, O/E converter  150  may include a photodiode configured to generate electrical RF signal  152  to be proportional to the intensity (optical power, squared amplitude of the electric field) of signal  142 ′, i.e., without mixing the received signal  142 ′ with light of a local optical source. 
       FIG.  2    graphically illustrates certain characteristics of an example optical signal  142  that can be transmitted in communication system  100  ( FIG.  1   ) according to an embodiment. In this particular example, optical data transmitter  104  is configured to generate optical signal  142  using a 64-QAM constellation  210 , the constellation symbols of which are shown as points on an IQ plane  202 , wherein the I and Q coordinate axes are the in-phase and quadrature-phase dimensions, respectively, of the IQ plane. In an example embodiments, the constellation-symbol amplitudes in constellation  210  in each of the I and Q dimensions thereof can be represented by the integers −7, −5, −3, −1, +1, +3, +5, and +7. 
     A three-dimensional (3D) bar chart  220  also shown in  FIG.  2    graphically illustrates relative probability of occurrence of the different constellation symbols of constellation  210  in optical signal  142 . In particular, chart  220  indicates that electronic encoder  110  may operate to generate a constellation-symbol stream for being carried by optical signal  142  such that constellation symbols of a first transmit energy occur with higher probability than constellation symbols of a second transmit energy that is greater than the first transmit energy. 
       FIGS.  3 A- 3 B  graphically illustrate example probability distributions that can be applied by electronic encoder  110  to the I or Q dimension of constellation  210  in some embodiments. Distribution  310  shown in  FIG.  3 A  is a Maxwell-Boltzmann (MB) distribution that can be expressed using Eq. (1):
 
 P ∝exp(−λ·| x|   2 )  (1)
 
where P is the probability of a constellation symbol; x is the amplitude of the constellation symbol in the I or Q dimension of the IQ plane  202 ; and λ is a positive real value. The value of λ is related to the rate H (e.g., entropy) of the shaping code used in electronic encoder  110  and can be selected, e.g., based on a desired code-rate value. Distribution  320  shown in  FIG.  3 B  is a generalized Maxwell-Boltzmann (GMB) distribution that can be expressed using Eq. (2):
 
 P ∝exp(−λ˜| x|   α )  (2)
 
where the parameter a is allowed to have any positive real value. The MB distribution is a special case of the GMB distribution, wherein α=2. In the GMB example shown in  FIG.  3 B , α=3.5. As the value of a increases above α=2, the GMB distribution becomes closer to a uniform distribution, e.g., as visually evident from a comparison of distributions  310  and  320  and the flatter middle portion of the latter. For the same shaping-code rate H, distribution  320  typically results in a larger average energy of optical signal  142  than distribution  310 . For example, for H=2.6 bit/symbol, the average-energy difference between distributions  310  and  320  is approximately 0.15 dB.
 
     Distributions  310  and  320  are also characterized by different respective values of kurtosis (hereafter denoted as K). More specifically, for H=2.6, distribution  310  has K=0.44, whereas distribution  320  has K=0.31. These kurtosis values have been calculated using the absolute-valued signals that have a positive mean value (or, equivalently, using only the positive portion of the constellation). Kurtosis values calculated without the absolute-value operator, e.g., for signals having the zero mean value, may be different. 
     In probability theory and statistics, kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued variable. Kurtosis can be used, e.g., to quantify the shape of a probability distribution. Herein, kurtosis refers to the standardized fourth-order moment of the probability distribution. 
       FIG.  4    shows a table of kurtosis values for several example modulation formats that can be used in system  100  in some embodiments. Therein, PS stands for “probabilistically shaped” and indicates the use of an MB distribution. The shown kurtosis values have been calculated using absolute-valued signals, as mentioned above. Different MB distributions corresponding to the different PS 16-QAM constellation entries in the table of  FIG.  4    correspond to different respective values of A, which are manifested by the different respective values of the shaping-code rate H shown therein. The entries of the table of  FIG.  4    indicate that, for the same constellation, e.g., 16-QAM, kurtosis can be a nonlinear function of the shaping-code rate H. The entries of the table further indicate that kurtosis may also depend on the constellation size. 
     For an additive white Gaussian noise (AWGN) channel, an optimal distribution of QAM constellation symbols may be an MB distribution, such as the distribution  310  ( FIG.  3 A ). However, for at least some optical fiber channels, a nonlinear interference noise (NLIN) may also be present in addition to the AWGN. If optical-signal power increases, then the NLIN increases. If kurtosis increases, then the NLIN increases. As such, kurtosis of the employed modulation format may be used as an approximate measure of the relative amount of NLIN in the corresponding optical channel. However, it should be noted that, for a given modulation format, kurtosis does not vary with the optical-signal power. Both AWGN and NLIN detrimentally reduce the effective SNR at the receiver. 
     As illustrated by the table of  FIG.  4   , at least some MB distributions may have higher kurtosis values than the corresponding uniform distribution(s). As a result, the use of MB distributions in some communication systems, e.g., systems employing highly nonlinear optical fiber links, such as the legacy dispersion-managed links that are typically operated with a relatively high optical launch power, may impose a significant SNR penalty on the received optical signal, e.g., optical signal  142 ′ (see  FIG.  1   ), due to the NLIN. At least some GMB distributions may in some cases reduce the SNR penalty with respect to that of the corresponding MB distribution(s), e.g., as indicated by the above-mentioned kurtosis values corresponding to distributions  310  and  320 . 
       FIG.  5    graphically illustrates an 8-ary Pulse-Amplitude-Modulation (8-PAM) constellation  510  that can be used to program electronic encoder  110  according to an embodiment. The eight constellation points of constellation  510  are all located on the I-axis of the I-Q plane. Each of the constellation points is used to encode three bits. An example of 3-bit bit-words assigned to different constellation points of constellation  510  is shown above the I-axis in  FIG.  5   . The relative amplitudes corresponding to the different constellation points of constellation  510  are shown below the I-axis in  FIG.  5    and can be represented by the integers −7, −5, −3, −1, +1, +3, +5, and +7. 
