Patent Publication Number: US-2023164017-A1

Title: Bits-to-Symbols Mapping for Amplitude Modulation

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority to Patent Application EP 21210156.2, filed in the European Patent Office on Nov. 24, 2021. 
     BACKGROUND 
     Field 
     Various example embodiments relate to communication equipment and, more specifically but not exclusively, to digital modulation. 
     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. 
     A constellation diagram is a graphical representation of a modulated signal generated using a digital-modulation scheme, such as Pulse Amplitude Modulation (PAM), Quadrature Amplitude Modulation (QAM), Phase Shift Keying (PSK), or the like. Such a constellation diagram may display the modulated signal as a scatter plot of constellation points on the complex IQ plane. The distance of a particular constellation point from the origin of the IQ plane represents the amplitude in the symbol space. 
     The number N of constellation points (also referred to as constellation symbols) in the constellation diagram may be selected to be a base-2 integer, i.e., N=2 k , where k is a positive integer. Each of such constellation symbols can then be used to encode exactly k bits, i.e., each constellation symbol can be assigned a corresponding unique k-bit word, often referred to as a label. Different communication systems may employ different constellation-labeling schemes for inter-converting bitstreams and symbol streams. 
     SUMMARY OF SOME SPECIFIC EMBODIMENTS 
     Disclosed herein are various embodiments of architectures for inter-converting bitstreams and symbol streams of PAM and/or QAM constellations of different sizes, e.g., that are not base-2 integers. Some of such constellations may be Gray-coded, and the constellation mapping may be performed to achieve an equiprobable distribution of different constellation symbols. Some embodiments may be compatible with FEC schemes. In an example embodiment, a transmitter DSP may employ a conventional constellation mapper preceded by an electronic encoder programmed to exclude some constellation-symbol labels from the bitstream applied to the mapper. In different embodiments, the electronic encoder may employ a CCDM and/or a long-division operation to select some amplitudes of the constellation and to exclude others. At least some embodiments are beneficially capable of achieving a smaller gap to the Shannon limit than comparable conventional solutions. 
     In various embodiments, the above architectures can be used in fiber-optic, wireless, and wireline data-communication systems. 
     According to an example embodiment, provided is an apparatus, comprising an optical data transmitter including: an optical modulator; and an electronic controller to operate the optical modulator to optically output a sequence of symbols in response to a bitstream, the electronic controller being configured to select the symbols by Gray-coded mapping of a 4B 2 -QAM constellation, where B≠2 k , B and k are positive integers, and B is greater than one. 
     According to another example embodiment, provided is an apparatus, comprising a data transmitter including: a front-end circuit; and an electronic controller to operate the front-end circuit to output a data-modulated signal carrying a sequence of symbols; and wherein the electronic controller comprises: a constellation mapper to select symbols for the sequence in response to a bitstream and based on symbol labels of a constellation; and an electronic encoder to generate the bitstream in response to a data stream such that the bitstream carries labels of a subset of symbols of the constellation. 
    
    
     
       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    graphically illustrates example Gray coding of an 8-PAM constellation; 
         FIG.  2    shows a block diagram of a PAM constellation mapper according to an embodiment; 
         FIG.  3    shows a block diagram of an electronic encoder that can be used in the PAM constellation mapper of  FIG.  2    according to an embodiment; 
         FIG.  4    shows a block diagram of an electronic encoder that can be used in the PAM constellation mapper of  FIG.  2    according to another embodiment; 
         FIG.  5    shows a block diagram of an electronic encoder that can be used in the PAM constellation mapper of  FIG.  2    according to yet another embodiment; 
         FIG.  6    shows a block diagram of a QAM constellation mapper according to an embodiment; 
         FIGS.  7 A- 7 D  graphically illustrate several QAM constellations that can be implemented using the QAM constellation mapper of  FIG.  6    according to example embodiments; 
         FIGS.  8 A- 8 B  show block diagrams of an electronic encoder that can be used in the PAM constellation mapper of  FIG.  2    according to yet another embodiment; 
         FIGS.  9 A- 9 B  show block diagrams of a PAM constellation demapper according to an embodiment; 
         FIG.  10    shows a block diagram of a communication system in which at least some embodiments can be practiced; 
         FIG.  11    shows a block diagram of an optical data transmitter that can be used in the communication system of  FIG.  10    according to an embodiment; 
         FIG.  12    shows a block diagram of an optical data receiver that can be used in the communication system of  FIG.  10    according to an embodiment; and 
         FIG.  13    graphically illustrates example performance improvements that may be expected according to an embodiment. 
     
    
    
     Herein, the same or similar reference numbers typically refer to similar steps, functions, structures, and/or signals of various embodiments. 
     DETAILED DESCRIPTION 
     In digital communications, Gray codes play an important role in error correction. For example, in a PAM or QAM modulation scheme, Gray coding may be used to label constellation symbols such that the bit-words conveyed by adjacent constellation symbols differ from one another by only one bit. A data-transmission system designed to use this type of labeling together with a forward-error-correction (FEC) scheme capable of correcting single-bit errors may thus very effectively correct transmission errors that cause symbol measurements to deviate into the areas corresponding to adjacent constellation symbols. Such error correction can beneficially make the corresponding transmission system less susceptible to noise. 
       FIG.  1    graphically illustrates example Gray coding of a 2 k -PAM constellation  100 . In this particular example, k=3 and N=2 k =8. There are N/2 (=4) different absolute amplitude values, {A n }, expressed as follows: 
         A   n =1+2( n− 1)  (1)
 
     where n=1, . . . , N/2. Each of the eight constellation points represents one of the eight signed amplitudes±A n , i.e., one of −7, −5, −3, −1, 1, 3, 5, and 7. Each of the corresponding eight labels has k (=3) bits. The most-significant bit (MSB) of each label can be referred to as the sign bit. The two least-significant bits (LSBs) of each label can be referred to as the amplitude label. 
     In the example shown in  FIG.  1   , the binary values of the labels are generated using a reflected double-Gray mapping scheme. According to this scheme, the constellation points located in the positive I-half of constellation  100  have: (i) the sign-bit value of “one” and (ii) amplitude labels generated using conventional Gray mapping. The labels of the constellation points located in the negative I-half of constellation  100  are generated by flipping the sign bit of the corresponding constellation-point labels of the positive I-half. With this type of labeling, 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. 
     A person of ordinary skill in the pertinent art will readily understand that a similar approach can be used for generating labels of a constellation or a constellation portion that uses the Q dimension of the IQ plane. A person of ordinary skill in the pertinent art will also be aware of the fact that many different implementations of Gray coding can be used in commercial products and various off-the-shelf PAM and QAM Gray-code mappers. 
