PATENT ABSTRACT
In a communication system, Huffman coding techniques are used to obtain shaping gains for an improvement in data transmission rates. More particularly, a novel method of Huffman shaping is described that achieves a shaping gain of greater than 1 dB. The shaping gain results in a higher data rate transmission in a communication system where transmitted power is constrained.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application makes reference to, and claims priority to and the benefit of, U.S. provisional application Ser. No. 60/224,733 filed Aug. 11, 2000. 
       INCORPORATION BY REFERENCE 
       [0002]    The above-referenced U.S. provisional application Ser. No. 60/224,733 is hereby incorporated herein by reference in its entirety. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0003]    N/A 
       BACKGROUND OF THE INVENTION 
       [0004]    Current data communication systems rarely approach highest possible rate, i.e., the rate corresponding to Shannon channel capacity. For example, voiceband modems complying with ITU-T recommendation V.90 employ uncoded modulation for downstream transmission. The nominal downstream rate of 56 kbit/s is thereby almost never achieved, although under practical channel conditions the capacity rate can exceed 56 kbit/s. 
         [0005]    The difference between the signal-to-noise ratio (SNR) required to accomplish a given rate with a given practical coding and modulation scheme and the SNR at which an ideal capacity-achieving scheme could operate at the same rate is known as “SNR gap to capacity”. At spectral efficiencies of 3 bit per signal dimension or higher, uncoded modulation with equiprobable PAM (pulse amplitude modulation) and QAM (quadrature amplitude modulation) symbols exhibit an SNR gap of 9 dB at a symbol error probability of 10 −6 . In the case of V.90 downstream transmission, the SNR gap can correspond to a rate loss of up to 12 kbit/s. 
         [0006]    This overall 9 dB gap is generally comprised of a “shaping gap” portion and a “coding gap” portion. The “shaping gap” portion (approximately 1.5 dB) is caused by the absence of constellation shaping (towards a Gaussian distribution). The remaining “coding gap” portion (approximately 7.5 dB) stems from the lack of sequence coding to increase signal distances between permitted symbol sequences. 
         [0007]    Two different techniques are used, generally in combination, to reduce the overall 9 dB gap. The first technique addresses the “coding gap” portion, and uses one of several coding techniques to achieve coding gains. One of these techniques is trellis-coded modulation. More recent techniques employ serial- or parallel-concatenated codes and iterative decoding (Turbo coding). These latter techniques can reduce the coding gap by about 6.5 dB, from 7.5 dB to about 1 dB. 
         [0008]    Once a coding gain is achieved, the second technique, referred to as shaping, can be used to achieve an even further gain. This type of gain is generally referred to as a shaping gain. Theoretically, shaping is capable of providing an improvement (i.e., shaping gain) of up to 1.53 dB. 
         [0009]    Two practical shaping techniques have been employed in the prior art to achieve shaping gains, namely, trellis shaping and shell mapping. With 16-dimensional shell mapping, such as employed in V.34 modems, for example, a shaping gain of about 0.8 dB can be attained. Trellis shaping can provide a shaping gain of about 1 dB at affordable complexity. Accordingly, between 0.5 and 0.7 dB of possible shaping gain remains untapped by these prior art shaping methods. 
         [0010]    Further limitations and disadvantages of conventional and traditional approaches will become apparent to one of skill in the art, through comparison of such systems with the present invention as set forth in the remainder of the present application with reference to the drawings. 
       BRIEF SUMMARY OF THE INVENTION 
       [0011]    Aspects of the present invention may be found in a method of communicating data in a communication system. The method generally comprises accepting and randomizing (scrambling) data from a source of user data, such as a computer, for example. The randomized data are accumulated until a Huffman codeword is recognized, at which time the Huffman codeword is mapped into a channel symbol. Then the channel symbol is applied to an input of a communication channel. In the field of source coding, the above operation is known as Huffman decoding. 
         [0012]    The encoding operation described above may be combined with further channel encoding operations such as, for example, trellis coded modulation or some form of serial- or parallel-concatenated coding to achieve coding gain in addition to shaping gain. In addition, channel symbols can be modulated in various ways before they are applied to the input of the communication channel. 
         [0013]    In one embodiment of the invention, the channel encoding operation described above is performed in combination with a framing operation to achieve transmission of data at a constant rate. 
         [0014]    Next, on the receiver side of the communication channel, a channel symbol is received from an output of the communication channel after suitable demodulation and channel decoding. Once obtained, the channel symbol is converted into the corresponding Huffman codeword. The data sequence represented by concatenated Huffman codewords is de-randomized (descrambled) and delivered to a sink of user data. 
         [0015]    In one embodiment of the invention, a deframing operation is performed, which provides for data delivery to the data sink at constant rate. 
         [0016]    The method of the present invention results in a symbol constellation and a probability distribution of symbols in this constellation that exhibits a shaping gain of greater than 1 dB. The shaping gain may be, for example, 1.35 dB or 1.5 dB, depending on the specific design 
         [0017]    In general, a communication system according to the present invention comprises a communication node that performs a “Huffman decoding” operation to generate channel symbols with a desired probability distribution. 
         [0018]    These and other advantages and novel features of the present invention, as well as details of an illustrated embodiment thereof, will be more fully understood from the following description and drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0019]      FIG. 1  is a block diagram of a generic communication system that may be employed in connection with the present invention. 
           [0020]      FIG. 2  illustrates additional detail regarding the transmitters of  FIG. 1  according to the present invention. 