     In the illustrated embodiment, the constellation-point labeling is in accordance with a reflected double-Gray mapping scheme, in which the constellation points located in the positive I-half of constellation  510  have binary-amplitude labels (i.e., the labels that do not include the sign bit) generated using conventional double-Gray mapping, while the constellation points located in the negative I-half of the constellation have binary labels generated by flipping the sign bits of the corresponding constellation points located in the positive I-half. With this type of mapping, the amplitude labels of the constellation points are symmetric, and the sign bits of the constellation points are anti-symmetric with respect to the origin of the I-axis. 
     In some alternative embodiments, a similar approach can be used to generate binary labels (i.e., assigned bit-words) for a constellation or a constellation portion that uses the Q dimension of the complex I-Q plane. For example, constellation-point labeling for the 64-QAM constellation  210  ( FIG.  2   ) can be implemented using the labeling shown in  FIG.  5   , with such labeling being applied to both I and Q dimensions of constellation  210 . 
     For illustration purposes and without any implied limitations, the description of some example embodiments is given herein below in reference to a 4-PAM constellation. A 4-PAM constellation is similar to constellation  510 , but has four distinct constellation points distributed along a 1-dimensional line. Herein, we assume that the constellation points are arranged equidistantly with respect to each other and symmetrically around the origin (zero) and can be represented by the integers −3, −1, 1, and 3. An extension of the presented description to any 2 m -PAM constellation is relatively straightforward, where m is an integer greater than one. Further extensions to other possible dimensions of optical signal  142 , e.g., time, polarization, spatial mode, and frequency/wavelength, will be apparent to a person of ordinary skill in the pertinent art without any undue experimentation. For example, two 2 m -PAM symbols can be combined to construct a 2 2m -QAM symbol by modulating each of the I and Q dimensions of the optical signal  142  independently with a respective 2 m -PAM symbol. A person of ordinary skill in the art will understand that, for example, the 64-QAM constellation  210  of  FIG.  2    corresponds to m=3. 
     As an illustrative example, let us consider a single-stage shaping encoder that produces a length-n symbol sequence X=[x 1  . . . x n ] in response to a length-k binary sequence B=[b 1  . . . b k ]. The total energy of the sequence X can be calculated as: 
                          X        2     =       ∑     i   =   1       i   =   n             ❘   &#34;\[LeftBracketingBar]&#34;       x   i       ❘   &#34;\[RightBracketingBar]&#34;       2               (   3   )               
For example, for 4-PAM, if n=8 and X=[1, 1, 1, 1, 1, 3, 3, 3], then Eq. (3) results in ∥x∥ 2 =32 (=1+1+1+1+1+9+9+9). In general, such a shaping encoder can be configured to apply a shaping code that, for a selected fixed n, can make the total energy ∥X∥ 2  of the sequence X to be not larger than a selected fixed threshold energy E max , i.e., any sequence X of length n at the output of the shaping encoder fulfills the inequality:
 
∥ X∥   2   ≤E   max   &lt;U   n   (4)
 
where U n  is the average energy of the unshaped length-n symbol sequence X, i.e., a sequence in which different constellation symbols occur with an approximately equal probability. For the above example of 4-PAM and n=8, said average energy is U n =40.
 
     Since Eq. (3) is analogous to the expression for the squared Euclidean norm of the vector X=(x 1  . . . x n ), wherein x 1 , . . . , x n  are the Cartesian coordinates of the vector in the n-dimensional space, it can be said that all vectors X produced by such a shaping encoder are enclosed by the n-dimensional sphere of radius √{square root over (E max )} centered on the origin of the coordinate system. Such shaping codes are sometimes referred to in the literature as “sphere shaping” codes. Representative examples of algorithms that can be adapted for implementing sphere shaping are described, e.g., in (1) Patrick Schulte, Georg Böcherer, “Constant Composition Distribution Matching,” IEEE TRANSACTIONS ON INFORMATION THEORY, 2016, vol. 62, No. 1, pp. 430-434; and (2) Yunus Can Gültekin, et al., “Approximate Enumerative Sphere Shaping,” 2018 IEEE International Symposium on Information Theory (ISIT), pp. 676-680, both of which are incorporated herein by reference in their entirety. 
     For a given constellation (e.g.,  510 ,  FIG.  5   ) and a fixed code rate H (=k/n bits per symbol), a sphere-shaping code produces a probability distribution whose kurtosis depends on the block length n of the code. For example, a shorter-length (e.g., length-n 1 ) sphere-shaping code typically produces a probability distribution with a smaller kurtosis than a longer-length (e.g., length-n 2 , n 2 &gt;n 1 ) sphere-shaping code. However, the shorter-length sphere-shaping code is typically less energy-efficient than the longer-length sphere-shaping code or a corresponding MB code. Circuit-implementation complexity typically increases with the block length n of the sphere-shaping code. 
     It would be desirable to provide, e.g., for communication system  100 , a shaping encoder that has at least some of the following characteristics: (i) has relatively low circuit-implementation complexity and/or relatively low cost; (ii) results in relatively high energy efficiency for transmission of payload data; (iii) provides a relatively high shaping gain for NLIN-impaired optical channels while being also capable of providing good performance for substantially linear optical channels; (iv) lends itself to convenient interfacing with an FEC encoder; (v) enables optical-signal transmission over a relatively large distance, e.g., larger than the reach of the corresponding unshaped optical signal; and (iv) can support relatively high data throughput. 
       FIG.  6    shows a block diagram of a multistage shaping encoder  600  that can be used in electronic encoder  110  according to an embodiment. Shaping encoder  600  has a nested architecture that is illustratively shown in  FIG.  6    as comprising L nested stages  602   1 - 602   L , where L is an integer greater than two. In some alternative embodiments, shaping encoder  600  may have only two stages, i.e., correspond to L=2. The first encoder stage  602   1  comprises a shaping encoder  610   1 . Each subsequent encoder stage  602   i  (where i=2, . . . , L) comprises the previous encoder stage  602   i-1  serially connected with a constellation demapper  620   i  and a shaping encoder  610   i . For example, encoder stage  602   2  comprises encoder stage  602   1  serially connected with constellation demapper  620   2  and shaping encoder  610   2 , and so on. 
