Comparison of TCM + RS Coding and Turbo Codes for Spectrally ...

the TCM concatenated with RS coding for systemsrequiring high spectral efficiency and low delay, restrictingthe delay to a level that is manageable in amodern communication system. In Section 2 we givea short introduction to the different coding schemeswe are focusing on. Then we define the system and itsmain parameters in Section 3. Section 4 is devoted tothe performance of the different systems and finallywe draw some conclusions in Section 5.2. CODING SCHEMES2.1. Trellis coded modulation concatenated withReed Solomon codingTCM was introduced by Ungerboeck in 1976 [10].He showed that for bandlimited channels, substantialcoding gain could be achieved by convolutional codingof signal levels (rather than coding of binary sourcelevels). He merged coding and modulation throughthe use of high-order signal constellations, increasingthe Euclidean distance between signal sequences.Reed Solomon codes were introduced in a paperby Reed and Solomon in 1960 [3]. They operate overfinite fields and are described by consecutive powersof a root of a primitive polynomial or consecutivezero coordinates in the Galois Field Fourier Transformof code words. They possess a design algorithmto achieve an arbitrary level of error correction. ReedSolomon codes are special cases of BCH codes whichpossess more rigid structure and good distance properties.Reed and Solomon published an inefficient decodingalgorithm in their 1960 paper. Berlekamp publishedan efficient decoder in 1967 [11].In this paper we will use the notation RS(N,K)to describe a Reed Solomon code with N symbols inthe code word generated from K information symbols.This code will be capable of correcting (N − K)/2symbol errors in each block of length N.Figure 1: Schematic view of the concatenated TCM+ RS system.Figure 2: Schematic view of the parallel concatenatedconvolutional turbo coding system.2.2. Parallel concatenated convolutional turbocodesPCC turbo codes are the type of codes introducedby Berrou, Glavieux, and Thitimajshima in 1993 [12].The encoder consists of two or more constituent recursiveconvolutional encoders in parallel, joined throughinterleavers. Proper design of the interleavers in theturbo encoder could result in a code with rather largeminimum distance, despite the fact that the constituentcodes are convolutional codes with low minimum distances.Output is taken from the input data andthe output of the convolutional encoders. Outputbits are usually mapped to a signal constellation withGray mapping. To obtain high spectral efficiency,puncturing of the turbo code is one possibility. An-other approach is to use high rate, non-punctured constituentcodes where only systematic bits from thesecond component encoder are punctured. We applythe latter technique in our PCC turbo codes.The optimal decoder is computationally intractable.The iterative turbo decoding algorithm is a low complexitysub-optimal decoding algorithm that providesnear Shannon limit performance across additive whiteGaussian noise (AWGN) channels. The drawback ofthe non-puncturing encoding scheme is the increasedcomplexity of the binary iterative turbo decoding algorithm.This can be overcome by considering decodingalgorithms working on a trellis of the dualcode [13].