           [0021]      FIG. 3  shows shaping gain versus rate for PAM and QAM sq  constellations of different sizes, in accordance with the present invention. 
           [0022]      FIG. 4  plots shaping gains versus rate for square and lowest-energy 1024-QAM constellations, in accordance with the present invention. 
           [0023]      FIG. 5  depicts the mean and standard deviation of the rate in bit/dimension and the shaping gain accomplished for a nominal rate of R=4 bit/dimension with QAM le  constellations of different sizes, in accordance with the present invention. 
           [0024]      FIG. 6  illustrates a 128-QAM le  constellation with Huffman shaping for a nominal rate of 3 bit/dimension, in accordance with the present invention. 
           [0025]      FIG. 7  illustrates one embodiment of a generic method for achieving constant rate and recovering from bit insertions and deletions. 
           [0026]      FIG. 8  illustrates the probability of pointer overflow as a function of framing buffer size in accordance with the present invention. 
           [0027]      FIG. 9  illustrates one embodiment of the design of a Huffman code in accordance with the present invention. 
           [0028]      FIG. 10  is a block diagram of one embodiment of a communication system that operates in accordance with the method of present invention. 
           [0029]      FIG. 11  is another embodiment of the design of a Huffman code in accordance with the present invention, when a framer/deframer is utilized. 
           [0030]      FIG. 12  is a block diagram of another embodiment of a communication system that operates in accordance with the method of present invention, utilizing a framer/deframer. 
           [0031]      FIG. 13  illustrates one operation of a system that employs Huffman shaping in accordance with the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0032]      FIG. 1  is a block diagram of a generic communication system that may be employed in connection with the present invention. The system comprises a first communication node  101 , a second communication node  111 , and a channel  109  that communicatively couples the nodes  101  and  111 . The communication nodes may be, for example, modems or any other type of transceiver device that transmits or receives data over a channel. The first communication node  101  comprises a transmitter  105 , a receiver  103  and a processor  106 . The processor  106  may comprise, for example, a microprocessor. The first communication node  101  is communicatively coupled to a user  100  (e.g., a computer) via communication link  110 , and to the channel  109  via communication links  107  and  108 . 
         [0033]    Similarly, the second communication node  111  comprises a transmitter  115 , a receiver  114  and a processor  118 . The processor  118 , like processor  106 , may comprise, for example, a microprocessor. The second communication node  111  is likewise communicatively coupled to a user  120  (again a computer, for example) via communication link  121 , and to the channel  109  via communication links  112  and  113 . 
         [0034]    During operation, the user  100  can communicate information to the user  120  using the first communication node  101 , the channel  109  and the second communication node  111 . Specifically, the user  100  communicates the information to the first communication node  101  via communication link  110 . The information is transformed in the transmitter  105  to match the restrictions imposed by the channel  109 . The transmitter  105  then communicates the information to the channel  109  via communication link  107 . The receiver  114  of the second communication node  111  next receives, via communication link  113 , the information from the channel  109 , and transforms it into a form usable by the user  120 . Finally, the information is communicated from the second communication node  111  to the user  120  via the communication link  121 . 
         [0035]    Communication of information from the user  120  to the user  100  may also be achieved in a similar manner. In either case, the information transmitted/received may also be processed using the processors  106 / 118 . 
         [0036]      FIG. 2  illustrates additional detail regarding the transmitters of  FIG. 1  according to the present invention. The functions of transmitter  201  may be decomposed into those of a source encoder  203  and a channel encoder  205 . Generally, the source encoder  203  is a device that transforms the data produced by a source (such as the user  100  or user  120  of  FIG. 1 ) into a form convenient for use by the channel encoder  205 . For example, the source may produce analog samples at a certain rate, such as, for example, 8000/s, as in a telephone application. The source encoder  203  then may perform the function of analog-to-digital conversion, converting each analog sample into an 8-bit binary code. The output of the source encoder  203  then would be a binary sequence of digits presented to the input of the channel encoder  205  at a rate of 8×8000=64,000 bit/s. The output of the source encoder  203  is passed to the channel encoder  205 , where the data are transformed into symbols that can be transmitted on the channel. For example, the data may be transformed using pulse-amplitude modulation (PAM), whereby successive short blocks of data bits of length N are encoded as analog pulses having one of 2 N  allowable amplitudes. 
         [0037]    In most communication systems, the data presented to the channel encoder are assumed to be completely random. This randomness is normally assured by the inclusion of a scrambler designed into the system. In the previous example of PAM, random data would lead to each 2 N  of the allowable amplitudes being equally likely. That is, each of them occurs with probability 2 −N . It turns out that employing equally likely pulse amplitudes leads to a small inefficiency in the use of the power in the signal that is transmitted into the channel. In fact, as mentioned above, if the amplitude distribution can be made more nearly Gaussian, then up to 1.53 dB of transmitted power can be saved for the same level of error performance at the receiver. 
         [0038]    Accordingly, a shaping function is provided in  FIG. 2  by a shaper  207 , which alters the statistical distribution of the values presented to modulator  209 . Shaping the transmitted signal generally means controlling the distribution of transmitted signal values to make the signal appear more Gaussian in character. The shaper  207  comprises a Huffman decoder  211  and a mapper  213 . The design of the Huffman decoder  211  depends upon the characteristics of the channel. 