     For illustration purposes, multistage shaping encoder  600  is described below as having been programmed to use a 2 m -PAM constellation, e.g., constellation  510  ( FIG.  5   ). 
     In an example embodiment, shaping encoder  610   1  operates to apply a first sphere-shaping code to an input bitstream  608   1  to generate an output amplitude stream  612   1  of positive amplitudes. The first sphere-shaping code has a rate H 1 =k 1 /n 1  and a threshold energy E 1  and is applied to bitstream  608   1  in a segment-by-segment manner. More specifically, in response to a first received segment of k 1  bits of bitstream  608   1 , shaping encoder  610   1  generates a corresponding sequence of n 1  positive amplitudes for amplitude stream  612   1 . In response to a next non-overlapping segment of k 1  bits of bitstream  608   1 , shaping encoder  610   1  generates a corresponding next sequence of n 1  positive amplitudes for amplitude stream  612   1 , which is appended to the previously outputted sequence of n 1  positive amplitudes, and so on. Amplitude stream  612   1  may include only the positive amplitudes of the operative 2 m -PAM constellation. For example, if constellation  510  ( FIG.  5   ) is the operative constellation, then amplitude stream  612   1  may include only the amplitudes 1, 3, 5, and 7 (see  FIG.  5   ). If the operative constellation is 4-PAM, then amplitude stream  612   1  may include only the amplitudes 1 and 3. The first sphere-shaping code typically causes a smaller amplitude to occur in amplitude stream  612   1  with a higher relative probability than a larger amplitude. 
     Demapper  620   2  operates to convert amplitude stream  612   1  into a corresponding bitstream  608   2  using the amplitude bits of the binary labels of the corresponding constellation points. In the example of constellation  510  ( FIG.  5   ), the amplitude bits are the two least significant bits (LSBs) of each binary label. In this case, each amplitude 1 is converted by demapper  620   2  into the binary “11” for bitstream  608   2 ; each amplitude 3 is converted by demapper  620   2  into the binary “10” for bitstream  608   2 ; each amplitude 5 is converted by demapper  620   2  into the binary “01” for bitstream  608   2 ; and each amplitude 7 is converted by demapper  620   2  into the binary “00” for bitstream  608   2 . 
     Shaping encoder  610   2  operates to apply a second sphere-shaping code to bitstream  608   2  to generate an output amplitude stream  612   2  of positive amplitudes. The second sphere-shaping code has a code rate H 2 =k 2 /n 2  and a threshold energy E 2  and is applied to bitstream  608   2  in a segment-by-segment manner. More specifically, in response to a first segment of k 2  bits of bitstream  608   2 , shaping encoder  610   2  generates a corresponding sequence of n 2  positive amplitudes for amplitude stream  612   2 . In response to a next non-overlapping segment of k 2  bits of bitstream  608   2 , shaping encoder  610   2  generates a corresponding next sequence of n 2  positive amplitudes for amplitude stream  612   2 , which is appended to the previously outputted sequence of n 2  positive amplitudes, and so on. Similar to amplitude stream  612   1 , amplitude stream  612   2  may include only the positive amplitudes of the operative 2 m -PAM constellation. The second sphere-shaping code typically causes a smaller amplitude to occur in amplitude stream  612   2  with a higher relative probability than a larger amplitude. 
     In some embodiments, the input block lengths for the first and second sphere-shaping codes may be the same, i.e., k 1 =k 2 . 
     In an example embodiment, n 2 &gt;n 1  and n 2 /n 1 =ρ, where ρ is an integer greater than one. The threshold energy E 2  is selected such as to cause a segment of amplitude stream  612   2  to have a smaller transmit energy than that of an equal-length segment of amplitude stream  612   1 , e.g., according to the following inequality:
 
 E   2   &lt;ρE   1   (5)
 
A functional relationship of any next stage  602   i  with the corresponding preceding stage  602   i-1  is generally analogous to the above-described functional relationship between the stages  602   2  and  602   1 .
 
     Demapper  620   L  operates to convert amplitude stream  612   L-1  received form stage  604   L-1  into a corresponding bitstream  608   L  using the amplitude bits of the binary labels of the corresponding constellation points. Shaping encoder  610   L  operates to apply an L-th sphere-shaping code to bitstream  608   L  to generate an output amplitude stream  612   L  of positive amplitudes. The L-th sphere-shaping code has a rate H L =k L /n L  and a threshold energy E L  and is applied to bitstream  608   L  in a segment-by-segment manner. The threshold energy E L  is selected such as to cause a segment of amplitude stream  612   L  to have a smaller transmit energy than that of an equal-length segment of amplitude stream  612   L-1 . 
     Shaping encoder  600  further comprises a multiplier  630  that operates to transform amplitude stream  612   L  into a corresponding amplitude stream  632  carrying signed amplitudes, i.e., positive and negative amplitudes. The conversion is performed in response to a bitstream  628 , in which binary “zeros” and “ones” have about equal probability of occurrence. In response to a binary “zero” received via bitstream  628 , multiplier  630  multiplies the corresponding received amplitude of amplitude stream  612   L  by −1, thereby outputting a negative amplitude for amplitude stream  632 . In response to a binary “one” received via bitstream  628 , multiplier  630  multiplies the corresponding received amplitude of amplitude stream  612   L  by +1, thereby outputting a positive amplitude for amplitude stream  632 . The resulting amplitude stream  632  can then be used to modulate the optical carrier, as already indicated above. 