     Some data-transmission systems may benefit from the use of PAM modulation formats characterized by non-integer information rates, e.g., the information rates between those provided by conventional 4-PAM (2 bits/symbol), 8-PAM (3 bits/symbol), 16-PAM (4 bits/symbol), and 32-PAM (5 bits/symbol). Some data-transmission systems may benefit from the use of QAM modulation formats characterized by intermediate information rates, e.g., information rates between those provided by conventional 16-QAM (4 bits/symbol), 64-QAM (6 bits/symbol), 256-QAM (8 bits/symbol), and 1024-QAM (10 bits/symbol). Some of such intermediate information rates may be integer, e.g., 5 or 7 bits/symbol, while others may be non-integer. Some of such data-transmission systems may require the transmitted QAM symbols to form a rectangular or square QAM constellation labeled using Gray coding. Some of such data-transmission systems may also require different transmitted symbols to have substantially the same relative probability of occurrence, i.e., to have an equiprobable distribution. 
     One example data-transmission system that may benefit from some of the above-indicated combinations of features is an inter-datacenter network employing medium-reach, terrestrial fiber-optic links. Disadvantageously, conventional modulation formats may not provide at least some of the above-indicated combinations of features. 
       FIG.  2    shows a block diagram of a PAM constellation mapper  200  according to an embodiment. Mapper  200  comprises an electronic x-label-exclusion encoder  210  and a 2 k -PAM Gray mapper  220  serially connected as indicated in  FIG.  2   . The positive integers k and x are the configuration parameters of mapper  200 . 
     In an example embodiment, mapper  220  can be a conventional 2 k -PAM Gray mapper. In operation, a conventional 2 k -PAM Gray mapper converts an input bitstream into an output stream of signed amplitudes of the corresponding Gray-coded 2 k -PAM constellation. The conversion is performed by: (i) parsing the input bitstream into k-bit words; and (ii) substituting each of the k-bit words by the corresponding signed amplitude A n  in accordance with the labeling used in the operative constellation (e.g., see  FIG.  1   ). When the input bitstream is an equiprobable (e.g., random or pseudorandom) bit sequence, i.e., has a 50/50 average probability of occurrence of binary “ones” and “zeros,” the resulting output stream of signed amplitudes generated by the mapper has an equiprobable distribution of the signed amplitudes. 
     Electronic encoder  210  is programmed to convert an input bitstream  202  into a corresponding bitstream  212 , which is a concatenation of codewords of the encoder. There are (2 k -x) different codewords. When input bitstream  202  is an equiprobable bit sequence, the resulting bitstream  212  has an equiprobable average distribution of the codewords, i.e., each of the codewords of encoder  210  has the same average probability of occurrence in bitstream  212 . In response to this bitstream  212 , mapper  220  operates to generate an output stream  222  of signed amplitudes corresponding to the codewords. The equiprobable average distribution of the codewords of encoder  210  in bitstream  212  causes output stream  222  to have an equiprobable distribution of the signed amplitudes corresponding to those codewords. 
     Each of the codewords of electronic encoder  210  is k-bit long and belongs to a selected subset of the labels of the 2 k -PAM constellation of mapper  220 . As already indicated above, the full set of labels of such 2 k -PAM constellation has N=2 k  different labels. Encoder  210  is programmed to exclude x of those labels from being the codewords thereof. Thus, depending on the number x, the subset of labels selected by electronic encoder  210  for generating bitstream  212  may, e.g., consist of: (N-2) labels for x=2; (N-4) labels for x=4; (N-6) labels for x=6, and so on. The excluded labels do not appear in bitstream  212 . In an example embodiment, the labels excluded from the codewords of electronic encoder  210  may be the labels corresponding to one or more specific absolute amplitude values A n (also see Eq. (1)). Several specific examples described in more detail below in reference to constellation  100  ( FIG.  1   ) further illustrate the label-exclusion scheme implemented in electronic encoder  210 . 
     First, let us consider an embodiment in which electronic encoder  210  is programmed to exclude labels corresponding to a single absolute amplitude value, e.g., A 4  (=7), of constellation  100 . The corresponding signed amplitudes are ±A 4  (=±7). The two labels, 000 and 100, corresponding to those signed amplitudes (see  FIG.  1   ) are not included in (i.e., are excluded from) the codewords of encoder  210 . Thus, in this particular embodiment, x=2. The N-x (=6) codewords of encoder  210  are 001, 010, 011, 111, 110, and 101 (also see  FIG.  1   ). Bitstream  212  generated by this particular embodiment of encoder  210  carries some sequence of these six codewords, with the exact composition of the sequence depending on the composition of input bitstream  202 . 
     Although the largest absolute amplitude value (i.e., A 4 ) of constellation  100  is selected for the label exclusion in this particular example embodiment, a different absolute amplitude value, e.g., any one of A 1 , A 2 , and A 3 , may alternatively be selected for the label exclusion in an alternative embodiment. 
     Second, let us consider an alternative embodiment in which electronic encoder  210  is programmed to exclude labels corresponding to two absolute amplitude values, e.g., A 3  (=5) and A 4  (=7), of constellation  100 . The corresponding signed amplitudes are ±A 3  (=±5) and ±A 4  (=±7). The four labels, 001, 101, 000, and 100, corresponding to those signed amplitudes (also see  FIG.  1   ) are not included in (i.e., are excluded from) the codewords of electronic encoder  210 . Thus, in this particular embodiment, x=4. The N-x (=4) codewords of electronic encoder  210  are 010, 011, 111, and 110 (also see  FIG.  1   ). Bitstream  212  generated by this particular embodiment of electronic encoder  210  carries some sequence of these four codewords, with the exact composition of the sequence depending on the composition of input bitstream  202 . 
     Although the two largest absolute amplitude values (i.e., A 3 , A 4 ) of constellation  100  are selected for the label exclusion in the latter example embodiment, any other pair of absolute amplitude values may alternatively be selected for the label exclusion in an alternative embodiment. The two absolute amplitude values selected for the label exclusion do not need to be consecutive. 
     For embodiments corresponding to larger (e.g., corresponding to k&gt;3) PAM constellations, three or more absolute amplitude values may similarly be selected for the label exclusion in electronic encoder  210  (e.g., see  FIG.  4   ). 
     Several example circuits that can be used to implement electronic encoder  210  are described in more detail below in reference to  FIGS.  3 - 5  and  8   . 
     In some alternative embodiments, the 2 k -PAM constellation of mapper  220  may not necessarily be Gray-coded. 
       FIG.  3    shows a block diagram of electronic encoder  210  ( FIG.  2   ) according to an embodiment. This particular embodiment of electronic encoder  210  corresponds to k=3 and x=2. Bitstreams  202  and  212  are explicitly shown in  FIG.  3    to better illustrate the relationship between the circuits shown in  FIGS.  2  and  3   . 
     As shown in  FIG.  3   , electronic encoder  210  comprises a demultiplexer (DMUX)  310  having an input port and three output ports, labeled A, B, and C, respectively. The input port of DMUX  310  is connected to receive bitstream  202  (also see  FIG.  2   ). DMUX  310  operates to demultiplex bitstream  202  into bitstreams  312 ,  314 , and  316 , which are outputted from ports A, B, and C, respectively. 