         [0039]    In the Huffman decoder  211 , the sequence of scrambled binary data bits is parsed into Huffman codewords. The codewords are then mapped into modulation symbols. The Huffman code is designed to let the modulation symbols assume approximately a sampled Gaussian distribution. 
         [0040]    Unlike trellis shaping or shell mapping, Huffman shaping is not a constant-rate-encoding scheme. Moreover, decoding errors can lead to bit insertion or deletion in the decoded binary data sequence. This may be acceptable for many systems, such as, for example, those in which variable-length packets are transmitted in burst mode with an Ethernet-like medium access protocol. In some cases, continuous transmission at constant rate is desirable, such as, for example, those involving variable-rate encoded voice and video streams over constant rate channels. A constant rate and recovery from bit insertions and deletions may be achieved, and the framing overhead may be kept to a value equivalent to a SNR penalty of 0.1 dB, for example, utilizing the method of the present invention. 
         [0041]    The following mathematical foundation of Huffman shaping is based upon M-ary PAM data transmission, but the concept clearly applies to two- and higher-dimensional modulation as well. 
         [0042]    Let A M  be a symmetric M-ary PAM constellation of equally spaced symbols. Adjacent symbols are spaced by 2, and M may be even or odd (usually M will be even): 
         [0000]      A M   ={a   i =−( M− 1)+2 i,  0 ≦i≦M− 1} 
         [0000]      e.g.: A 8 ={−7,−5,−3,−1,+1,+3,+5,+7}, A 5 ={−4,−2,0,+2,+4}  (1) 
         [0043]    If symbols are selected independently with probabilities p={p i , 0≦i≦M−1}, the symbol entropy H(p) (=rate) and the average symbol energy E(p) become: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0044]    Shaping gain G S (p) expresses a saving in average symbol energy achieved by choosing symbols from A M  with probabilities p rather than selecting equiprobable symbols from a smaller constellation A M′ , where M′=2 H(p)  (M′&lt;M, ignoring that M′ may not be an integer): 
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         [0045]    The maximum shaping gain is obtained by the probability distribution p={tilde over (p)}, which minimizes E(p) subject to the constraints R=H(p) and Σ i=0   M−1 p i =1. Differentiation of 
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         [0000]    with respect to the probabilities p i  yields the conditions 
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         [0000]    The parametric solution of (6), with the Lagrange multipliers λ 1 ,λ 2  transformed into the new variables α,s, becomes 
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         [0046]    The optimum distribution {tilde over (p)} is thus found to be a Gaussian distribution sampled at the symbol values of A M . This solution can also be obtained by maximizing the rate R=H(p) subject to the constraints E(p)=S and Σ i=0   M−1 p i =1. The value of α follows from Σ i=0   M−1 p i =1. The value of s may be chosen to achieve a given rate R≦log 2 (M) or a given average symbol energy S≦E M . 
         [0047]    If M and R are increased, the optimum shaping gain tends towards the ultimate shaping gain G S   ∞ =πe/6=1.423 (1.53 dB). This gain can be derived as the ratio of the variance of a uniform density over a finite interval and the variance of a Gaussian density, both with the same differential entropy. 
         [0048]    One can see that (7) does not only hold for regular symmetric PAM constellations, but gives the optimum shaping probabilities for arbitrary one- and higher-dimensional symbol constellations as well. 
         [0049]    In general, given a sequence of M-ary source symbols which occur independently with probability distribution p, a traditional Huffman coding approach encodes the source symbols into binary codewords of variable lengths such that (a) no codeword is a prefix of any other codeword (prefix condition), and (b) the expected length of the codewords is minimized. 
         [0050]    An optimum set of codewords is obtained by Huffman&#39;s algorithm. More particularly, let a i  be a source symbol that occurs with probability p i . The algorithm associates a i  with a binary codeword c i  of length l i  such that 2 −l     i   ≈p i . The algorithm guarantees that Σ i=0   M−1 2 −l     i   =1 (Kraft&#39;s inequality is satisfied with equality), and that the expected value of the codeword length, L=Σ i=0   M−1 p i l i , approaches the entropy of the source symbols within one bit [10]: 
         [0000]        H ( p )≦ L&lt;H ( p )+1.  (8) 
         [0051]    In the limit for large H(p), the concatenated Huffman codewords yield a binary sequence of independent and equiprobable zeroes and ones with rate R=L≅H(p) bit per source symbol. However, for certain probability distributions L may be closer to H(p)+1 than H(p) because of quantization effects inherent in the code construction. If H(p) is small, the difference between L and H(p) can be significant. The rate efficiency may be improved by constructing a Huffman code for blocks of K&gt;1 source symbols. Then, (8) takes the form H(p)≦L(K)/K=L≦H(p)+1/K, where L(K) is the expected length of the Huffman codewords associated with K-symbol blocks. The code comprises M K  codewords and the rate expressed in bit per source symbol will generally be within 1/K bit from H(p). 
         [0052]    With the Huffman shaping method of the present invention, the traditional encoding approach is reversed. A Huffman code is generated for the optimum probability distribution {tilde over (p)} of the modulation symbols in a given M-ary constellation. In the transmitter, the sequence of data bits is suitably scrambled so that perfect randomness can be assumed. The scrambled sequence is buffered and segmented into Huffman codewords, as in traditional Huffman decoding. A codeword c i  is encountered with probability 2 −l     i   {tilde over (p)} i  and mapped into modulation symbol a i . In the receiver, when a symbol a i  is detected codeword c i  is inserted into the binary output stream. 
         [0053]    For the general case of K-dimensional modulation (K=1: PAM, K=2: QAM), it is appropriate to express rates and symbol energies per dimension, while a i , {tilde over (p)} i , and l i  relate to K-dimensional symbols. 
         [0054]    The mean value  R   h  and the standard deviation σ R   h  of the number of bits encoded per symbol dimension become 
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                                     KR 
                                     h 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0055]    The average symbol energy per dimension S h  and the shaping gain G s   h  of the Huffman-shaped symbol sequence are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       S 
                       h 
                     
