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for a discrete memoryless source can be made to approach zero by considering Cartesian products of the source. This implies that the relative entropy between the Maxwell—Boltzmann distribution and its dyadic approximation can be made to approach zero by considering Cartesian products of the basic constellation. Convergence in relative entropy implies £1 convergence of the probabilities [1, Section 12.6]; hence, the performance obtained from our dyadic approximations can be made to approach arbitrarily closely to the performance ob- tained by using the optimal Maxwell-Boltzmann distribution. Numerical calculations confirm the performance improvement obtained by working with Cartesian products of the basic constellations.

Table 2 for a discrete memoryless source can be made to approach zero by considering Cartesian products of the source. This implies that the relative entropy between the Maxwell—Boltzmann distribution and its dyadic approximation can be made to approach zero by considering Cartesian products of the basic constellation. Convergence in relative entropy implies £1 convergence of the probabilities [1, Section 12.6]; hence, the performance obtained from our dyadic approximations can be made to approach arbitrarily closely to the performance ob- tained by using the optimal Maxwell-Boltzmann distribution. Numerical calculations confirm the performance improvement obtained by working with Cartesian products of the basic constellations.

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Fig. 2. Normalized gain of spherical constellations drawn from various dense lattices with signal point selection performed according to the Maxwell—Boltzmann distribution.
Fig. 2. Normalized gain of spherical constellations drawn from various dense lattices with signal point selection performed according to the Maxwell—Boltzmann distribution.
Fig. 3. Comparison of the approximate and actual normalized gain of spherical constellations based on various .V-dimensional lattices. The dotted curves are obtained by applying continuous approximations for gain.
Fig. 3. Comparison of the approximate and actual normalized gain of spherical constellations based on various .V-dimensional lattices. The dotted curves are obtained by applying continuous approximations for gain.
fs < Bp. Note also that since the primary channel operates at a fixed rate, it can operate as a “standard” channel, and is not affected by the system problems associated with variable rate transmission. The “opportunistic secondary channels” of [2] and the “in-band coding method” of [21] (see also [17]-{20]) are examples of nonuniform signaling schemes that use prefix codes to separate data into primary and secondary channels, although these schemes are not described in terms of prefix codes.
fs < Bp. Note also that since the primary channel operates at a fixed rate, it can operate as a “standard” channel, and is not affected by the system problems associated with variable rate transmission. The “opportunistic secondary channels” of [2] and the “in-band coding method” of [21] (see also [17]-{20]) are examples of nonuniform signaling schemes that use prefix codes to separate data into primary and secondary channels, although these schemes are not described in terms of prefix codes.
Fig. 8. Illustrating the performance of dyadic approximations to the Maxwell-Boltzmann distribution obtained from the Huffman procedure.
Fig. 8. Illustrating the performance of dyadic approximations to the Maxwell-Boltzmann distribution obtained from the Huffman procedure.
PERFORMANCE OF HUFFMAN-CODED SIGNAL CONSTELLATIONS BasED on Z? of shaping gain y, itself is not our aim; rather, we have attempted to maximize the combination -y,7y¢. Further numeri- cal calculations obtained from dyadic approximations to the Maxwell—Boltzmann distribution with parameter 1 > Aopt show that, in some cases, the effective dimension Neg can be made to increase significantly above the values shown in Tables I-III. Note also that PAR2 and CER2 values shown in Tables I-IH are all quite reasonable, especially when com- pared to the PAR2 and CER2 of large-dimensional Voronoi constellations [6]. The PAR2 and CER2 can, in principle, 9¢ improved by sacrificing some shaping gain. Indeed, nu- merical calculations show that dyadic approximations to the Maxwell—Boltzmann distribution with parameter A < opt will, in general, result in improved PAR2 and CER2, with some corresponding sacrifice in overall gain. We have also applied this procedure to multidimensional constellations, with similar results. However, since many multidimensional con- stelJations are best implemented as coset codes (see [10] and [11}), it may be preferable to use a nonuniform signal point selection scheme that is suited for a coset code (as described in the next section) rather than a direct mapping of the words of a prefix code onto the constellation points.
PERFORMANCE OF HUFFMAN-CODED SIGNAL CONSTELLATIONS BasED on Z? of shaping gain y, itself is not our aim; rather, we have attempted to maximize the combination -y,7y¢. Further numeri- cal calculations obtained from dyadic approximations to the Maxwell—Boltzmann distribution with parameter 1 > Aopt show that, in some cases, the effective dimension Neg can be made to increase significantly above the values shown in Tables I-III. Note also that PAR2 and CER2 values shown in Tables I-IH are all quite reasonable, especially when com- pared to the PAR2 and CER2 of large-dimensional Voronoi constellations [6]. The PAR2 and CER2 can, in principle, 9¢ improved by sacrificing some shaping gain. Indeed, nu- merical calculations show that dyadic approximations to the Maxwell—Boltzmann distribution with parameter A < opt will, in general, result in improved PAR2 and CER2, with some corresponding sacrifice in overall gain. We have also applied this procedure to multidimensional constellations, with similar results. However, since many multidimensional con- stelJations are best implemented as coset codes (see [10] and [11}), it may be preferable to use a nonuniform signal point selection scheme that is suited for a coset code (as described in the next section) rather than a direct mapping of the words of a prefix code onto the constellation points.
Fig. 9. The sequence of subconstellations {A, B, C, D} is determined by an Ungerboeck code. Signal point selection within each subconstellation is performed using a simplex prefix code.
Fig. 9. The sequence of subconstellations {A, B, C, D} is determined by an Ungerboeck code. Signal point selection within each subconstellation is performed using a simplex prefix code.
signal point selector § is no longer memoryless in this case. Other partitions of Z?, e.g., the 8-way partition Z?/2RZ? or the 16-way partition Z?/4Z?, allow the use of more powerful coset codes, many of which are listed in [10, Tables IV, V, IX, X, XI]. Unlike the 4-way partition 27/277, the cosets in these partitions do not all have the same theta series. For example, the 8-way partition of Z? + (1/2, 1/2), has two classes of cosets [typified by A = 2RZ? + (1/2, 1/2) and B = 2RZ? + (3/2, 1/2)] with weight enumerators
signal point selector § is no longer memoryless in this case. Other partitions of Z?, e.g., the 8-way partition Z?/2RZ? or the 16-way partition Z?/4Z?, allow the use of more powerful coset codes, many of which are listed in [10, Tables IV, V, IX, X, XI]. Unlike the 4-way partition 27/277, the cosets in these partitions do not all have the same theta series. For example, the 8-way partition of Z? + (1/2, 1/2), has two classes of cosets [typified by A = 2RZ? + (1/2, 1/2) and B = 2RZ? + (3/2, 1/2)] with weight enumerators
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MathematicsComputer ScienceInformation TheoryElectrical and Electronic EngineeringAmplitude ModulationMaxwell Boltzmann DistributionBoltzmann distributionGaussian ChannelPossibility DistributionCoded Modulation
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