     In some embodiments, at least some segments of bitstream  628  may include parity bits, e.g., generated by applying a suitable FEC code to bitstream  608   1 . In some embodiments, bitstream  628  can be generated using some of the methods and/or circuits disclosed in U.S. Pat. Nos. 10,727,951, 10,523,400, 10,200,231, and 10,091,046, all of which are incorporated herein by reference in their entirety. 
     In an example embodiment, appropriate selection of L sphere-shaping codes for the L stages of shaping encoder  600  advantageously enables amplitude stream  632  to have a probability distribution that has a desired value of kurtosis, e.g., a value that provides nearly optimal performance for a given communication link  106 . In particular, the probability distribution of amplitude stream  632  can be adjusted, by changing the code selection, to properly mitigate the adverse effects of the given amounts of AWGN and NLIN in link  106 . 
     In some embodiments, shaping encoder  600  can beneficially be implemented using a less-complex digital circuit than that of a comparable single-stage shaping encoder configured to realize a GMB distribution (see Eq. (2)). 
       FIG.  7    shows a block diagram of a multistage shaping encoder  700  that can be used in electronic encoder  110  according to an alternative embodiment. Shaping encoder  700  has a serial architecture and is illustratively shown in  FIG.  6    as comprising L serially connected shaping encoders  610   1 ,  710   2 - 710   L , where L is an integer greater than two. In some alternative embodiments, multistage shaping encoder  700  may have only two serially connected shaping encoders, i.e., encoders  610   1 ,  710   2  for L=2. 
     Shaping encoder  610   1  is already described above in reference to  FIG.  6   . Each of shaping encoders  710   2 - 710   L  differs from shaping encoder  610   1  in that a shaping encoder  710  receives an amplitude stream and outputs another amplitude stream, whereas shaping encoder  610   1  receives a bitstream and outputs an amplitude stream. For example, shaping encoder  710   2  operates to apply a second sphere-shaping code to amplitude stream  612   1  to generate an amplitude stream  712   2  of positive amplitudes. The second sphere-shaping code has a fractional rate R 2 =n 1 /n 2 &lt;1 and a threshold energy E 2  and is applied to amplitude stream  612   1  in a segment-by-segment manner. The threshold energy E 2  is selected such as to cause a segment of amplitude stream  712   2  to have a smaller transmit energy than that of an equal-length segment of amplitude stream  612   1 . 
     Shaping encoder  710   L  operates to apply an L-th sphere-shaping code to amplitude stream  712   L-1  to generate an output amplitude stream  712   L  of positive amplitudes. The L-th sphere-shaping code has a fractional rate R L =n L-1 /n L &lt;1 and a threshold energy E L  and is applied to amplitude stream  712   L-1  in a segment-by-segment manner. The threshold energy E L  is selected such as to cause a segment of amplitude stream  712   L  to have a smaller transmit energy than that of an equal-length segment of amplitude stream  712   L-1 . That is, 
     
       
         
           
             
               
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     Shaping encoder  700  further comprises a multiplier  630  that operates to transform amplitude stream  712   L  into a corresponding amplitude stream  732  carrying signed amplitudes. The conversion is performed in response to bitstream  628 , e.g., as already described above in reference to  FIG.  6   . 
       FIG.  8    shows a block diagram of a shaping encoder  800  that can be used in electronic encoder  110  according to yet another embodiment. Shaping encoder  800  implements the functionality of shaping encoder  700  using a segment matcher  810  and a codebook store  820 . In operation, segment matcher  810  receives a length-k 1  segment of bitstream  608   1 , finds a match to this segment in codebook store  820 , retrieves a corresponding length-n L  amplitude sequence from the codebook store, and outputs the retrieved amplitude sequence to generate a length-n L  segment for amplitude stream  712   L . 
     Codebook store  820  has stored therein a shaping codebook constructed to compress the multistage encoding of shaping encoder  700  into a single search-and-match operation, but otherwise produces the same amplitude stream  712   L  as shaping encoder  700  in response to the same bitstream  608   1 . In an example embodiment, a shaping codebook for codebook store  820  can be constructed, e.g., as described below in reference to an example shaping codebook  900  corresponding to a 4-PAM constellation (i.e., m=2), L=2, k 1 =36, n 1 =4, n 2 =60, E 1 =20, E 2 =124, H=k 1 /n 2 =0.6 bit/amplitude. A person of ordinary skill in the pertinent art will be able to use a similar codebook-construction procedure to construct other shaping codebooks corresponding to other sets of the above-listed parameters. 
       FIGS.  9 A- 9 D  illustrate an example codebook-construction procedure that can be used to construct shaping codebook  900  for codebook store  820  according to an embodiment. 
       FIG.  9 A  shows all possible length-4 amplitude sequences corresponding to m=2. 
       FIG.  9 B  illustrates a codebook-construction step in which the amplitude sequences of  FIG.  9 A  are sorted and pruned to remove the sequences that do not satisfy the E 1 =20 energy constraint, i.e., to remove the length-4 sequences for which ∥X∥ 2 &gt;20. The remaining sequences form a sequence set  910 , which has N=11 different length-4 amplitude sequences. 
       FIG.  9 C  illustrates a codebook-construction step in which all possible combinations of amplitude sequences from the set  910  ( FIG.  9 B ) of length 60 are enumerated. The total number of such combinations is 11 15 =N C , where C=n 2 /n 1 =15. In the decimal format, 11 15  4.18×10 15 . This number of amplitude sequences can be used to encode └log 2 11 15 ┘=51 bits. 
       FIG.  9 D  illustrates a codebook-construction step in which the amplitude sequences of  FIG.  9 C  are sorted and pruned to remove the amplitude sequences that do not satisfy the E 2 =124 energy constraint, i.e., to remove the length-60 amplitude sequences for which ∥X∥ 2 &gt;124. Some more length-60 amplitude sequences for which ∥X∥ 2 =124 may need to be removed from the sorted list of amplitude sequences to limit the remaining number of amplitude sequences in a set  920  exactly to 2 36 ≈6.87×10 10 . These remaining amplitude sequences can then be mapped, one-to-one, to 2 36  different length-36 bit sequences in any suitable manner. The resulting mapping of 2 36  different length-36 bit sequences to 2 36  different length-60 amplitude sequences can serve as shaping codebook  900 , which can be loaded into the codebook store  820  ( FIG.  8   ) and used as described above. 
