1462 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 10, OCTOBER 1999

The Constellation-Shaping Algorithm Using Closed-Form Expressionsfor the Number of Ring Combinations

Corneliu Eugen D. Sterian,Senior Member, IEEE

Abstract—The V.34 high-speed modem combines a numberof advanced techniques including constellation shaping by shellmapping. This technique makes use of some functions represent-ing the number of ring combinations with a given cost. In thiscorrespondence, closed-form expressions for these functions aregiven and used in the constellation shaping by the shell-mappingalgorithm.

Index Terms—Constellation shaping, ring, shell mapping.

I. INTRODUCTION

T HE V.34 modem contains a sophisticated “tool box”that seamlessly combines a number of advanced tech-

niques [1]–[3]. Two of these innovative techniques are thefour-dimensional (4-D) trellis-coded modulation (TCM) andconstellation shaping. The first one has been introduced in apaper by Wei [4]. A 2 -D point is simply the concatenationof two-dimensional (2-D) points, which are transmitted byquadrature amplitude modulation in successive signalingintervals. For transmitting bits per signaling interval ac-cording to Wei, this amounts to adding only redundantouter points, a power of two, to the points of the basicconstellation, which contains theinner points.The subset ofinner points is partitioned into so-calledinner groupsofsize each; therefore, the 2-D constituent constellation ofthe 2 -D signal set has equal-sized groups amongwhich the outer point group has the largest average energy. Inthe original work of Wei, the inner groups have equal averageenergy, but actually this is not necessary; they could be equal-sized concentric rings as well. As the outer points are used

times less often than the inner points, this resultsin a constellation shaping. In order to select the inner andouter groups with the requested frequency, the trellis-codedmodulator of Wei includes a so-called2N-D block encoder.This is the forefather of theshell mapperin the V.34 modem,and its mapping frame consists of a single 2-D symbolinterval. A small step toward the shell mapping has beenmade in [5], where the 2-D TCM method of Wei has beenextended to , not a power of two. In order to maintain themapping frame equal to a single 2-D interval, two outerrings of equal size, instead of only a single ring, are used.The constellation shaping based on concentric rings has beenintroduced by Calderbank and Ozarow [6]. However, as much

Paper approved by M. Fossorier, the Editor for Coding and CommunicationTheory of the IEEE Communications Society. Manuscript received November19, 1998; revised March 31, 1999.

The author is at Sos Pantrelimon 233, Bl. 68, Sc. B, Ap. 46, Sector 2,Bucharest, Romania.

Publisher Item Identifier S 0090-6778(99)07787-9.

Fig. 1. 96-point 2-D signal constellation partitioned intoM = 12 ringsnumbered from 0 to 11. The number over a signal point is the ring label.

as we know, the first reference on shell mapping in the openliterature is a paper by Fortier, Ruiz, and Cioffi [7].

II. SHELL-MAPPING ALGORITHM IN A NEW FORM

Number the rings from 0 to in increasingorder of the average energy of the rings. As an example,Fig. 1 shows a 96-point 2-D signal constellation partitionedinto rings numbered from 0 to 11. Each of the

rings is assigned a cost , whichapproximates the average energy of the points in the ring. Tosimplify the implementation, has been chosen. Thischoice is actually the one which is used in the V.34 modemand thus has practical importance. From a theoretical point ofview, the optimal choice would have been the average energyof the points inside the ring, but this is not easy to manage.Define ashell as a definite combination of rings. The costof a shell is the sum of the costs of thecomponent rings.

In a V.34 modem, . Therefore, there are ringcombinations, but only input bit combinations.To minimize the average cost, shell mapping selects thecombinations that have the least total cost. As in [1]–[3], define

as the number of -ring combinations that have costand 8. Define also as the number of eight-

ring combinations with cost less than. Note that all of these

0090–6778/99$10.00 1999 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 10, OCTOBER 1999 1463

functions have as a parameter. Since the closed form ofthese functions was not available, for every possible value of

, they were precomputed using the method of the generatingfunction of costs and recorded in a read-only memory. Theclosed form of the functions is given in the Appendix. Forthe derivation of these formulas, which is somewhat tedious,interested readers are kindly invited to write to the author.Using these closed-form expressions, we have reformulatedthe shell-mapping algorithm as follows.