                     = 
                     
                       
                         1 
                         K 
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             M 
                             - 
                             1 
                           
                         
                          
                         
                           
                             2 
                             
                               - 
                               
                                 l 
                                 i 
                               
                             
                           
                            
                           
                             
                                
                               
                                 a 
                                 i 
                               
                                
                             
                             2 
                           
                            
                           
                               
                           
                            
                           energy 
                            
                           
                               
                           
                            
                           per 
                            
                           
                               
                           
                            
                           dimension 
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               ≈ 
                               
                                 
                                   1 
                                   K 
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                       p 
                                       ~ 
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     G 
                     s 
                     h 
                   
                   = 
                   
                     
                       
                         
                           2 
                           
                             2 
                              
                             
                               
                                 R 
                                 _ 
                               
                               h 
                             
                           
                         
                         - 
                         1 
                       
                       
                         3 
                         × 
                         
                           E 
                           h 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0056]    The corresponding quantities obtained with optimum shaping probabilities {tilde over (p)} will be denoted, respectively, by {tilde over (R)} and σ {tilde over (R)}  (bit/dimension), {tilde over (S)} (energy per dimension), and {tilde over (G)} s  (optimum shaping gain). 
         [0057]    For numerical evaluations, uncoded modulation with M-PAM (M=2m) and M-QAM (M=4m) constellations have been considered. The M-QAM constellations are either square constellations M-QAM sq =√{square root over (M)}−PAM×√{square root over (M)}−PAM, or lowest-energy constellations M-QAM le  comprising the M points in the set {(1+2i,1+2k), i, kεZ} nearest to the origin. The symmetries of the symbol constellations are enforced on the Huffman codes. In the PAM case, m codewords are constructed for positive symbols and then extended by a sign bit. Similarly, in the QAM case m codewords are constructed for symbols in the first quadrant and extended by two quadrant bits. The results of different numerical evaluations are depicted in  FIGS. 3 ,  4 , and  5 . 
         [0058]      FIG. 3  shows shaping gain versus rate for PAM and QAM sq  constellations of different sizes, in accordance with the present invention. The solid curves indicate the shaping gains obtained with the optimum shaping probabilities {tilde over (p)}. Every rate in the interval 1≦R≦log 2 (M)/K can be accomplished (bit per dimension). The shaping gains vanish at R=1 (constellations reduced to BPSK or QPSK) and R=log 2 (M)/K (equiprobable M-QAM). The optimum shaping gains practically reach the ultimate shaping gain of 1.53 dB at R=4 bit per dimension for ≧32-PAM and ≧1024-QAM sq  constellations. With the Huffman shaping method of the present invention, not every rate can be realized because of quantization effects in the construction of Huffman codes. For PAM, shaping gains of up to ≈1.35 dB are achieved at some rates above 3 bit per dimension. The effects of quantization are significantly reduced in the QAM cases. With ≧256-QAM sq  constellations shaping gains within 0.1 dB from the ultimate shaping gain of 1.53 dB are consistently obtained at rates above 3 bit per dimension. 
         [0059]      FIG. 4  plots shaping gains versus rate for square and lowest-energy 1024-QAM constellations, in accordance with the present invention. Minor differences occur in the region of diminishing shaping gains, at rates above 4.5 bit/dimension. The shaping gain of equiprobable 1024-QAM le  (R=5 bit/dimension) is 0.2 dB. 
         [0060]      FIG. 5  depicts the mean and standard deviation of the rate in bit/dimension and the shaping gain accomplished for a nominal rate of R=4 bit/dimension with QAM le  constellations of different sizes, in accordance with the present invention. The nominal rate is at least closely achieved with Huffman shaping (with optimum shaping it is exactly achieved). The standard deviation increases with increasing constellation size to a final value of ≈1 bit/dimension. The optimum shaping gain and the Huffman shaping gain increase rapidly when the initial 256-QAM constellation is enlarged. The respective final shaping gains of ≈1.5 dB and ≈1.4 dB are practically achieved with M=512 (512-QAM le : 1.495 dB and 1.412 dB, 1024-QAM le : 1.516 dB and 1.432 dB). 
         [0061]      FIG. 6  illustrates a 128-QAM le  constellation with Huffman shaping for a nominal rate of 3 bit/dimension, in accordance with the present invention. The codeword lengths ranging from 5 to 12 bits are indicated for the first-quadrant symbols.  R   h =2.975 (σ R   h =0.919) bit/dimension and G s   h =1.378 dB ({tilde over (G)} s =1.443 dB) are achieved. The symbol energies, optimum shaping probabilities, codeword probabilities and lengths, and the codewords of the first quadrant symbols are listed below. The codewords for the first-quadrant symbols end with 00. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Huffman code words tabulated against 
               
               
                 their index 
               
             
          
           
               
                   
                 i 
                 |a i | 2   
                 {tilde over (p)} i   
                 p i   h  = 2 −l     i     
                 l i   
                 c i   
               
               
                   
                   
               
             
          
           
               
                   
                 0 
                 2 
                 0.03872 
                 0.03125 
                 5 
                 00000 
               
               
                   
                 1 
                 10 
                 0.02991 
                 0.03125 
                 5 
                 10000 
               
               
                   
                 2 
                 10 
                 0.02991 
                 0.03125 
                 5 
                 01100 
               
               
                   
                 3 
                 18 
                 0.02311 
                 0.03125 
                 5 
                 11100 
               
               
                   
                 4 
                 26 
                 0.01785 
                 0.01563 
                 6 
                 010000 
               
               
                   
                 5 
                 26 
                 0.01785 
                 0.01563 
                 6 
                 001100 
               
               
                   
                 6 
                 34 
                 0.01379 
                 0.01563 
                 6 
                 110000 
               
               
                   
                 7 
                 34 
                 0.01379 
                 0.01563 
                 6 
                 101100 
               
               
                   
                 8 
                 50 
                 0.00823 
                 0.00781 
                 7 
                 1010000 
               
               
                   
                 9 
                 50 
                 0.00823 
                 0.00781 
                 7 
                 0101100 
               
               
                   
                 10 
                 50 
                 0.00823 
                 0.00781 
                 7 
                 0101000 
               
               
                   
                 11 
                 58 
                 0.00636 
                 0.00781 
                 7 
                 1101100 
               
               
                   
                 12 
                 58 
                 0.00636 
                 0.00781 
                 7 
                 1101000 
               
               
                   
                 13 
                 74 
                 0.00379 
                 0.00391 
                 8 
                 10101100 
               
               
                   
                 14 
                 74 
                 0.00379 
                 0.00391 
                 8 
                 10101000 
               
               
                   
                 15 
                 82 
                 0.00293 
                 0.00195 
                 9 
                 001001000 
               
               
                   
                 16 
                 82 
                 0.00293 
                 0.00195 
                 9 
                 001000100 
               
               
                   
                 17 
                 90 
                 0.00226 
                 0.00195 
                 9 
                 001010100 
               
               
                   
                 18 
                 90 
                 0.00226 
                 0.00195 
                 9 
                 001010000 
               
               
                   
                 19 
                 98 
                 0.00175 
                 0.00195 
                 9 
                 001011100 
               
               
                   
                 20 
                 106 
                 0.00135 
                 0.00098 
                 10 
                 0010011100 
               
               
                   
                 21 
                 106 
                 0.00135 
                 0.00098 
                 10 
                 0010011000 
               
               
                   
                 22 
                 122 
                 0.00081 
                 0.00049 
                 11 
                 00100000100 
               
               
                   
                 23 
                 122 
                 0.00081 
                 0.00049 
                 11 
                 00100000000 
               
               
                   
                 24 
                 130 
                 0.00062 
                 0.00049 
                 11 
                 00101101000 
               
               
                   
                 25 
                 130 
                 0.00062 
                 0.00049 
                 11 
                 00101100100 
               
               
                   
                 26 
                 130 
                 0.00062 
                 0.00049 
                 11 
                 00101100000 
               
               
                   
                 27 
                 130 
                 0.00062 
                 0.00049 
                 11 
                 00100001100 
               
               
                   
                 28 
                 146 
                 0.00037 
                 0.00024 
                 12 
                 001000010100 
               
               
                   
                 29 
                 146 
                 0.00037 
                 0.00024 
                 12 
                 001000010000 
               
               
                   
                 30 
                 162 
                 0.00022 
                 0.00024 
                 12 
                 001011011000 
               
               
                   
                 31 
                 170 
                 0.00017 
                 0.00024 
                 12 
                 001011011100 
               
               
                   
                   
               
             
          
         
       
     
         [0062]      FIG. 7  illustrates one embodiment of a generic method for achieving constant rate and recovering from bit insertions and deletions. Data frames of N b  bits are embedded into symbol frames of N s  modulation symbols. Every sequence of bits transmitted within a symbol frame begins with a S&amp;P (synch &amp; pointer) field of n sp =n s +n p  bits, where n s  is the width of a synch subfield and n p  is the width of a pointer subfield. The synch subfield enables the receiver to acquire symbol-frame synchronization. In principle, sending a known pseudo-random binary sequence with one bit (n s =1) in every S&amp;P field is sufficient (as in T1 systems). The pointer subfield of the n th  symbol frame expresses the offset in bits of the n th  data frame from the S&amp;P field. 
         [0063]    With reference to  FIG. 7 , in the 1 st  symbol frame, the 1 st  data frame follows the S&amp;P field with zero offset. The S&amp;P field and 1 st  data frame are parsed into Huffman codewords, which are then mapped into modulation symbols indexed by 1, 2, 3, . . . N s . The end of the 1 st  data frame is reached before the N s   th  modulation symbol has been determined. The data frame is padded with fill bits until the N s   th  modulation symbol is obtained. The 2 nd  data frame follows the S&amp;P field of the 2 nd  symbol frame again with zero offset. Now the last symbol of the 2 nd  symbol frame is found before the 2 nd  data frame is completely encoded. The S&amp;P field of the 3 rd  symbol frame is inserted and encoding of the remaining part of the 2 nd  data frame is then continued, followed by encoding the 3 rd  data frame. The pointer in the S&amp;P field indicates the offset of the 3 rd  data frame from the S&amp;P field. The 3 rd  data frame can again not completely be encoded in the 3 rd  symbol frame. The 4 th  data frame becomes completely encoded in the 4 th  symbol frame and is padded with fill bits, and so on. The pointer information in the S&amp;P fields enables a receiver to recover from bit insertion and deletion errors. 
         [0064]    To determine the overhead in framing bits per symbol, first let B n  be the number of bits that are encoded into the N s  symbols of the n th  symbol frame. As mentioned above, the mean and standard deviation of the number of bits encoded per symbol dimension are R h  and σ R   h , respectively, as given by (9) and (10). Then B=N s KR h  is the mean and σ B =√{square root over (N s K)}σ R   h  the standard deviation of B n . For large N s , the probability distribution of B n  will accurately be approximated by the Gaussian distribution 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Pr 
                        
                       
                         ( 
                         
                           
                             B 
                             n 
                           
                           = 
                           x 
                         
                         ) 
                       
                     
                     ≅ 
                     
                       
                         1 
                         
                           
                             
                               2 
                                
                               π 
                             
                           
                            
                           
                             σ 
                             B 
                           
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             - 
                             
                               
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     B 
                                   
                                   ) 
                                 
                                 2 
                               
                               
                                 2 
                                  
                                 
                                   σ 
                                   B 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     x 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   2 
                   , 
                   3 
                   , 
                   … 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0065]    Next, let P n  be the pointer value in the S&amp;P field of the n th  symbol frame. The pointer values will remain bounded if B&gt;n sp +N b . Equivalently, the average number of fill bits per frame, n fill , is nonzero: 
         [0000]        n   fill   =B− ( n   sp   +N   b )&gt;0.  (14) 
         [0066]    Moreover, in a practical implementation the pointer values remain limited to the values that can be represented in the n p -bit pointer subfield, i.e. 0≦P n ≦2 n     p   −1. Parameters are chosen such that the probability of P n &gt;2 n     p   −1 becomes negligible. From  FIG. 7 , one can verify the recursive relation 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     n 
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   sp 
                                 
                               
                               + 
                               
                                 P 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               + 
                               
                                 N 
                                 b 
                               
                             
                             ≤ 
                             
                               B 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               n 
                               sp 
                             
                             + 
                             
                               P 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                             + 
                             
                               N 
                               b 
                             
                             - 
                             
                               B 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             otherwise 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The temporal evolution of the pointer probabilities then becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Pr 
                        
                       
                         ( 
                         
                           
                             P 
                             n 
                           
                           = 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           x 
                           ≥ 
                           0 
                         
                       
                        
                       
                         
                           Pr 
                            
                           
                             ( 
                             
                               
                                 P 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               = 
                               x 
                             
                             ) 
                           
                         
                          
                         
                           Pr 
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ≥ 
                               
                                 
                                   n 
                                   sp 
                                 
                                 + 
                                 
                                   N 
                                   b 
                                 
                                 + 
                                 x 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Pr 
                        
                       
                         ( 
                         
                           
                             P 
                             n 
                           
                           = 
                           y 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             x 
                             ≥ 
                             
                               0 
                                
                               
                                   
                               
                                
                               and 
                             
                           
                           
                             x 
                             ≥ 
                             
                               y 
                               - 
                               
                                 n 
                                 sp 
                               
                               - 
                               
                                 N 
                                 b 
                               
                             
                           
                         
                       
                        
                       
                         
                           Pr 
                            
                           
                             ( 
                             
                               
                                 P 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               = 
                               x 
                             
                             ) 
                           
                         
                          
                         
                           Pr 
                            
                           
                             ( 
                             
                               
                                 B 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               = 
                               
                                 
                                   n 
                                   sp 
                                 
                                 + 
                                 
                                   N 
                                   b 
                                 
                                 + 
                                 x 
                                 - 
                                 y 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     y 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   3 
                   , 
                   
                     … 
                      
                     
                         
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0067]    (equation (17) changed to fit within page margins) 
         [0068]    The steady-state distribution Pr(P=x)=Pr(P n→∞ =x) can be determined numerically (mathematically speaking, Pr(P=x) is the eigensolution of (16) and (17) associated with eigenvalue one). Pr(P=x) and Pr(P≧x) are plotted in  FIG. 8  for the following case.
       Lowest-energy 512-QAM, nominal rate R=4 bit/dimension   Huffman code design:             R h =4.015, σ R   h =0.927 bit/dimension; shaping gain G s   h =1.412 dB.   Assume N s =512 QAM symbols/symbol, N b =4094 bit/data frame, n sp =12 (n s =1, n p =11)             B=4111.36, σ B =29.66, n fill =5.36 bit/symbol frame.       
 
         [0074]    The pointer field allows for a maximum pointer value of 2047.  FIG. 6  shows that Pr(P&gt;2047) is well below 10 −10 . The pointer values exhibit a Paré to distribution, i.e., log(Pr(P≧x)) decreases linearly for large x. 
         [0075]    A framing overhead of (n sp +n fill )/N s =0.034 bit/QAM symbol is found, which is equivalent to an SNR penalty of 0.102 dB. The final net shaping gain becomes 1.412−0.102=1.310 dB. 
         [0076]    Based on the above mathematical foundation of Huffman shaping, in one embodiment of the invention, the method of the present invention may generally comprise two parts. The first is related to the design of the Huffman code to be employed on a given channel, and the second is related to the operation of the Huffman shaper in the transmitter. While the above mathematical foundation of Huffman shaping assumes a PAM implementation; extension to higher-dimensional modulation are also possible. 
         [0077]      FIG. 9  illustrates one embodiment of the design of a Huffman code in accordance with the present invention. The modulation scheme is characterized by parameters M, α, and s (see (7) and accompanying text above) acquired in block  901 , from which are derived the constellation levels {a i ;i=0, 1, . . . , M−1} also in block  901 . The probability p i  is then calculated for each a i  in step  903  for i=0, 1, . . . , M−1. Finally, a Huffman code for the symbols {a i } and their corresponding probabilities {p i } is constructed in block  905 . 
         [0078]      FIG. 10  is a block diagram of one embodiment of a communication system that operates in accordance with the method of present invention. Upon completion of the construction of the Huffman code in  FIG. 9 , a Huffman shaper is employed. Referring to  FIG. 10 , Huffman shaper  1001  is loaded with information from a table similar to Table 1 above. The Huffman shaper information comprises one entry for each valid Huffman codeword and a corresponding entry for the channel symbol into which that Huffman codeword is mapped. The information is also sent to the receiver, using means available in the training procedure for the system. Then Huffman shaping proceeds during data transmission. 
         [0079]    Specifically, referring again to  FIG. 10 , data source  1003  generates (typically binary, but this is not required) data symbols at an adjustable rate controlled by the Huffman shaper  1001 . The data symbols are converted to pseudo-random form in a scrambler  1005 . The Huffman shaper  1001  generally comprises two parts, namely, a Huffman parser  1007  and a mapper  1009 . The Huffman parser  1007  accumulates outputs from the scrambler  1005 , symbol by symbol (e.g., bit by bit), until it accumulates a valid Huffman codeword. This codeword forms the input to the mapper  1009 . The mapper  1009  generates the channel symbol that corresponds to the Huffman codeword and passes the channel symbol to modulator  1011 , under the control of the modulator clock  1013 . The modulator clock  1013  defines the timing of the system. If required by the modulator clock  1013 , the Huffman shaper  1001  controls the rate at which it accumulates output symbols from the scrambler  1005 , in order to meet the demands of the modulator clock  1013 . 
         [0080]    Slicer/decision element  1015  maps the symbol received from the channel  1017  into its best estimate of the channel symbol transmitted by the remote transmitter. The Huffman encoder  1019  maps the estimated received channel symbol into a Huffman codeword, which is passed to the descrambler  1021 . The descrambler  1021  inverts the operation of the scrambler  1005 , and the resulting received sequence of data symbols is passed to the user  1023 . 
         [0081]    The Huffman shaper  1001  is modeled as being able to control the rate at which data are input to the shaper (see reference numeral  1025  of  FIG. 10 ). More colloquially, present-day communication systems often operate in an environment where a large buffer of data are available for transmission, and data can be removed from that buffer at any rate appropriate for the transmission medium. Therefore, ascribing an adjustable rate capability to the Huffman shaper  1001  does not burden the method of the present invention with functionality that is not already present in practical situations. 
         [0082]    As described above, a system that employs Huffman shaping carries a variable number of bits per modulation symbol. Therefore channel errors can introduce data in the receiver that is incorrect bit-by-bit, and that actually may contain the wrong number of bits as well. That is, referring to  FIG. 10 , if a channel symbol different from the one introduced at the input to the modulator  1011  is received at the output of the slicer/decision element  1015 , then both the bits and the number of bits passed to the Huffman encoder  1019  may be incorrect. To compensate for this potential effect, a framer/deframer may be introduced. 
         [0083]      FIG. 11  is another embodiment of the design of a Huffman code in accordance with the present invention, when a framer/deframer is utilized. Again, a PAM implementation is assumed, but extensions to higher-dimensional modulation are also possible. Referring to  FIG. 11 , the modulation scheme is characterized by parameters M, α, s, N b , N s , n s , and n p  acquired in block  1001 , from which are derived the constellation levels {a i ;i=0, 1, . . . , M−1} (block  1101 ). Parameters N b , N s , n s , and n p  define, respectively, the number of data bits, the number of modulation symbols, the number of synch bits, and the number of pointer bits in each symbol frame. The probability p i  then is calculated for each a i  in block  1103  for i=0, 1, . . . , M−1. Finally, a Huffman code for the symbols {a i } and their corresponding probabilities {p i } is constructed in block  1105 . 
         [0084]      FIG. 12  is a block diagram of another embodiment of a communication system that operates in accordance with the method of present invention, utilizing a framer/deframer. Upon completion of the construction of the Huffman code in  FIG. 11 , a Huffman shaper is employed. Referring to  FIG. 12 , Huffman shaper  1201  is loaded with information from a table similar to Table 1 above. The Huffman shaper information consists of one entry for each valid Huffman codeword and a corresponding entry for the channel symbol into which that Huffman codeword is mapped. A framer  1203  is loaded with parameters N b , N s , n s , and n p . The information is also sent to the receiver using means available in the training procedure for the system. In the receiver a deframer  1205  is loaded with the same parameters, N b , N s , n s , and n p . Then Huffman shaping proceeds during data transmission. 
         [0085]    Specifically, referring to  FIG. 12 , data source  1207  generates data symbols at an adjustable rate controlled by the Huffman shaper  1201 . The data symbols are converted to pseudo-random form in a scrambler  1209 . The scrambler  1209  output is collected in the framer  1203 , which arranges transmitted data in groups of N b  bits per symbol frame, N s  modulation symbols per symbol frame, n s  synch bits per frame and n p  pointer bits per frame as discussed above. The Huffman shaper  1201  generally comprises of two parts, a Huffman parser  1211  and the mapper  1213 . The Huffman parser  1211  accumulates outputs from the framer  1203 , symbol by symbol, until it accumulates a valid Huffman codeword. This codeword forms the input to the mapper  1213 . The mapper  1213  generates the channel symbol that corresponds to the Huffman codeword and passes the channel symbol to the modulator  1215  under the control of the modulator clock  1217 . The modulator clock  1217  defines the timing of the system. If required by the modulator clock  1217 , the Huffman shaper  1201  controls the rate at which it accumulates output symbols from the scrambler  1209  in order to meet the demands of the modulator clock  1217  (see reference numeral  1218  in  FIG. 12 ). 
         [0086]    The slicer/decision element  1219  maps the symbol received from the channel  1221  into its best estimate of the channel symbol transmitted by the remote transmitter. The Huffman encoder  1223  maps the estimated received channel symbol into a Huffman codeword. In this embodiment, switch  1225  is in position A. The deframer  1205  is able to distinguish individual received modulation symbols by means of the demodulator clock  1227  signal from the demodulator  1229 . It uses the received symbol frame as well as the synch and pointer bits to construct a serial data stream corresponding to the output of the scrambler  1209 . This output is passed to the descrambler  1231 , which inverts the operation of the scrambler  1209 , and the resulting received sequence of data symbols is passed to the user  1233 . 
         [0087]    In still another embodiment of the invention, the Huffman code constructed in a slightly modified fashion. This embodiment uses a one-dimensional form of the Huffman code described above. Specifically, a Huffman code is constructed for only the positive modulation symbols. After a Huffman code word has been collected in the transmitter by the Huffman decoder, the decoder uses its next input bit to define the sign of the modulation symbol corresponding to the collected Huffman code word. An inverse procedure is applied in the receiver. Again, a PAM implementation is assumed, but extension to higher-dimensional modulation is also possible. 
         [0088]    Referring to  FIG. 11 , the modulation scheme is characterized by parameters M, α, s, N b , N s , n s , and n p  acquired in block  1101 , from which are derived the constellation levels {a i ;i=0, 1, . . . , M−1} (block  1101 ). Parameters N b , N s , n s , and n p  define, respectively, the number of data bits, the number of modulation symbols, the number of synch bits, and the number of pointer bits in each symbol frame. The probability p i  is then calculated for each nonnegative a i  in block  1103  for i=0, 1, . . . , M−1. Finally, a Huffman code for the nonnegative symbols {a i } and their corresponding probabilities {p i } is constructed in block  1105 . 
         [0089]    Upon completion of the construction of the Huffman code in  FIG. 11 , a Huffman shaper is employed. Referring to  FIG. 12 , Huffman shaper  1201  is loaded with information from a table similar to Table 1 above. The Huffman shaper information consists of one entry for each valid Huffman codeword and a corresponding entry for the channel symbol into which that Huffman codeword is mapped. The framer  1203  is loaded with parameters N b , N s , n s , and n p . The information is also sent to the receiver using means available in the training procedure for the system. In the receiver the deframer  1205  is loaded with the same parameters, N b , N s , n s , and n p . Then Huffman shaping proceeds during data transmission. 
         [0090]    Specifically, data source  1207  generates data symbols at an adjustable rate controlled by the Huffman shaper  1201 . The data symbols are converted to pseudo-random form in scrambler  1209 . The scrambler  1209  output is collected in the framer  1203 , which arranges transmitted data in groups of N b  bits per symbol frame, N s  modulation symbols per symbol frame, n s  synch bits per frame and n p  pointer bits per frame, as discussed above. The Huffman shaper  1201  generally comprises two parts, the Huffman parser  1211  and the mapper  1213 . The Huffman parser  1211  accumulates outputs from the framer  1203 , symbol by symbol, until it accumulates a valid Huffman codeword. The Huffman parser  1211  then accumulates one additional input bit and appends it to the Huffman codeword. This Huffman codeword with the appended bit forms the input to the mapper  1213 . The mapper  1213  generates the channel symbol that corresponds to the Huffman codeword, and uses the appended bit to define the sign of the channel symbol. It then passes the channel symbol to the modulator  1215  under the control of the modulator clock  1217 . 
         [0091]    The slicer/decision element  1219  maps the magnitude of the symbol received from the channel  1221  into its best estimate of the magnitude of the channel symbol transmitted by the remote transmitter. It also estimates the sign of the received symbol. The channel symbol magnitude is passed to the Huffman encoder  1223 , which maps the estimated received channel symbol magnitude into a Huffman codeword and presents the output at the A input of switch  1225 . The sign of the received symbol is presented at the B input of switch  1225  by means of connection sign information  1235 . Switch  1225 , normally in the A position; is switched to the B position after each received Huffman code word, in order to accept the sign information  1235  from the slicer/decision element  1219 . The deframer  1205  is able to distinguish individual received modulation symbols by means of the demodulator clock  1227  signal from the demodulator  1229 . It uses the received symbol frame as well as the synch and pointer bits to construct a serial data stream corresponding to the output of the scrambler  1209 . This output is passed to the descrambler  1231 , which inverts the operation of the scrambler  1209 , and the resulting received sequence of data symbols is passed to the user  1233 . 
         [0092]      FIG. 13  illustrates one operation of a system that employs Huffman shaping in accordance with the present invention. A transmitter  1301  accepts user data (block  1303 ). The transmitter  1301  may also perform a framing operation ( 1307 ) to provide a means to recover from possible errors that may be introduced in the channel. 
         [0093]    The transmitter  1301  then implements Huffman shaping. Specifically, the transmitter  1301  accumulates source data until a Huffman codeword is recognized (block  1309 ), and then maps the resulting Huffman codeword into a channel symbol (block  1311 ). The transmitter then performs a modulation operation (block  1313 ), which optionally includes sequence coding to increase the signal distances between permitted symbol sequences. Finally, the modulated signal is applied to the input of the communications channel (block  1315 ). 
         [0094]    The receiver  1317  accepts the received signal from the channel output (block  1319 ), and demodulates it (block  1321 ). Demodulation generally includes such operations as timing tracking and equalization. The received signal is then subjected to a decision operation, which may optionally include sequence decoding (block  1323 ). The Huffman shaping (blocks  1309  and  1311 ) is inverted by applying the received signal to the input of a Huffman encoder (block  1325 ). The receiver  1317  then performs a deframing operation (block  1327 ), and communicates the received data to the user (block  1331 ). 
         [0095]    Based on the foregoing discussion, it should be apparent that in one embodiment of the invention, once data is received from a data source, the sequence of binary data bits is randomized by a scrambling operation and bits are mapped into channel symbols such that the channel symbols occur with a probability distribution suitable for achieving shaping gain. This is accomplished by accumulating scrambled data bits until a Huffman codeword is recognized, at which time the Huffman codeword is mapped into a channel symbol. Then the channel symbol is applied to the input of a communication channel. The probability of recognizing in the scrambled data sequence a particular Huffman codeword of length L bits is 2 −L . Hence, the channel symbol associated with that particular Huffman codeword will be transmitted with probability 2 −L . Note that this channel encoding operation via Huffman codes corresponds in the field of source coding to Huffman decoding. 
         [0096]    In one embodiment of the invention, the channel encoding operation described above is performed in combination with a framing operation to achieve transmission of data at a constant rate. In addition, channel symbols can be modulated in various ways before they are applied to the input of the communication channel. 
         [0097]    Next, on the receiver side of the communication channel a channel symbol is obtained at the demodulator output. The channel symbol is converted into the corresponding Huffman codeword. The sequence of bits represented by concatenated Huffman codewords is descrambled and delivered to the data sink. The described channel decoding operation corresponds in the field of source coding to Huffman encoding. 
         [0098]    In one embodiment of the invention, a deframing operation is performed, which provides for data delivery to the data sink at constant rate. In addition, the deframing operation limits the effect of channel demodulation errors, which can cause a temporal shift of the received binary data sequence. This shift can occur when a channel symbol is erroneously decoded whose associated Huffman codeword differs in length from the Huffman codeword associated with the correct channel symbol. 
         [0099]    The method of the present invention results in a symbol constellation and a probability distribution of symbols in this constellation that exhibits a shaping gain of greater than 1 dB. The shaping gain may be, for example, 1.35 dB or 1.5 dB, depending on the specific design. More specifically, for PAM constellations, shaping gains of up to ≈1.35 dB are achieved for some rates. For QAM constellations, shaping gains within 0.1 dB from the ultimate shaping gain are consistently obtained for rates of &gt;3 bit per dimension. 
         [0100]    In general, a communication system according to the present invention comprises a communication node that performs a Huffman decoding operation to generate channel symbols with a desired probability distribution. 
         [0101]    Many modifications and variations of the present invention are possible in light of the above teachings. Thus, it is to be understood that, within the scope of the appended claims, the invention may be practiced otherwise than as described hereinabove.