       FIG.  10    shows a block diagram of a shaping decoder  1000  that can be used in electronic decoder  160  according to an embodiment. Shaping decoder  1000  is compatible with shaping encoders  700  and  800  and performs amplitude-sequence decoding that is inverse to the encoding performed by shaping encoders  700  and  800 . For example, in response to amplitude stream  732  received from the upstream circuitry of receiver  108 , shaping decoder  1000  operates to recover bitstream  608   1 , as indicated in  FIG.  10   . 
     Shaping decoder  1000  comprises a demultiplexer  1030 , a sequence matcher  1010 , and a copy of codebook store  820 . Demultiplexer  1030  operates to demultiplex amplitude stream  732  (which carries signed amplitudes, as explained above in reference to  FIGS.  7 ,  8   ) into bitstream  628  (which carries sign bits of the signed amplitudes) and amplitude stream  712   L  (which carries unsigned amplitudes). Sequence matcher  1010  operates to receive a length-n L  amplitude sequence of amplitude stream  712   L , finds a match to this sequence in codebook store  820 , retrieves a corresponding length-k 1  bit-word from the codebook store, and outputs the retrieved bit-word to generate a length-k 1  segment for bitstream  608   1 . 
     The codebook store  820  of shaping decoder  1000  has therein the same shaping codebook as that of the codebook store  820  of the counterpart shaping encoder  800 . For example, the codebook store  820  of shaping decoder  1000  may be loaded with a copy of the above-described shaping codebook  900 . A person of ordinary skill in the art will understand that, in some embodiments, shaping decoder  1000  may be adapted in a relatively straightforward manner to perform decoding that is inverse to the encoding of shaping encoder  700 . 
       FIG.  11    graphically illustrates example SNR improvements that can be achieved according to an embodiment. The SNR data shown in  FIG.  11    represent computer simulations of a representative legacy dispersion-managed communication link  106 . The optical launch power of −7 dBm corresponds to an approximately optimum launch power of the unshaped QPSK-modulated optical signal, which is typical of the legacy design target. A curve  1102  graphically shows SNR results corresponding to a 1-stage shaping algorithm implemented using the Constant Composition Distribution Matching (CCDM) described in the above-cited Schulte reference, with the shaping-code rate of H=0.6, the amplitude sequence lengths n=5,120, and the CCDM being configured to implement the MB distribution for a 4-PAM constellation. A curve  1104  graphically shows SNR results corresponding to a 1-stage shaping algorithm implemented using the Enumerative Sphere Shaping (ESS) described in the above-cited Gültekin reference, with the shaping-code rate of H=0.6 and the amplitude sequence lengths n=60 for the same 4-PAM constellation. A curve  1106  graphically shows SNR performance data corresponding to a 2-stage shaping (2-SS) algorithm implemented using a shaping codebook analogous to the above-described shaping codebook  900  but constructed for n 2 =20. The higher SNR values of curve  1106  attest to the better performance of the 2-stage shaping algorithm with respect to either of the above-mentioned 1-stage shaping algorithms. For example, at the optical launch power of −7 dBm and transmission distance of 2400 km, the 2-stage sphere-shaping algorithm achieves ˜1.5-dB gain in effective SNR relative to the MB CCDM shaping algorithm. 
       FIG.  12    shows a block diagram of optical transmitter  104  that can be used in system  100  ( FIG.  1   ) according to an embodiment. 
     In operation, transmitter  104  receives input stream  102  of payload data and applies it to a digital signal processor (DSP)  112 , which implements, inter alia, the electronic encoder  110  ( FIG.  1   ). DSP  112  processes input data stream  102  to generate digital signals  114   1 - 114   4 . In an example embodiment, DSP  112  may perform, one or more of the following: (i) de-multiplex input stream  102  into two sub-streams, each intended for optical transmission using a respective one of orthogonal (e.g., X and Y) polarizations of optical output signal  142 ; (ii) encode each of the sub-streams using one or more suitable codes, e.g., as outlined above; and (iii) convert each of the two resulting sub-streams into a corresponding sequence of constellation symbols. In each signaling interval (also referred to as a symbol period or time slot), signals  114   1  and  114   2  carry digital values that represent the in-phase (I) component and quadrature (Q) component, respectively, of a corresponding constellation symbol intended for transmission using a first (e.g., X) polarization of light. Signals  114   3  and  114   4  similarly carry digital values that represent the I and Q components, respectively, of the corresponding constellation symbol intended for transmission using a second (e.g., Y) polarization of light. 
     E/O converter  140  operates to transform digital signals  114   1 - 114   4  into a corresponding modulated optical output signal  142 . More specifically, drive circuits  118   1  and  118   2  transform digital signals  114   1  and  114   2 , as known in the art, into electrical analog drive signals I X  and Q X , respectively. Drive signals I X  and Q X  are then used, in a conventional manner, to drive an I-Q modulator  124   X . In response to drive signals I X  and Q X , I-Q modulator  124   X  operates to modulate an X-polarized beam  122   X  of light supplied by a laser source  120  as indicated in  FIG.  12   , thereby generating a modulated optical signal  126   X . 
     Drive circuits  118   3  and  118   4  similarly transform digital signals  114   3  and  114   4  into electrical analog drive signals I Y  and Q Y , respectively. In response to drive signals I Y  and Q Y , an I-Q modulator  124   Y  operates to modulate a Y-polarized beam  122   Y  of light supplied by laser source  120  as indicated in  FIG.  12   , thereby generating a modulated optical signal  126   Y . A polarization beam combiner  128  operates to combine modulated optical signals  126   X  and  126   Y , thereby generating optical output signal  142  (also see  FIG.  1   ). 