     Bitstream  312  controls the output of a sign selector  320 . More specifically, in response to a binary “zero” of bitstream  312 , selector  320  outputs −1 for a digital stream  322 . In response to a binary “one” of bitstream  312 , selector  320  outputs 1 for digital stream  322 . When bitstream  312  is an equiprobable bitstream, the values of 1 and −1 occur in digital stream  322  with the equal probability of 1/2. 
     Bitstream  314  controls the output of a level selector  330 . More specifically, in response to a binary “zero” of bitstream  314 , selector  330  outputs the value of L1 for a digital stream  332 . In response to a binary “one” of bitstream  314 , selector  330  outputs the value of L2 for digital stream  332 . When bitstream  314  is an equiprobable bitstream, the values of L1 and L2 occur in digital stream  332  with the equal probability of 1/2. 
     Bitstream  316  is applied to a distribution matcher (DM)  370 . In an example embodiment, DM  370  can be a binary Constant Composition Distribution Matcher (CCDM). As known in the pertinent art, distribution matching can be used to transform an equiprobable (e.g., random or pseudo-random) bit sequence into a bit sequence with a desired probability ratio for binary “zeros” and “ones.” In this particular embodiment, DM  370  is programmed to transform an equiprobable bitstream into a bitstream, wherein the probabilities (Pr) of occurrence of binary “zeros” and “ones” are 2/3 and 1/3, respectively. Thus, in operation, DM  370  transforms an equiprobable bitstream  316  into a bitstream  372 , wherein Pr(0)=2/3 and Pr(1)=1/3. A person of ordinary skill in the pertinent will readily understand how to make and use such a DM without any undue experimentation. 
     Bitstream  372  controls the output of a level selector  340 . More specifically, in response to a binary “zero” of bitstream  372 , selector  340  outputs, for a digital stream  342 , the level value provided by digital stream  332 . In response to a binary “one” of bitstream  372 , selector  330  outputs the value of L3 for digital stream  342 . Due to the above-indicated configurations of DM  370  and level selectors  330  and  340 , digital stream  342  has the values of L1, L2, and L3 occurring therein with equal probabilities, i.e., Pr(L1)=Pr(L2)=Pr(L3)=1/3. 
     In an example embodiment, the values of L1, L2, and L3 can be selected as follows: L1=A 1 , L2=A 2 , and L3=A 3 . Thus, in this particular embodiment, A 4  is excluded. There are two corresponding labels in constellation  100 , i.e., 000 and 100, corresponding to −A 4  and +A 4 , respectively. These two labels are excluded from the codewords of this particular embodiment of encoder  210 . 
     In alternative embodiments, any one of A 1 , A 2 , and A 3  may be excluded. 
     Encoder  210  further comprises a multiplier  350  and an 8-PAM demapper  360 . In operation, multiplier  350  generates a stream  352  of signed amplitudes by multiplying the values provided by digital streams  322  and  342 . 8-PAM demapper  360  then operates to convert each of the signed amplitudes of stream  352  into a corresponding 3-bit label of the operative constellation, thereby generating bitstream  212 . When the operative constellation is constellation  100  and the selected levels are L1=A 1 , L2=A 2 , and L3=A 3 , bitstream  212  carries a sequence of the labels 001, 010, 011, 111, 110, and 101, with the exact composition of this sequence depending on the composition of bitstream  202 . 
     In some embodiments of mapper  200  employing the above-described electronic encoder  210  of  FIG.  3   , the 8-PAM mapper  220  ( FIG.  2   ) and 8-PAM demapper  360  ( FIG.  3   ) may not necessarily be programmed to use identically labeled 8-PAM constellations. 
       FIG.  4    shows a block diagram of electronic encoder  210  ( FIG.  2   ) according to another embodiment. This particular embodiment of electronic encoder  210  corresponds to k=4 and x=6. Bitstreams  202  and  212  are explicitly shown in  FIG.  4    to better illustrate the relationship between the circuits of  FIGS.  2  and  4   . 
     As shown in  FIG.  4   , electronic encoder  210  comprises a DMUX  410  having an input port and three output ports, labeled A, B, and C. The input port of DMUX  410  is connected to receive bitstream  202  (also see  FIG.  2   ). DMUX  410  operates to demultiplex bitstream  202  into bitstreams  412 ,  414 , and  416 , which are outputted from ports A, B, and C, respectively. 
     Bitstream  412  controls the output of a sign selector  420 . More specifically, in response to a binary “zero” of bitstream  412 , selector  420  outputs −1 for a digital stream  422 . In response to a binary “one” of bitstream  412 , selector  420  outputs 1 for digital stream  422 . When bitstream  412  is an equiprobable bitstream, the values of 1 and −1 occur in digital stream  422  with the equal probability of 1/2. 
     Bitstream  414  controls the output of a level selector  430 . More specifically, in response to a binary “00” of bitstream  414 , selector  430  outputs the value of L1 for a digital stream  432 . In response to a binary “01” of bitstream  414 , selector  430  outputs the value of L2 for digital stream  432 . In response to a binary “10” of bitstream  414 , selector  430  outputs the value of L3 for digital stream  432 . In response to a binary “11” of bitstream  414 , selector  430  outputs the value of L4 for digital stream  432 . When bitstream  414  is an equiprobable bitstream, the values of L1, L2, L3, and L4 occur in digital stream  432  with the equal probability of 1/4. 
     Bitstream  416  is applied to a DM  470 . In an example embodiment, DM  470  can be a binary CCDM. In this particular embodiment, DM  470  is programmed to transform an equiprobable bitstream into a bitstream, wherein the probabilities of occurrence of binary “zeros” and “ones” are 4/5 and 1/5, respectively. Thus, in operation, DM  470  transforms an equiprobable bitstream  416  into a bitstream  472 , wherein Pr(0)=4/5 and Pr(1)=1/5. A person of ordinary skill in the pertinent will readily understand how to make and use such a DM without any undue experimentation. 
     Bitstream  472  controls the output of a level selector  440 . More specifically, in response to a binary “zero” of bitstream  472 , selector  440  outputs, for a digital stream  442 , the level value provided by digital stream  432 . In response to a binary “one” of bitstream  472 , selector  430  outputs the value of L5 for digital stream  442 . Due to the above-indicated configurations of DM  470  and level selectors  430  and  440 , digital stream  442  has the values of L1, L2, L3, L4, and L5 occurring therein with equal probabilities, i.e., Pr(L1)=Pr(L2)=Pr(L3)=Pr(L4)=Pr(L5)=1/5. 
     In an example embodiment, the values of L1, L2, L3, L4, and L5 can be selected as follows: L1=A 1 , L2=A 2 , L3=A 3 , L4=A 4 , and L5=A 5  (also see Eq. (1)). Thus, in this particular embodiment A 6 , A 7 , and A 8  are excluded. There are six (x=6) corresponding labels in the operative 16-PAM constellation, which correspond to −A 6 , −A 7 , −A 8 , +A 6 , +A 7 , and +A 8 , respectively. Those six labels are excluded from the codewords of this particular embodiment of electronic encoder  210 . 