On the set of eight-ring combinations, or shells, introducea relation of order as follows.

1) An eight-ring combination ( ) of total costprecedes (is “smaller than”) another eight-ring combi-

nation ( ) of total cost if .2) If two eight-ring combinations have the same total cost,

i.e., , form the numbers and written inbase as follows:

(1)

(2)

If , then the eight-ring combination of total costprecedes the eight-ring combination of total cost.The shell-mapping algorithm implemented in the V.34 mo-

dem establishes a bijection from the set of the firstbinarynumbers to the ordered set defined as above of the firsteight-ring combinations out of a total of .

In the hand-shaking period, after the two modems haveagreed as to which value of to use, compute at most eightvalues of for , such that

. Store these values in a small random-accessmemory.

Write the input bits to be sent as the binary number. After comparing to at most eight thresholds

find that positive integer for which

(3)

Then, the total cost is such that

(4)

Using a single formula, compute in this range of valuesuntil

(5)

Then, the total cost of the eight-ring combinations corre-sponding to equals . There are eight-ring combina-tions of total cost . Find that for which

(6)

Then, the costs of the eight rings are simply the coefficientsin (1).

III. CONCLUSIONS

The functions representing the number of ring combinationswith a given cost have been given in a closed form (seeAppendix), which was not known in the open literature.

Using these closed-form expressions, a new description of theconstellation shaping by the shell-mapping algorithm has beengiven. This can simplify the implementation, saving memoryand processing time.

The nice form of the functions suggests that similar formulasmay be obtained for other number of ring combinations, forinstance 16, which may be useful in the future.

APPENDIX

FUNCTIONS , AND IN CLOSED FORM

The function represents the number of rings that havecost , namely

if and otherwise

(A.1)

The function represents the number of two-ring com-binations that have cost. We have two cases.

Case 1— :

Case 2— :

(A.2)

The function represents the number of four-ringcombinations that have cost. We have four cases.

Case 1— :

Case 2— :

Case 3— :

Case 4— and :

(A.3)

The function represents the number of eight-ringcombinations that have the cost. We have eight cases, asshown in (A.4) at the top of the next page.

Define as

(A.5)

There are eight cases, as shown in (A.6) at the bottom ofthe next page.

1464 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 10, OCTOBER 1999

if

if

if

if

if

if

if

if and (A.4)

if

if

if

if

if

if

if

if and (A.6)

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 10, OCTOBER 1999 1465

REFERENCES

[1] ITU-T, “V.34—A modem operating at data signaling rates of up to28,800 bit/s for use on the general switched telephone network and onleased point-to-point 2-wire telephone-type circuits,” Sept. 1994.

[2] M. V. Eyuboglu, G. D. Forney, Jr., P. Dong, and G. Long, “Advancedmodulation techniques for V. fast,”Europ. Trans. Telecommun.,vol. 4,pp. 243–256, July 1993.

[3] G. D. Forney, Jr., L. Brown, M. V. Eyuboglu, J. L. Moran, III, “TheV.34 high-speed modem standard,”IEEE Commun. Mag.,vol. 34, pp.28–33, Dec. 1996.

[4] L.-F. Wei, “Trellis-coded modulation with multidimensional constel-lations,” IEEE Trans. Inform. Theory,vol. IT-33, pp. 483–501, July1987.

[5] C. E. D. Sterian, “Wei-type trellis-coded modulation with 2N -dimensional rectangular constellation forN not a power of two,”IEEE Trans. Inform. Theory,vol. 43, pp. 750–758, Mar. 1997.

[6] A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on theGaussian channel,”IEEE Trans. Inform. Theory,vol. 36, pp. 726–740,July 1990.

[7] P. Fortier, A. Ruiz, and J. M. Cioffi, “Multidimensional signal setsthrough the shell construction for parallel channels,”IEEE Trans.Commun.,vol. 40, pp. 500–512, Mar. 1992.