       FIG.  13    shows a block diagram of optical receiver  108  that can be used in system  100  ( FIG.  1   ) according to an embodiment. 
     O/E converter  150  comprises an optical hybrid  159 , light detectors  161   1 - 161   4 , analog-to-digital converters (ADCs)  166   1 - 166   4 , and an optical local-oscillator (OLO) source  156 . Optical hybrid  159  has (i) two input ports labeled S and R and (ii) four output ports labeled  1  through  4 . Input port S receives optical signal  142 ′ (also see  FIG.  1   ). Input port R receives an OLO signal  158  generated by OLO source  156 . OLO signal  158  has an optical-carrier wavelength (frequency) that is sufficiently close to that of signal  142 ′ to enable coherent (e.g., intradyne) detection of the latter signal. OLO signal  158  can be generated, e.g., using a relatively stable laser whose output wavelength (frequency) is approximately the same as the carrier wavelength (frequency) of optical signal  142 . 
     In an example embodiment, optical hybrid  159  operates to mix input signal  142 ′ and OLO signal  158  to generate different mixed (e.g., by interference) optical signals (not explicitly shown in  FIG.  13   ). Light detectors  161   1 - 161   4  then convert the mixed optical signals into four electrical signals  162   1 - 162   4  that are indicative of complex values corresponding to two orthogonal-polarization components of signal  142 ′. For example, electrical signals  162   1  and  162   2  may be an analog I signal and an analog Q signal, respectively, corresponding to a first (e.g., horizontal, h) polarization component of signal  142 ′. Electrical signals  162   3  and  162   4  may similarly be an analog I signal and an analog Q signal, respectively, corresponding to a second (e.g., vertical, v) polarization component of signal  142 ′. Note that the orientation of the h and v polarization axes at receiver  108  may not coincide with the orientation of the X and Y polarization axes at transmitter  104 . 
     Each of electrical signals  162   1 - 162   4  is converted into digital form in a corresponding one of ADCs  166   1 - 166   4 . Optionally, each of electrical signals  162   1 - 162   4  may be amplified in a corresponding electrical amplifier (not explicitly shown) prior to the resulting signal being converted into digital form. Digital signals  168   1 - 168   4  produced by ADCs  166   1 - 166   4  are then processed by a DSP  170 , which implements, inter alia, electronic decoder  160  (see  FIG.  1   ). 
     In an example embodiment, in addition to the above-described decoding, DSP  170  may perform one or more of the following: (i) signal processing directed at dispersion compensation; (ii) signal processing directed at compensation of nonlinear distortions; (iii) electronic polarization de-multiplexing; and (iv) FEC decoding. 
     According to an example embodiment disclosed above, e.g., in the summary section and/or in reference to any one or any combination of some or all of  FIGS.  1 - 13   , provided is an apparatus comprising: a digital encoder (e.g.,  700 ,  FIG.  7   ) having, at least, first and second digital stages (e.g.,  610   1 ,  710   2 ,  FIG.  7   ) to produce a stream of symbols (e.g.,  712 ,  FIG.  7   ) from a bitstream (e.g.,  608   1 ,  FIG.  7   ), the first digital stage being configured to separately encode segments of a same number of bits (e.g., k 1 ) of the bitstream into first sequences of symbols such that each of the first sequences has a same first length (e.g., n 1 ) and a respective total energy of the symbols therein lower than a threshold (e.g., E 1 +δE), the second digital stage being configured to encode blocks of the first sequences into second sequences of symbols, each of the blocks having a same number of first sequences therein, each of the second sequences having a same second length (e.g., n 2 ); and wherein a total energy of symbols of any of the second sequences divided by the second length is smaller than the threshold divided by the first length. Herein, δE denotes an infinitesimal value, e.g., 0.01% of E 1 . 
     In some embodiments of the above apparatus, the threshold is smaller than an average energy (e.g., U n , Eq. (4)) of an unshaped sequence of symbols of the first length. 
     In some embodiments of any of the above apparatus, the second length is an integer multiple of the first length, the integer being two or more. 
     In some embodiments of any of the above apparatus, the first digital stage is configured to produce the first sequences such that symbols of larger energy are less common therein than symbols of lower energy. 
     In some embodiments of any of the above apparatus, the second stage is configured to produce the second sequences such that symbols of larger energy are less common therein than symbols of lower energy. 
     In some embodiments of any of the above apparatus, the threshold is smaller than an average energy of the symbols times the first length, the average being taken over a corresponding constellation of the symbols. 
     In some embodiments of any of the above apparatus, the digital encoder is configured to create the stream of symbols from the second sequences by using some of the bits of the bitstream as signs for the symbols. 
     In some embodiments of any of the above apparatus, the apparatus further comprises an optical modulator (e.g.,  124 ,  FIG.  12   ) and electronic driver thereof (e.g.,  118 ,  FIG.  12   ); and wherein the digital encoder is configured to cause the electronic driver to operate the optical modulator to produce a modulated optical carrier carrying the second sequences of symbols. 
     In some embodiments of any of the above apparatus, the second length is an integer multiple of the first length, the integer being three or more. 
     In some embodiments of any of the above apparatus, the second digital stage comprises a constellation demapper (e.g.,  620   2 ,  FIG.  6   ) connected to receive the first sequences of symbols. 