     In alternative embodiments, any three of A 1 -A 8  may be excluded. 
     Electronic encoder  210  further comprises a multiplier  450  and a 16-PAM demapper  460 . In operation, multiplier  450  generates a stream  452  of signed amplitudes by multiplying the values provided by digital streams  422  and  442 . 16-PAM demapper  460  then operates to convert each of the signed amplitudes of stream  452  into a corresponding 4-bit label of the operative constellation, thereby generating bitstream  212 . 
     In some embodiments of mapper  200  employing the electronic encoder  210  of  FIG.  4   , the 16-PAM mapper  220  ( FIG.  2   ) and 16-PAM demapper  460  ( FIG.  4   ) may not necessarily be programmed to use identically labeled 16-PAM constellations. 
       FIG.  5    shows a block diagram of electronic encoder  210  ( FIG.  2   ) according to yet another embodiment. This particular embodiment of electronic encoder  210  corresponds to k=4 and x=4. Bitstreams  202  and  212  are explicitly shown in  FIG.  5    to better illustrate the relationship between the circuits of  FIGS.  2  and  5   . 
     As shown in  FIG.  5   , electronic encoder  210  comprises a DMUX  510  having an input port and four output ports, labeled A-D. The input port of DMUX  510  is connected to receive bitstream  202  (also see  FIG.  2   ). DMUX  510  operates to demultiplex bitstream  202  into bitstreams  512 ,  514 ,  516 , and  518 , which are outputted from ports A, B, C, and D, respectively. 
     Bitstream  512  controls the output of a sign selector  520 . More specifically, in response to a binary “zero” of bitstream  512 , selector  520  outputs −1 for a digital stream  522 . In response to a binary “one” of bitstream  512 , selector  520  outputs 1 for digital stream  522 . When bitstream  512  is an equiprobable bitstream, the values of 1 and −1 occur in digital stream  522  with the equal probability of 1/2. 
     Bitstream  514  controls the output of a level selector  530 . More specifically, in response to a binary “00” of bitstream  514 , selector  530  outputs the value of L1 for a digital stream  532 . In response to a binary “01” of bitstream  514 , selector  530  outputs the value of L2 for digital stream  532 . In response to a binary “10” of bitstream  514 , selector  530  outputs the value of L3 for digital stream  532 . In response to a binary “11” of bitstream  514 , selector  530  outputs the value of L4 for digital stream  532 . When bitstream  514  is an equiprobable bitstream, the values of L1, L2, L3, and L4 occur in digital stream  532  with the equal probability of 1/4. 
     Bitstream  516  is applied to a DM  570 . In an example embodiment, DM  570  can be a binary CCDM. In this particular embodiment, DM  570  is programmed to transform an equiprobable bitstream into a bitstream, wherein the probabilities of occurrence of binary “zeros” and “ones” are 4/5 and 1/5, respectively. Thus, in operation, DM  570  transforms an equiprobable bitstream  516  into a bitstream  572 , wherein Pr(0)=4/5 and Pr(1)=1/5. A person of ordinary skill in the pertinent will readily understand how to make and use such a DM without any undue experimentation. 
     Bitstream  572  controls the output of a level selector  540 . More specifically, in response to a binary “zero” of bitstream  572 , selector  540  outputs, for a digital stream  542 , the level value provided by digital stream  532 . In response to a binary “one” of bitstream  572 , selector  530  outputs the value of L5 for digital stream  542 . Due to the above-indicated configurations of DM  570  and level selectors  530  and  540 , digital stream  542  has the values of L1, L2, L3, L4, and L5 occurring therein with equal probabilities, i.e., Pr(L1)=Pr(L2)=Pr(L3)=Pr(L4)=Pr(L5)=1/5. 
     Bitstream  518  is applied to a DM  580 . In an example embodiment, DM  580  can be a binary CCDM. In this particular embodiment, DM  580  is programmed to transform an equiprobable bitstream into a bitstream, wherein the probabilities of occurrence of binary “zeros” and “ones” are 5/6 and 1/6, respectively. Thus, in operation, DM  580  transforms an equiprobable bitstream  518  into a bitstream  582 , wherein Pr(0)=5/6 and Pr(1)=1/6. A person of ordinary skill in the pertinent will readily understand how to make and use such a DM without any undue experimentation. 
     Bitstream  582  controls the output of a level selector  544 . More specifically, in response to a binary “zero” of bitstream  582 , selector  544  outputs, for a digital stream  546 , the level value provided by digital stream  542 . In response to a binary “one” of bitstream  572 , selector  544  outputs the value of L6 for digital stream  546 . Due to the above-indicated configurations of DMs  570  and  580  and level selectors  530 ,  540 , and  544 , digital stream  546  has the values of L1, L2, L3, L4, L5, and L6 occurring therein with equal probabilities, i.e., Pr(L1)=Pr(L2)=Pr(L3)=Pr(L4)=Pr(L5)=Pr(L6)=1/6. 
     In an example embodiment, the values of L1, L2, L3, L4, L5, and L6 can be selected as follows: L1=A 1 , L2=A 2 , L3=A 3 , L4=A 4 , L5=A 5 , and L6=A 6  (also see Eq. (1)). Thus, in this particular embodiment A 7  and A 8  are excluded. There are four (x=4) corresponding labels in the operative 16-PAM constellation, which correspond to −A 7 , −A 8 , +A 7 , and +A 8 , respectively. These four labels are excluded from the codewords of this particular embodiment of electronic encoder  210 . 
     In alternative embodiments, any two of A 1 -A 8  may be excluded. 
     Electronic encoder  210  further comprises a multiplier  550  and a 16-PAM demapper  560 . In operation, multiplier  550  generates a stream  552  of signed amplitudes by multiplying the values provided by digital streams  522  and  546 . 16-PAM demapper  560  then operates to convert each of the signed amplitudes of stream  552  into a corresponding 4-bit label of the operative constellation, thereby generating bitstream  212 . 
     In some embodiments of mapper  200  employing the above-described electronic encoder  210  of  FIG.  5   , the 16-PAM mapper  220  ( FIG.  2   ) and 16-PAM demapper  560  ( FIG.  5   ) may not necessarily be programmed to use identically labeled 16-PAM constellations. 
     Based on the embodiments of  FIGS.  3 - 5   , a person of ordinary skill in the pertinent art will be able to make and use other embodiments of encoder  210 , e.g., corresponding to other values of k and/or x, without any undue experimentation. 
       FIG.  6    shows a block diagram of a QAM constellation mapper  600  according to an embodiment. Mapper  600  comprises a DMUX  610 , PAM constellation mappers  200   I  and  200   Q , and a real-to-complex (R/C) converter  620  interconnected as indicated in  FIG.  6   . 
     In operation, DMUX  610  demultiplexes an input bitstream  602 , thereby generating input bitstreams  2021  and  202   Q  for mappers  200   I  and  200   Q , respectively. The relative bit rates of bitstreams  202   I  and  202   Q  depend on the configuration parameters of mappers  200   I  and  200   Q . More specifically, mapper  200   I  may be programmed using an N I -PAM constellation and employ a corresponding embodiment of electronic encoder  210  configured to exclude x I  labels of that constellation. Mapper  200   Q  may similarly be programmed using an N Q -PAM constellation and employ a corresponding embodiment of electronic encoder  210  configured to exclude x Q  labels of that constellation. Depending on the embodiment, the numbers N I  and N Q  may be the same or different. The numbers x I  and x Q  may also be the same or different. The relative bit rates with which DMUX  610  demultiplexes input bitstream  602  are thus selected such that the symbol rates of digital output streams  222   I  and  222   Q  generated by mappers  200   I  and  200   Q , respectively, are equal to one another. 
     R/C converter  620  operates to convert the real-valued streams  222   I  and  222   Q  into a corresponding complex-valued stream  622 , e.g., an optical carrier modulated to carry the I stream  222   I  on an in-phase component thereof and to carry the Q stream  222   Q  on a quadrature-phase component thereof. For each complex value a+jb of stream  622 , stream  222   I  provides the real part a, while stream  222   Q  provides the imaginary part b. A person of ordinary skill in the art will understand that different complex values of stream  622  represent different symbols of a QAM constellation. The effective size of this QAM constellation is (N 1 −x I )×(N Q −x Q ), e.g., as illustrated below in reference to the examples shown in  FIGS.  7 A- 7 D . Due to the above-indicated characteristics of the mappers  200 , different symbols of the QAM constellation will occur in stream  622  with equal probabilities when input bitstream  602  is an equiprobable bitstream. 
     In some embodiments of optical data transmitters, R/C converter  620  may be absent. 
       FIGS.  7 A- 7 D  graphically illustrate several QAM constellations that can be implemented using mapper  600  according to example embodiments. More specifically,  FIG.  7 A  corresponds to the embodiment of mapper  600  corresponding to k I =k Q =3 and X I =X Q =2.  FIG.  7 B  corresponds to the embodiment of mapper  600  corresponding to k I =k Q =4 and X I =X Q =2.  FIG.  7 C  corresponds to the embodiment of mapper  600  corresponding to k I =k Q =4 and X I =X Q =4.  FIG.  7 D  corresponds to the embodiment of mapper  600  corresponding to k I =k Q =4 and X I =X Q =6. Herein, k I  is the number of bits in the labels of the PAM constellation corresponding to the I-dimension, and k Q  is the number of bits in the labels of the PAM constellation corresponding to the Q-dimension. 
     The illustrated embodiments provide “square” effective QAM constellations  712 ,  722 ,  732 , and  742 . Herein, the term “square” means that the effective constellation has an equal number of rows and columns in which the constellation points are arranged on the rectangular grid of the IQ plane. However, embodiments are not so limited. More specifically, alternative embodiments may provide various “rectangular” effective QAM constellations. Herein, the term “rectangular” means that the effective constellation has a first number of rows and a second number of columns, in which the constellation points are arranged on the rectangular grid of the IQ plane, with the first and second numbers being different from one another. 
     Square effective QAM constellations may be described using the number B expressed as follows: 
         B =(2 k   −x )/2  (2)
 
     where k=k I =k Q  and x=X I =X Q . The number of constellation points in such a square effective QAM constellation is 4 B 2 . For the square effective QAM constellations  712 ,  722 ,  732 , and  742  of  FIGS.  7 A- 7 D , the number B is 3, 7, 6, and 5, respectively. 
       FIG.  7 A  graphically shows a Gray-coded 64-QAM constellation  710 , with the labels shown next to the constellation points thereof. There are 64 constellation points, each having a label of k I +k Q =6 bits.  28  of the 64 constellation points are excluded by the electronic encoders  210  of mappers  200   I  and  200   Q  (also see  FIGS.  2  and  6   ). The remaining 36 constellation points are enclosed by dashed line  712 . When input bitstream  602  is an equiprobable bitstream, different symbols of the square effective 36-QAM constellation  712  will occur in stream  622  with equal probabilities. The 28 excluded symbols of constellation  710  are not present in stream  622 . 
       FIG.  7 B  graphically shows a Gray-coded 256-QAM constellation  720 , with the labels shown next to the constellation points thereof. There are 256 constellation points, each having a label of k I +k Q =8 bits.  60  of the 256 constellation points are excluded by the electronic encoders  210  of mappers  200   I  and  200   Q  (also see  FIGS.  2  and  6   ). The remaining 196 constellation points are enclosed by dashed line  722 . When input bitstream  602  is an equiprobable bitstream, different symbols of the square effective 196-QAM constellation  722  will occur in stream  622  with equal probabilities. The 60 excluded symbols of constellation  720  are not present in stream  622 . 
       FIG.  7 C  also graphically shows Gray-coded 256-QAM constellation  720 , with the labels shown next to the constellation points thereof. 112 of the 256 constellation points are excluded by the electronic encoders  210  of mappers  200   I  and  200   Q  (also see  FIGS.  2  and  6   ). The remaining 144 constellation points are enclosed by dashed line  732 . When input bitstream  602  is an equiprobable bitstream, different symbols of the square effective 144-QAM constellation  732  will occur in stream  622  with equal probabilities. The 112 excluded symbols of constellation  720  are not present in stream  622 . 
       FIG.  7 D  also graphically shows Gray-coded 256-QAM constellation  720 , with the labels shown next to the constellation points thereof.  156  of the 256 constellation points are excluded by the electronic encoders  210  of mappers  200   I  and  200   Q  (also see  FIGS.  2  and  6   ). The remaining 100 constellation points are enclosed by dashed line  742 . When input bitstream  602  is an equiprobable bitstream, different symbols of the square effective 100-QAM constellation  742  will occur in stream  622  with equal probabilities. The 156 excluded symbols of constellation  720  are not present in stream  622 . 
       FIGS.  8 A- 8 B  show block diagrams of label-exclusion encoder  210  ( FIG.  2   ) according to yet another embodiment. More specifically,  FIG.  8 A  shows an overall block diagram of encoder  210 .  FIG.  8 B  shows a block diagram of an example circuit that can be used to implement a base-B converter  820  of encoder  210  shown in  FIG.  8 A . 
     Referring to  FIG.  8 A , encoder  210  comprises a DMUX  810 , base-B converter  820 , an FEC encoder  830 , a multiplexer (MUX)  834 , and a MUX  840 . In some embodiments, FEC encoder  830  and MUX  834  may be absent. In such embodiments, bitstream  836  is replaced by bitstream  812 . 
     DMUX  810  operates to demultiplex bitstream  202  into bitstreams  812  and  814 . The relative bit rates of bitstreams  812  and  814  depend on the specific embodiment and are selected such that the relative bit rates of bitstreams  822  and  836  are 1:(k-1). As such, the configuration of DMUX  810  depends on the numbers k and x and on the rate of FEC encoder  830 . 
     Base-conversion circuit  820  operates to convert bitstream  814  into a bitstream  822 , e.g., as described below in reference to  FIG.  8 B . The rate of conversion is such that, in response to a “message” having M·log 2  (B) bits delivered by bitstream  814 , circuit  820  outputs M×B bits for bitstream  822 . 
     In an example embodiment, FEC encoder  830  is a systematic FEC encoder. As known to those skilled in the pertinent art, a systematic FEC encoder operates to generate parity bits without altering the corresponding information bits. As such, bitstream  822  passes through FEC encoder  830  unaltered. The parity bits corresponding to bitstream  822  form bitstream  832 . MUX  834  then multiplexes bitstreams  812 , thereby generating bitstream  836  for MUX  840 . 
     MUX  840  operates to generate bitstream  212  by multiplexing bitstreams  822  and  836  such that one bit of bitstream  836  is pre-pended to the corresponding (k-1) bits of bitstream  822 . As a result, each of such k-bit words in the generated bitstream  212  is a codeword of electronic encoder  210  in the sense explained above in reference to  FIG.  2   . 
       FIG.  8 B  shows a block diagram of circuit  820  corresponding to B=3. As such, in response to a “message” having M·log 2  (3) bits delivered by bitstream  814 , circuit  820  outputs M ternary symbols for bitstream  822 . As shown, circuit  820  comprises a selector  850  and a “division-by-3” circuit  860 . 
     Selector  850  is controlled by a control signal  848 , which causes the selector to select input A thereof one time and input B thereof (M-1) subsequent times. A n  output  852  of selector  850  provides a dividend for circuit  860  to act on. A reminder of the division by 3 is outputted for bitstream  822 . A quotient  862  of the division by 3 is looped back to input B of selector  850 . A person of ordinary skill in the art will understand that, in this manner, circuit  860  performs long division by 3 on each “message” delivered by bitstream  814 . 
     In embodiments corresponding to B=5, 6, and 7 (see, e.g.,  FIGS.  7 B- 7 D ), the “division-by-3” circuit  860  is replaced by a similar one of “division-by-5,” “division-by-6,” and “division-by-7” circuits, respectively. 
       FIG.  9 A  shows a block diagram of a PAM constellation demapper  900  according to an embodiment. Demapper  900  is compatible with mapper  200  of  FIG.  2   . Demapper  900  comprises a 2 k -PAM Gray demapper  920  and an electronic decoder  910  serially connected as indicated in  FIG.  9   . The positive integers k and x are the configuration parameters of demapper  900 . In some embodiments, two instances of PAM constellation demapper  900  can be used to construct a corresponding QAM constellation demapper, e.g., using the concept illustrated by  FIG.  6   . 
     In operation, demapper  900  receives a stream  902  of digitized measurements of the received PAM signal. Demapper  920  uses the operative 2 k -PAM constellation, in a conventional manner, to convert stream  902  into a bitstream  212 ′. In the absence of errors, bitstream  212 ′ is the same as bitstream  212  of  FIG.  2   . Errors (if any) can be corrected, e.g., using an FEC decoder (not explicitly shown in  FIG.  9   ), provided that the transmitted signal is FEC-encoded at the corresponding transmitter. For example, such an FEC decoder may be incorporated into decoder  910  in a manner consistent with the encoder embodiment shown in  FIG.  8   . 
     Decoder  910  operates to processes bitstream  212 ′ to recover bitstream  202  (also see  FIG.  2   ). In an example embodiment, the processing implemented in decoder  910  is inverse to the processing implemented in encoder  210  ( FIG.  2   ). 
       FIG.  9 B  shows a block diagram of an example embodiment of decoder  910  corresponding to k=3 and x=2. As shown, decoder  910  comprises a systematic FEC decoder  930 , a slicer  940 , a binary DM  950 , and a MUX  960 . 
     FEC decoder  930  operates to correct errors (if any) in bitstream  212 ′ using the corresponding FEC code. After discarding the parity bits, FEC decoder  930  outputs sign bits via bitstream  932  and outputs amplitude levels via data stream  934 . Slicer  940  then processes data stream  934  to generate bitstreams  942  and  944  in the following manner:
         (i) if the input level is L1, then a binary “0” is outputted for bitstream  942 , and a binary “0” is outputted for bitstream  944 ;   (ii) if the input level is L2, then a binary “1” is outputted for bitstream  942 , and a binary “0” is outputted for bitstream  944 ; and   (iii) if the input level is L3, then there is no output for bitstream  942 , and a binary “1” is outputted for bitstream  944 .
 
DM  950  operates to convert bitstream  944  into bitstream  952  by performing an operation that is inverse to that of DM  370  ( FIG.  3   ). MUX  960  operates to properly multiplex bitstreams  932 ,  942 , and  952 , thereby recovering bitstream  202 .
       

       FIG.  10    shows a block diagram of a communication system  1000  in which at least some embodiments can be practiced. System  1000  comprises a data transmitter  1010  and a data receiver  1030  configured to communicate via a communication medium  1020 . Depending on the embodiment, communication medium  1020  may include a wireless medium, a wire line or cable, an optical fiber, etc. 
     Data transmitter  1010  comprises a digital signal processor (DSP)  12  and a front-end circuit  16 . In operation, DSP  12  transforms one or more bitstreams into one or more data-encoded digital signals suitable for driving front-end circuit  16 . Front-end circuit  16  then operates to convert the one or more data-encoded digital signals received form DSP  12  into signals suitable for transmission via communication medium  1020 . 
     Data receiver  1030  comprises a front-end circuit  72  and a DSP  70 . In operation, front-end circuit  72  performs measurements on the signal(s) received, via communication medium  1020 , from data transmitter  1010  and provides corresponding streams of digital samples of said measurements to DSP  70 . DSP  70  then operates to process the digital-sample streams of said measurements to recover the one or more bitstreams encoded by data transmitter  1010  in the signals transmitted to the communication medium  1020 . 
     In an example embodiment, DSP  12  may include mapper  200 . DSP  70  may include demapper  900 . Example optical embodiments of data transmitter  1010  and data receiver  1030  that can be used when communication medium  1020  comprises a fiber-optic link are described in more detail below in reference to  FIGS.  11 - 12   . 
       FIG.  11    shows a block diagram of an optical data transmitter  1010  that can be used in system  1000  ( FIG.  10   ) according to an embodiment. 
     For illustration purposes, transmitter  1010  is shown to receive bitstream  602 , which is applied DSP  12 . DSP  12  processes bitstream  602  to generate digital signals  14   1 - 14   4 . In an example embodiment, DSP  12  may perform, inter alia, one or more of the following: (i) encode input data stream  602  using a suitable code; (ii) parse the resulting encoded data stream into a sequence of bit-words; (iii) for each bit-word, determine a corresponding constellation symbol of the operative QAM constellation; (iv) generate a digital drive signal carrying the constellation symbol. For example, in each modulation time slot, signals  14   1  and  14   2  may carry digital values that represent the I component and Q component, respectively, of a QAM constellation symbol intended for transmission using a first (e.g., X) polarization of light. Signals  14   3  and  14   4  may similarly carry digital values that represent the I and Q components, respectively, of a QAM constellation symbol intended for transmission using a second (e.g., Y) polarization of light. 
     In some embodiments, DSP  12  may include four instances of circuit  200  connected in parallel, each being in the processing chain connected to generate a respective one of digital signals  14   1 - 14   4 . In some other embodiments, DSP  12  may include two instances of circuit  600 , i.e., one per polarization. Such instances of circuit  600  may or may not include circuit  620 . 
     Front-end circuit  16  is an electrical-to-optical (E/O) converter that operates to transform digital signals  14   1 - 14   4  into a corresponding modulated optical output signal  30 . More specifically, drive circuits  18   1  and  18   2  transform digital signals  14   1  and  14   2 , as known in the pertinent 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 optical I-Q modulator  24   X . In response to drive signals I X  and Q X , optical I-Q modulator  24   X  operates to modulate an X-polarized beam  22   X  of light supplied thereto by a laser source  20  as indicated in  FIG.  11   , thereby generating a modulated optical signal  26   X . 
     The output wavelength of laser source  20  is wavelength λ 0 . The optical output power of laser source  20  can be set and/or changed in response to a control signal  84 . 
     Drive circuits  18   3  and  18   4  similarly transform digital signals  14   3  and  14   4  into electrical analog drive signals I Y  and Q Y , respectively. In response to drive signals I Y  and Q Y , an optical I-Q modulator  24   Y  operates to modulate a Y-polarized beam  22   Y  of light supplied by laser source  20  as indicated in  FIG.  11   , thereby generating a modulated optical signal  26   Y . A polarization beam combiner (PBC)  28  operates to combine modulated optical signals  26   X  and  26   Y , thereby generating the optical output signal  30 , said optical output signal being a polarization-division-multiplexed (PDM) signal. Optical output signal  30  may then be directed for transmission to optical fiber  1020 . 
       FIG.  12    shows a block diagram of an optical data receiver  1030  that can be used in system  1000  ( FIG.  10   ) according to an embodiment. 
     Front-end circuit  72  of receiver  1030  is an optical-to-electrical (O/E) converter comprising an optical hybrid  60 , light detectors  61   1 - 61   4 , analog-to-digital converters (ADCs)  66   1 - 66   4 , and an optical local-oscillator (OLO) source  56 . Optical hybrid  60  has (i) two input ports labeled S and R and (ii) four output ports labeled 1 through 4. Input port S receives optical signal  30  from optical fiber  1020 . Input port R receives an OLO signal  58  generated by OLO source (e.g., laser)  56 . OLO signal  58  has an optical-carrier wavelength (frequency) that is sufficiently close to that of signal  30  to enable coherent (e.g., intradyne) detection of the latter optical signal. 
     In an example embodiment, optical hybrid  60  operates to mix optical signal  30  and OLO signal  58  to generate different mixed (e.g., by interference) optical signals (not explicitly shown in  FIG.  12   ). Light detectors  61   1 - 61   4  then convert the mixed optical signals into four electrical signals  62   1 - 62   4  that are indicative of complex values corresponding to two orthogonal-polarization components of optical signal  30 . For example, electrical signals  62   1  and  62   2  may be indicative of an analog I signal and an analog Q signal, respectively, or linearly independent mixtures thereof corresponding to a first (e.g., horizontal, h) polarization component of optical signal  30 . Electrical signals  62   3  and  62   4  may similarly be indicative of an analog I signal and an analog Q signal, respectively, or linearly independent mixtures thereof corresponding to a second (e.g., vertical, v) polarization component of optical signal  30 . 
     Each of electrical signals  62   1 - 62   4  is converted into digital form in a corresponding one of ADCs  66   1 - 66   4 . Optionally, each of electrical signals  62   1 - 62   4  may be low-pass filtered and amplified in a corresponding electrical amplifier (not explicitly shown) prior to the resulting signal being converted into digital form. Digital signals  68   1 - 68   4  produced by ADCs  66   1 - 66   4 , respectively, are then processed by a DSP  70  to recover the data of the original bitstream  602  applied to transmitter  1010 . 
     In an example embodiment, DSP  70  may perform, inter alia, one or more of the following: (i) signal processing directed at dispersion compensation; (ii) signal processing directed at compensation of nonlinear distortions; (iii) electronic compensation for polarization rotation and polarization de-multiplexing; (iv) compensation of frequency offset between OLO  56  and laser source  20 ; (v) error correction based on FEC-encoding (if any) performed at DSP  12 ; (vi) mapping of a set of complex values conveyed by digital signals  68   1 - 68   4  onto the operative QAM constellations to determine a corresponding constellation symbol thereof, and (vii) concatenating the labels of the constellation symbols determined through said mapping to reconstruct data stream  202 . 
     In some embodiments, DSP  70  may include four instances of circuit  900  connected in parallel and being in the processing chain connected to process digital signals  68   1 - 68   4 . 
       FIG.  13    graphically illustrates example improvements that the inventors believe may be achievable according to an embodiment. More specifically, curve  1306  shows the Shannon limit, for optical data transmission in system  1000 , as a function of signal-to-noise ratio (SNR). Curve  1302  shows a throughput that the inventors believe may be achievable using a conventional Gray-coded 32-QAM constellation, wherein the constellation symbols are arranged on the IQ plane in 6 rows having 4, 6, 6, 6, 6, and 4 symbols respectively, with each quadrant having 8 identically arranged symbols. Curve  1304  shows a throughput that the inventors believe may be obtainable using a Gray-coded 36-QAM constellation  712  of  FIG.  7 A . Thus, the inventors believe that a 36-QAM constellation  712  can outperform a conventional 32-QAM constellation, as indicated by the smaller distance of curve  1304  to the Shannon limit  1306  compared to that of curve  1302 . 
     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 an optical data transmitter (e.g.,  1010 ,  FIG.  11   ) including: an optical modulator (e.g.,  24 ,  FIG.  11   ); and an electronic controller (e.g.,  12 ,  FIG.  11   ) to operate the optical modulator to optically output a sequence of symbols in response to a bitstream (e.g.,  602 ,  FIG.  11   ), the electronic controller being configured to select the symbols by Gray-coded mapping of a 4B 2 -QAM constellation, where B≠2 k , B and k are positive integers, and B is greater than one. 
     In some embodiments of the above apparatus, the electronic controller is configured to cause each one of the symbols to carry 2 k bits, where B&lt;2 k-1 . 
     In some embodiments of any of the above apparatus, B=3 (e.g.,  712 ,  FIG.  7 A ) or B=5 (e.g.,  742 ,  FIG.  7 D ) or B=6 (e.g.,  732 ,  FIG.  7 A ) or B=7 (e.g.,  722 ,  FIG.  7 B ). 
     In some embodiments of any of the above apparatus, the electronic controller is configured to map a random or pseudorandom bitstream to different symbols of the constellation with about equal probability. 
     In some embodiments of any of the above apparatus, the electronic controller comprises: a constellation mapper (e.g.,  220 ,  FIG.  2   ) to select the symbols in response to a bitstream (e.g.,  212 ,  FIG.  2   ) and based on symbol labels; and an electronic encoder (e.g.,  210 ,  FIG.  2   ) to generate the bitstream in response to a data stream (e.g.,  202 ,  FIG.  2   ) such that bit-words of the bitstream are limited to the labels. 
     In some embodiments of any of the above apparatus, the electronic encoder comprises: a first distribution matcher (e.g.,  370 ,  470 ,  570 ,  FIGS.  3 - 5   ) to generate a first binary control signal (e.g.,  372 ,  472 ,  572 ,  FIGS.  3 - 5   ) having first fixed unequal probabilities of ones and zeros therein; and a first selector (e.g.,  340 ,  440 ,  540 ,  FIGS.  3 - 5   ) to select different amplitudes of the constellation in response to the first binary control signal. 
     In some embodiments of any of the above apparatus, the electronic encoder further comprises: a second distribution matcher (e.g.,  580 ,  FIG.  5   ) to generate a second binary control signal (e.g.,  582 ,  FIG.  5   ) having second fixed unequal probabilities of ones and zeros therein, the second fixed unequal probabilities being different from the first fixed unequal probabilities; and a second selector (e.g.,  544 ,  FIG.  5   ) to select various amplitudes of the constellation in response to the second binary control signal. 
     In some embodiments of any of the above apparatus, the electronic encoder is configured to select different amplitudes of the constellation based on a stream of remainders (e.g.,  822 ,  FIG.  8 B ) of a long-division operation. 
     In some embodiments of any of the above apparatus, the electronic encoder comprises an FEC encoder (e.g.,  830 ,  FIG.  8 A ). 
     In some embodiments of any of the above apparatus, the constellation is a square QAM constellation. 
     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 a data transmitter (e.g.,  1000 ,  FIG.  11   ) including: a front-end circuit (e.g.,  16 ,  FIG.  10   ); and an electronic controller (e.g.,  12 ,  FIG.  10   ) to operate the front-end circuit to output a data-modulated signal carrying a sequence of symbols; and wherein the electronic controller comprises: a constellation mapper (e.g.,  220 ,  FIG.  2   ) to select symbols for the sequence in response to a bitstream (e.g.,  212 ,  FIG.  2   ) and based on symbol labels of a constellation; and an electronic encoder (e.g.,  210 ,  FIG.  2   ) to generate the bitstream in response to a data stream (e.g.,  202 ,  FIG.  2   ) such that the bitstream carries labels of a subset of symbols of the constellation. Herein, the term “subset” is to be construed to mean some but not all elements of the corresponding set, e.g., not all of the symbols of the constellation. 
     In some embodiments of the above apparatus, the electronic encoder is configured to exclude at least two constellation-symbol labels of the constellation from appearing in the bitstream. 
     In some embodiments of any of the above apparatus, the constellation is a rectangular or square QAM constellation. 
     In some embodiments of any of the above apparatus, the symbols of the subset are Gray-coded. 
     In some embodiments of any of the above apparatus, the electronic controller is configured to cause each of the symbols of the subset to occur in the sequence with equal probability when the data stream is a random or pseudorandom bitstream. 
     In some embodiments of any of the above apparatus, the electronic encoder comprises: a first distribution matcher (e.g.,  370 ,  470 ,  570 ,  FIGS.  3 - 5   ) to generate a first binary control signal (e.g.,  372 ,  472 ,  572 ,  FIGS.  3 - 5   ) having first fixed unequal probabilities of ones and zeros therein; and a first selector (e.g.,  340 ,  440 ,  540 ,  FIGS.  3 - 5   ) to select different amplitudes of the constellation in response to the first binary control signal. 
     In some embodiments of any of the above apparatus, the electronic encoder further comprises: a second distribution matcher (e.g.,  580 ,  FIG.  5   ) to generate a second binary control signal (e.g.,  582 ,  FIG.  5   ) having second fixed unequal probabilities of ones and zeros therein, the second fixed unequal probabilities being different from the first fixed unequal probabilities; and a second selector (e.g.,  544 ,  FIG.  5   ) to select various amplitudes of the constellation in response to the second binary control signal. 
     In some embodiments of any of the above apparatus, the electronic encoder comprises an amplitude-selection circuit (e.g.,  820 ,  FIG.  8   ) to select different amplitudes of the constellation based on a stream of remainders of a long-division operation applied to a sub-stream (e.g.,  814 ,  FIG.  8   ) of the data stream. 
     In some embodiments of any of the above apparatus, the constellation is a PAM or QAM constellation. 
     In some embodiments of any of the above apparatus, the electronic controller (e.g.,  12 ,  FIG.  11   ) is configured to select each of the symbols from a set of 4 B 2  different QAM symbols, where B≠2 k  and B and k are positive integers. 
     In some embodiments of any of the above apparatus, the electronic controller is configured to cause each one of the different QAM symbols to carry 2 k bits, where B&lt;2 k-1 . 
     In some embodiments of any of the above apparatus, B=3 (e.g.,  712 ,  FIG.  7 A ) or B=5 (e.g.,  742 ,  FIG.  7 D ) or B=6 (e.g.,  732 ,  FIG.  7 A ) or B=7 (e.g.,  722 ,  FIG.  7 B ). 
     In some embodiments of any of the above apparatus, the electronic controller is configured to cause each of the 4B 2  different QAM symbols to occur in the sequence with equal probability when the data stream is a random or pseudorandom data stream. 
     In some embodiments of any of the above apparatus, the electronic encoder comprises an FEC encoder (e.g.,  830 ,  FIG.  8   ). 
     In some embodiments of any of the above apparatus, the front-end circuit comprises a laser (e.g.,  20 ,  FIG.  11   ) and an optical modulator (e.g.,  24 ,  FIG.  11   ) optically coupled to the laser. 
     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.,  1010 ,  FIG.  11   ) including: an optical modulator (e.g.,  24 ,  FIG.  11   ); and an electronic controller (e.g.,  12 ,  FIG.  11   ) to operate the optical modulator to optically output a sequence of symbols in response to a bitstream (e.g.,  602 ,  FIG.  11   ), the electronic controller being configured to select the symbols from a Gray-coded QAM constellation to obtain a 4B 2 -symbol constellation, where B≠2 k , B and k are positive integers, and B is 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. 
     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 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. 
     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. 
     “SUMMARY OF SOME SPECIFIC EMBODIMENTS” in this specification is intended to introduce some example embodiments, with additional embodiments being described in “DETAILED DESCRIPTION” and/or in reference to one or more drawings. “SUMMARY OF SOME SPECIFIC EMBODIMENTS” is not intended to identify essential elements or features of the claimed subject matter, nor is it intended to limit the scope of the claimed subject matter.