BER−curves, Monte Carlo simulationsBER−curves, Monte Carlo simulations and estimated10 −210 −310 −410 −510 −610 −710 −810 −910 −1 E b/N 0[dB]10 −10BER10 −15PCC code, interleaver 1PCC code, interleaver 2PCC code, interleaver 3128TCM + RS(255,243)256TCM + RS(255,211)13 13.5 14 14.5 15 15.5 16 16.5 17 17.510 −2010 −5 E b/N 0[dB]BERPCC code, interleaver 1, Part 1 + 2PCC code, interleaver 2, Part 1 + 2PCC code, interleaver 3, Part 1 + 2128TCM + RS(255,243), Part 1 + 2256TCM + RS(255,211), Part 1 + 213 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18Figure 3: Bit error rate as a function of E b /N 0 for thedifferent coding schemes. Monte Carlo simulations.η = 6.2 bit/s/Hz.occur within a block. This simplification has only minoreffect on the results.A BER below 10 −13 is required in many communicationnetworks. This is however not practicalto simulate in a Monte Carlo simulation. Figure 4therefore shows estimated curves for the error floorsof the turbo codes and estimated upper bounds forthe TCM+RS BER-curves (part 2 of curves). Thecurves from the Monte Carlo simulations are kept inthis plot for comparison (part 1 of curves). The errorfloors of the turbo codes are estimated with the unionbounding technique as described in [20]. The upperbound BER-curves for TCM+RS are estimated withthe technique presented in [14], p. 333. This techniquetakes the BER before RS decoding as input, andthese values were found in a Monte Carlo simulationof a system with TCM only. The method also assumesperfect interleaving (independent errors), and the actualBER will be higher than this upper bound in asystem with burst errors and poor interleaving. Witha good choice of interleaver, it is possible to achieveBERs close to or even below this upper bound also forinterleavers with moderate delay. As we can see fromFigure 4, the slope of the simulated TCM+RS BERcurveis not as steep as the estimated upper boundfor low BERs. We note that the lines cross at a BERabout 10 −7 . This is due to the delay restrictions whichresulted in the choice of the interleaver with B = 8and M = 31. The TCM + RS code will howevernot give an error floor, as is the case for the PCCturbo codes, and the curve is expected to follow theestimated upper bound quite well also for low BERs.From Figures 3 and 4 we can see that the 128TCM-4D combined with RS(255,243) has better performanceFigure 4: Bit error rate as a function of E b /N 0 forthe different coding schemes. Part 1 of curves: MonteCarlo simulations. Part 2 of curves: Estimated curvesfor low BERs. η = 6.2 bit/s/Hz.than the 256TCM-4D combined with RS(255,211).Even though the latter curve is steeper and will eventuallycross the other line, this will happen at a BERfar below the region of interest. For the same spectralefficiency a high rate RS code concatenated witha suitable TCM seems to be better than a more powerfulRS code concatenated with a more spectral efficientTCM. In the further discussion, only the 128TCM-4D + RS(255,243) will be considered.For the turbo codes we can see that it is possible tooptimize the code with respect to the error floor andachieve an error floor well below 10 −13 (interleavers2 and 3). This is achieved through a smart design ofthe bit-interleaver as described in [20]. A very lowerror floor will however degrade the performance inthe waterfall region (BER between 10 −3 and 10 −9 ).The turbo code has 0.6 dB higher E b /N 0 at BER =10 −6 with interleaver 3 compared to interleaver 1.When comparing the turbo codes to the TCM+RScode, we can see that the turbo codes give a gain of1.1 - 1.8 dB at BER = 10 −6 . The curves for the turbocodes have however an error floor, i.e. the curve is notas steep as the TCM+RS curve at low BERs. Atsome point the curves will therefore cross. The turbocode with the best performance at BER = 10 −6 (interleaver1) has the highest error floor, and crossesthe TCM+RS curve above BER = 10 −13 . The errorfloors of the turbo codes with interleavers 2 and3 cross the TCM+RS curve below a BER of 10 −19 .Consequently these turbo codes have better performancethan the TCM+RS code for all BERs of interest.The improvement is at the cost of higher complexity,which in turn might increase the delay when

the limited processing speed in an implementation isincluded in the delay budget. The performance of theTCM+RS code might be improved by increasing thenumber of states in the trellis code. This improvementis also at the cost of higher complexity, but thedelay will not change much.5. CONCLUSIONSThis paper compares the performance of concatenatedTCM+RS to that of PCC turbo codes for systemsrequiring very high spectral efficiency and low delay.We have shown that it is possible to design PCC turbocodes with better performance than TCM+RS for allrelevant BERs. When a low error floor is required,the gain is about 1.5 dB at BER = 10 −6 . Becauseof the error floor of the PCC turbo codes, the curveswill cross. However, with careful design of the PCCcode, the crossing is at a level of minor interest inmost applications. The improvement is at the cost ofhigher complexity, which in turn might increase thedelay when the limited processing speed in an implementationis included in the delay budget. The performanceof the TCM+RS code might be improvedby increasing the number of states in the trellis code.This improvement is also at the cost of higher complexity,but the delay will not change much.6. REFERENCES[1] G. Ungerboeck, “Trellis-coded modulation withredundant signal sets, part I: Introduction,”IEEE Communications Magazine, vol. 25, pp. 5–11, February 1987.[2] L.-F. Wei, “Trellis-coded modulation with multidimensionalconstellations,” IEEE Trans. onInformation Theory, vol. 33, pp. 483–501, July1987.[3] I. Reed and G. Solomon, “Polynomial codesover certain finite fields,” Journal of the Societyfor Industrial and Applied Mathematics, vol. 8,pp. 300–304, June 1960.[4] C. Berrou and A. Glavieux, “Near optimum errorcorrection coding and decoding: Turbo-codes,”IEEE Trans. on Communications, vol. 44,pp. 1261–1271, October 1996.[5] R. M. Pyndiah, “Near optimum decoding ofproduct codes: Block turbo codes,” IEEE Trans.on Communications, vol. 46, pp. 1003–1010, August1998.[6] R. G. Gallager, Low-Density Parity-Check Codes.Cambridge, Massachusettes: M.I.T Press, 1963.[7] D. J. C. MacKay, “Good error-correcting codesbased on very sparse matrices,” IEEE Trans. onInformation Theory, vol. 45, pp. 399–431, March1999.[8] J. Li, K. R. Narayanan, and C. N. Georghiades,“Product accumulate codes: A classof capacity-approaching, low complexitycodes,” IEEE Trans. on Information Theory,submitted for publication, June 2001,http://ee.tamu.edu/~krishna/.[9] T. Flo, P. Orten, and B. Risløv, “Evaluationof coding schemes for spectrally efficient lowdelayradio systems,” in Proc. of 3rd InternationalSymp. on Turbo Codes & Related Topics,(Brest, France), pp. 387–390, September 2003,preprint accepted for publication.[10] G. Ungerboeck, “On improving data-link performanceby increasing channel alphabet and introducingsequence coding,” in Proc. of IEEE InternationalSymp. on Information Theory (ISIT),(Ronneby, Sweden), June 1976.[11] E. Berlekamp, Algebraic coding theory. NewYork: McGraw-Hill, 1968.[12] C. Berrou, A. Glavieux, and P. Thitimajshima,“Near Shannon limit error-correcting coding anddecoding: Turbo-codes,” in Proc. of IEEE InternationalConference on Communications (ICC),(Geneva, Switzerland), pp. 1064–1070, vol. 2,May 1993.[13] S. Riedel, “MAP decoding of convolutional codesusing reciprocal dual codes,” IEEE Trans. on InformationTheory, vol. 44, pp. 1176–1187, May1998.[14] G. C. Clark, Jr. and J. B. Cain, Error-CorrectionCoding for Digital Communications. PlenumPress, third ed., 1988.[15] S. N. Crozier, “New high-spread high-distance interleaversfor turbo-codes,” 2001, preprint.[16] K. S. Andrews, C. Heegard, and J. Kozen, “Interleaverdesign methods for turbo codes,” inProc. of IEEE International Symp. on InformationTheory (ISIT), (Boston, MA), p. 420, August1998.[17] M. Breiling, S. Peeters, and J. Huber, “Interleaverdesign using backtracking and spreadingmethods,” in Proc. of IEEE International Symp.on Information Theory (ISIT), (Sorrento, Italy),p. 451, June 2000.

[18] “Multiplexing and channel coding (FDD),” 3rdGeneration Partnership Project, June 1999, 3GTS 25.212.[19] E. Rosnes and Ø. Ytrehus, “Improved algorithmsfor the determination of turbo code weight distributions,”IEEE Trans. on Communications, acceptedfor publication, 2003.[20] E. Rosnes and Ø. Ytrehus, “On lowering the errorfloor of bit-interleaved turbo-coded modulation,”2003, preprint.