     According to another example embodiment disclosed above, e.g., in the summary section and/or in reference to any one or any combination of some or all of  FIGS.  1 - 13   , provided is an apparatus comprising an optical data transmitter (e.g.,  104 ,  FIGS.  1 ,  12   ) that comprises an optical front end (e.g.,  140 ,  FIGS.  1 ,  12   ) and a digital shaping encoder (e.g.,  110 ,  FIG.  1   ;  112 ,  FIG.  12 ,  600   ,  FIG.  6   ;  700 ,  FIG.  7   ;  800 ,  FIG.  8   ), the digital shaping encoder being configured to: encode a bitstream (e.g.,  608   1 ,  FIGS.  6 ,  7 ,  8   ) into a stream (e.g.,  632 ,  FIG.  6   ;  732 ,  FIGS.  7 ,  8   ) of symbols of a constellation; and cause the optical front end to produce a modulated optical signal (e.g.,  142 ,  FIG.  1   ) carrying the stream of symbols; and wherein the shaping encoder is configured to generate a symbol sequence for the stream in response to a segment of the bitstream, the symbol sequence including an integer number (e.g., ρ=n 2 /n 1 ) of non-overlapping subsequences of a fixed length (e.g., n 1 ), any one of the subsequences having a transmit energy not exceeding a first threshold energy (e.g., E 1 ), the symbol sequence having a transmit energy not exceeding a second threshold energy (e.g., E 2 ) and being generated such that some symbols of higher energy have a lower probability of being generated by the digital shaping encoder than other symbols of lower energy, the first threshold energy being smaller than an average energy (e.g., U n , Eq. (4)) of an unshaped sequence of the fixed length of the symbols of the constellation, the second threshold energy being smaller than the integer number of the first threshold energies (e.g., Eq. (5)), the integer number being greater than one. 
     In some embodiments of the above apparatus, the digital shaping encoder comprises a first shaping stage (e.g.,  602   1 ,  FIG.  6   ) and at least a second shaping stage (e.g.,  602   2 ,  FIG.  6   ), the first shaping stage being configured to generate a first amplitude stream (e.g.,  612   1 ,  FIG.  6   ) in response to the bitstream, the first amplitude stream being a stream of non-overlapping amplitude sequences of the fixed length, each one of the amplitude sequences having an energy not exceeding the first threshold energy; and wherein the digital shaping encoder is configured to generate the symbol sequence based on the first amplitude stream. 
     In some embodiments of any of the above apparatus, the second shaping stage comprises a constellation demapper (e.g.,  620   2 ,  FIG.  6   ) connected to receive the first amplitude stream. 
     In some embodiments of any of the above apparatus, the second shaping stage is configured to generate a second amplitude stream (e.g.,  612   2 ,  FIG.  6   ) in response to the first amplitude stream, the second amplitude stream being a stream of non-overlapping amplitude sequences of a larger second fixed length, each one of the amplitude sequences of said second fixed length having an energy not exceeding the second threshold energy; and wherein the digital shaping encoder is configured to generate the symbol sequence based on the second amplitude stream. 
     In some embodiments of any of the above apparatus, the shaping encoder comprises a memory (e.g.,  820 ,  FIG.  8   ) having stored therein a codebook (e.g.,  900 , generated in accordance with  FIGS.  9 A- 9 D ) for converting different segments of the bitstream into corresponding different amplitude sequences, each of the amplitude sequences having an energy not exceeding the second threshold energy and having a length equal to a product of the integer number and the fixed length. 
     In some embodiments of any of the above apparatus, the digital shaping encoder comprises a first shaping stage (e.g.,  602   1 ,  FIG.  6   ) and at least a second shaping stage (e.g.,  602   2 ,  FIG.  6   ), the first shaping stage being nested in the second shaping stage, the first shaping stage being configured to apply a first shaping code to the segment of the bitstream, the second shaping stage being configured to apply a second shaping code to the segment of the bitstream, the second shaping code having a smaller code rate than the first shaping code (e.g., k/n 2 &lt;k/n 1 ). 
     In some embodiments of any of the above apparatus, the integer number is greater than ten. 
     According to yet another example embodiment disclosed above, e.g., in the summary section and/or in reference to any one or any combination of some or all of  FIGS.  1 - 13   , provided is an apparatus comprising an optical data transmitter (e.g.,  104 ,  FIGS.  1 ,  12   ) that comprises an optical front end (e.g.,  140 ,  FIGS.  1 ,  12   ) and a digital signal processor (e.g.,  110 ,  FIG.  1   ;  112 ,  FIG.  12   ), the digital signal processor being configured to: encode a bitstream (e.g.,  608   1 ,  FIGS.  6 ,  7 ,  8   ) to generate a stream (e.g.,  632 ,  FIG.  6   ;  732 ,  FIGS.  7 ,  8   ) of symbols of a constellation; and drive the optical front end to cause a modulated optical signal (e.g.,  142 ,  FIG.  1   ) generated by the optical front end to carry the symbols of the stream; and wherein the digital signal processor comprises a shaping encoder (e.g.,  600 ,  FIG.  6   ;  700 ,  FIG.  7   ;  800 ,  FIG.  8   ) configured to generate a symbol sequence for the stream in response to a segment of the bitstream, the symbol sequence including an integer number (e.g., ρ=n 2 /n 1 ) of non-overlapping subsequences of a fixed length (e.g., n 1 ), any one of the subsequences having a transmit energy not exceeding a first threshold energy (e.g., E 1 ), the symbol sequence having a transmit energy not exceeding a second threshold energy (e.g., E 2 ) and being generated such that some symbols of higher energy have a lower probability of being generated by the shaping encoder than other symbols of lower energy, the first threshold energy being smaller than an average energy (e.g., U n , Eq. (4)) of an unshaped sequence of the fixed length of the symbols of the constellation, the second threshold energy being smaller than the integer number of the first threshold energies (e.g., Eq. (5)), the integer number being greater than one. 
     While this disclosure includes references to illustrative embodiments, this specification is not intended to be construed in a limiting sense. Various modifications of the described embodiments, as well as other embodiments within the scope of the disclosure, which are apparent to persons skilled in the art to which the disclosure pertains are deemed to lie within the principle and scope of the disclosure, e.g., as expressed in the following claims. 
     Some embodiments can be embodied in the form of methods and apparatuses for practicing those methods. Some embodiments can also be embodied in the form of program code recorded in tangible media, such as magnetic recording media, optical recording media, solid state memory, floppy diskettes, CD-ROMs, hard drives, or any other non-transitory machine-readable storage medium, wherein, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the patented invention(s). Some embodiments can also be embodied in the form of program code, for example, stored in a non-transitory machine-readable storage medium including being loaded into and/or executed by a machine, wherein, when the program code is loaded into and executed by a machine, such as a computer or a processor, the machine becomes an apparatus for practicing the patented invention(s). When implemented on a general-purpose processor, the program code segments combine with the processor to provide a unique device that operates analogously to specific logic circuits. 
     Unless explicitly stated otherwise, each numerical value and range should be interpreted as being approximate as if the word “about” or “approximately” preceded the value or range. 
     It will be further understood that various changes in the details, materials, and arrangements of the parts which have been described and illustrated in order to explain the nature of this disclosure may be made by those skilled in the art without departing from the scope of the disclosure, e.g., as expressed in the following claims. 
     The use of figure numbers and/or figure reference labels in the claims is intended to identify one or more possible embodiments of the claimed subject matter in order to facilitate the interpretation of the claims. Such use is not to be construed as necessarily limiting the scope of those claims to the embodiments shown in the corresponding figures. 
     Although the elements in the following method claims, if any, are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence. 
     Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the disclosure. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments necessarily mutually exclusive of other embodiments. The same applies to the term “implementation.” 
     Unless otherwise specified herein, the use of the ordinal adjectives “first,” “second,” “third,” etc., to refer to an object of a plurality of like objects merely indicates that different instances of such like objects are being referred to, and is not intended to imply that the like objects so referred-to have to be in a corresponding order or sequence, either temporally, spatially, in ranking, or in any other manner. 
     Unless otherwise specified herein, in addition to its plain meaning, the conjunction “if” may also or alternatively be construed to mean “when” or “upon” or “in response to determining” or “in response to detecting,” which construal may depend on the corresponding specific context. For example, the phrase “if it is determined” or “if [a stated condition] is detected” may be construed to mean “upon determining” or “in response to determining” or “upon detecting [the stated condition or event]” or “in response to detecting [the stated condition or event].” 
     Also for purposes of this description, the terms “couple,” “coupling,” “coupled,” “connect,” “connecting,” or “connected” refer to any manner known in the art or later developed in which energy is allowed to be transferred between two or more elements, and the interposition of one or more additional elements is contemplated, although not required. Conversely, the terms “directly coupled,” “directly connected,” etc., imply the absence of such additional elements. The same type of distinction applies to the use of terms “attached” and “directly attached,” as applied to a description of a physical structure. For example, a relatively thin layer of adhesive or other suitable binder can be used to implement such “direct attachment” of the two corresponding components in such physical structure. 
     The described embodiments are to be considered in all respects as only illustrative and not restrictive. In particular, the scope of the disclosure is indicated by the appended claims rather than by the description and figures herein. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope. 
     A person of ordinary skill in the art would readily recognize that steps of various above-described methods can be performed by programmed computers. Herein, some embodiments are intended to cover program storage devices, e.g., digital data storage media, which are machine or computer readable and encode machine-executable or computer-executable programs of instructions where said instructions perform some or all of the steps of methods described herein. The program storage devices may be, e.g., digital memories, magnetic storage media such as a magnetic disks or tapes, hard drives, or optically readable digital data storage media. The embodiments are also intended to cover computers programmed to perform said steps of methods described herein. 
     The description and drawings merely illustrate the principles of the disclosure. It will thus be appreciated that those of ordinary skill in the art will be able to devise various arrangements that, although not explicitly described or shown herein, embody the principles of the disclosure and are included within its spirit and scope. Furthermore, all examples recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the disclosure and the concepts contributed by the inventor(s) to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the disclosure, as well as specific examples thereof, are intended to encompass equivalents thereof. 
     The functions of the various elements shown in the figures, including any functional blocks labeled as “processors” and/or “controllers,” may be provided through the use of dedicated hardware as well as hardware capable of executing software in association with appropriate software. When provided by a processor, the functions may be provided by a single dedicated processor, by a single shared processor, or by a plurality of individual processors, some of which may be shared. Moreover, explicit use of the term “processor” or “controller” should not be construed to refer exclusively to hardware capable of executing software, and may implicitly include, without limitation, digital signal processor (DSP) hardware, network processor, application specific integrated circuit (ASIC), field programmable gate array (FPGA), read only memory (ROM) for storing software, random access memory (RAM), and non volatile storage. Other hardware, conventional and/or custom, may also be included. Similarly, any switches shown in the figures are conceptual only. Their function may be carried out through the operation of program logic, through dedicated logic, through the interaction of program control and dedicated logic, or even manually, the particular technique being selectable by the implementer as more specifically understood from the context. 
     As used in this application, the term “circuitry” may refer to one or more or all of the following: (a) hardware-only circuit implementations (such as implementations in only analog and/or digital circuitry); (b) combinations of hardware circuits and software, such as (as applicable): (i) a combination of analog and/or digital hardware circuit(s) with software/firmware and (ii) any portions of hardware processor(s) with software (including digital signal processor(s)), software, and memory(ies) that work together to cause an apparatus, such as a mobile phone or server, to perform various functions); and (c) hardware circuit(s) and or processor(s), such as a microprocessor(s) or a portion of a microprocessor(s), that requires software (e.g., firmware) for operation, but the software may not be present when it is not needed for operation.” This definition of circuitry applies to all uses of this term in this application, including in any claims. As a further example, as used in this application, the term circuitry also covers an implementation of merely a hardware circuit or processor (or multiple processors) or portion of a hardware circuit or processor and its (or their) accompanying software and/or firmware. The term circuitry also covers, for example and if applicable to the particular claim element, a baseband integrated circuit or processor integrated circuit for a mobile device or a similar integrated circuit in server, a cellular network device, or other computing or network device. 
     It should be appreciated by those of ordinary skill in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the disclosure. Similarly, it will be appreciated that any flow charts, flow diagrams, state transition diagrams, pseudo code, and the like represent various processes which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown.