Higher State Trellis Coded Modulation for Asymmetric DigitalSubscriber Transceivers

Muhammad Arif, Student Member IEEE, Dr Noor M. Sheikh, Member IEEEElectrical Department, University of Engineering & Technology Lahore, Pakistan

arif.wahlaggmail.com

ABSTRACT

A 128 state 4D TCM code for ADSL assumingAWGN is analyzed. The performance is compared with 16state (3, 2, 5) linear 4D code developed by L. F. Wei [1].Simulation shows an improvement of 20% in spectralefficiency. We have also analyzed 8-DMT Symbol baseddecoding for Wei's code 16 state (3,2 ,5) in up streamdirection and achieved .9 db gain.Key Words: Spectral Efficiency. Trellis CodedModulation (TCM). Asymmetric Digital Subscriber Line(ADSL).

I. INTRODUCTION

The fundamental parameters that control the rateand quality of information transmission are channelbandwidth W and signal to noise ratio SNR. The rate oftransmission is a measure of efficient use of channelbandwidth which is related to spectral efficiency 7i. Bit errorrate BER is the measure of transmission quality. There is astrong trade off between SNR, W, 7 and BER, whichdepends on the type of modulation used.

The ADSL operates in bandwidth limited regionwhere bandwidth is restricted, powerful coding techniquesare required to improve the spectral efficiency withoutexpanding the bandwidth. Using Shannon's theorem for thechannel capacity C of a bandlimited additive whiteGaussian noise (AWGN) channel with bandwidth W , aparameter 7 called the spectral efficiency can be representedas

q = C /W = log 2(1 + SNR) bits/Hz/s (1)where W 1 / T , T being the interval per symbol andSNR = Es No = qEb /No is energy per symbol/noise

density, Eb being the energy per bit.

Eb INO>(27-11/) (2)

and 0<7<CIW.The value of 7 could be improved by using higher

level modulation schemes. This however would increase thecomplexity of implementation. A balance needs to beachieved depending on the application [2].

We analyze and discuss the performance of a higher

state Trellis Coded Modulation (TCM) for ADSLapplications.

The ADSL utilizes Discrete Multitone Modulation(DMT), which is a common form of multicarrier modulation.The basic of DMT is Quadrature Amplitude Modulation(QAM). In ADSL using DMT, the transfer channel (twist-pair copper phone line) is partitioned into equal bandwidth256 subchannel. All subchannels have a bandwidth of4.3125 kHz. The first subchannel is used by POTS (PlainOld Telephone Service) for the traditional voicetransmission. Neither ADSL nor POTS uses the subchannels2 - 6 (4.3125 - 25.875 kHz) for avoiding the interferencebetween them. Subchannels 7 - 32 (25.875 - 138 kHz) areused for ADSL upstream and subchannels 33 - 256 (138 -

1104 kHz) are used for ADSL downstream. According toShannon's noisy channel coding theorem, the maximumnumber of bits that can be carried by subchannel dependson its Signal to Noise ratio (SNR) based on its channelresponse and is given by

bi = log2(1 + SNR ) (3)The total bits transported by the channel can be expressed as

n2

btotal = Ebi (4)ni

For the upstream, n1 equals 7 and n2 equals 32. For the

downstream, n1 equals 33 and n2 equals 256. Therefore,the capacity of the channel, is given by[3]

n2C = EVJi = 4.3125btotal

n,

II. TRELLIS CODEDD MODULATION

Trellis coded modulation (TCM ) is an integratedmodulation and coding technique to devise an effectiveapproach for mapping the coded bits into signal points, insignal space, such that minimum squared Euclidean distance(MSED) is maximized to achieve significant coding gainwithout requiring extra bandwidth.

Ungerboeck [4-6] constructed trellis codes foramplitude modulation (AM), phase shift keying (PSK) andquadrature amplitude modulation (QAM) schemes. The firstimportant insight due to Ungerboeck was his realization thatfor the nonbinary signal constellation used in TCM,

(5)

minimum free squared distance d2free in Euclidean space,rather than minimum free Hamming distance dfree, is theprimary parameter that determines code performance. Histechnique of "mapping by set partitioning," as opposed tothe conventional Gray mapping used in uncoded modulation,turned out to be the key to achieving performance gains withTCM compared to uncoded modulation at high spectralefficiencies.

Due to the bandwidth constraints of the channel, theuse of voice modems over voice grade telephone lines waslimited to low data rates. A clear need developed for ascheme which would allow constellation expansion at thesame signaling rates, thus achieving higher data rates and yetprovide a coding gain to overcome the noise margin lost bythe closer packing of signals. TCM proved to be just such ascheme, and combined with some sophisticated signalprocessing techniques, resulted in a series of improvementswhich pushed the modem speeds up to 56 kbps full duplex.The 16 state, (3,2,5) linear four dimensional (4D) codedeveloped by L.F.Wei[4] and adopted by ITU-T as one ofthe codes for the v.34 standard has also been used in ADSL.

III. 128 STATE TCM CODE FOR ADSL

The maximum number of bits bi that can becarried by subchannel i depending on its SNR based on itschannel response is calculated according to (1). To sendbi information bits in each signaling interval using TCM, a

2D constellation of 2bi+ points is used. The inherent cost ofthis scheme is that the size of the 2D constellation is doubledover uncoded modulation scheme. This is due to the fact thata redundant bit is added every signaling interval. This costcan be reduced to half by combining 2 bi information bits oftwo DMT tones (subchannels) and using a 4D constellationwith a trellis code of rate k I k + 1.

For the purpose of simplicity we have assumed thenumber of information bits bi, 3 and 4 for the pair ofconsecutive DMT tones. The r 2 /3 convolutional codeadds one redundant bit which completes the pair of 2D16QAM points. The two constituent 2D points are combined

U2

to make 4D point to map eight bits. Of these eight bits one isredundant bit and seven information bits, which means therelevant uncoded modulation scheme is a non integralmodulation combination of 8 QAM and 16 QAM. This pointis marked as 8/16 QAM on uncoded modulation curve inFig.4, which will serve as basic reference to measure thecoding gain and spectral efficiency 7 of trellis codes. Theuncoded modulation has the spectral efficiency 7 = 3.5 atSNR of 14.5 db. Next we have implemented the 16 state,(3, 2, 5), (4D) code developed by L.F.Wei and used inADSL. This code serves as reference to our 128 state codefor comparison of reduction of required SNR andimprovement in spectral efficiency 7 . The spectralefficiency of this code is 7 = 5 for the down stream at thesame SNR and BER. The coding gain of this code is 3.6 dbfor up stream 2.4 db for down stream at the same BER and 7i.

We have used the rate r = 2 / 3 systematic recursiveconvolutional code (3, 2, 8) shown in Fig. 1 for our 128 statetrellis code shown in Fig. 2. The trellis encoder takes a set ofinformation bits U = [U1,U2,U3,...,u ] as its inputs.Because of the four dimensional nature of the encoder, eachword u is encoded into two binary words v and w, whichare modulated into two constellation points for thesuccessive subchannels. The TCM portion of the encoderhas three input information bits: two coded (u1 and U2) and

one uncoded (U3 ). The convolutional encoder produces

three output bits: u1 and u2 (information bits) as well as uo(a parity bit). Along with uncoded bit u3, they enter the 4D

signal mapper. The output bits of the encoder ( uo, u1,u2 )are used to select one of the eight 4D subsets. The subset ismapped to two indices that determine the Least SignificantBits (LSBs) of v and w. The mapping method is:

Vo - U3

V1 =UI ® U3

Wo =U2 U3WI = Uo ( Ul ( U2 (i U3

U2

H(0) = (203)8 = (10 000 011)2H(1) = (014)8 = (00 001 100)2H(2)= (042)8 = (00 100 010)2

flip horizontally (110 000 01)2]flip horizontally (001 100 00)2flip horizontally (010 001 00)J

Polynomials

128 State Rate 2/3 Convolutional Encoder for TCMFig. I

The remaining bits of u, which are the additional uncodedinformation bits needed to achieve the desired spectralefficiency 7, are directly mapped to the Most SignificantBits (MSBs) of v and w. These are used to select theparticular 4D signal point to be transmitted. The number ofthese bits depends on bi for particular successivesubchannels. For the example under discussion these arefour bits ( u4, U5,u 6, U7 ).

TCM decoder reads a pair of constellation pointsfrom a pair of successive DMT subchannels and takes themas its input. Soft-decision Viterbi decoding is used to decodethe 4D codes. The output of the Viterbi decoder is anestimated sequence of received constellation points. Afterthat, QAM decoder is applied to convert the constellationpoints into a set of bits. Fig. 3 illustrates the ViterbiUn

U,

U7 No

U6

U5

Wn

W3

W2

gain of up stream closer to coding gain of down stream i.e.2.4+.9 = 3.3 db. The cost of this modification is increaseddecoding delay but still it is less than the decoding delay ofdown stream direction. 8-DMT symbol based decodingresolves the problem of low channel capacity of up streamtransmission in ADSL.

For each received 4D Doints

First received2D points

Find closest pointin each 2D subsets

and its metric

Second received2D points

Find closest pointin each 2D subsets

and its metric

* V3

U4Find closest points in each 4D subsets and its metric

VI

WI

VO

WO

Fig. 2 128 State TCM

decoding process. The metric used in the decoding is theEuclidean distance. The 4D metric can be obtained byadding the two 2D subset metrics for the pair of 2D subsetscorresponding to that 4D subset.

IV. 8-DMT Symbol Based Decoding

The coding gain of 16 state, (3, 2, 5), 4D codedeveloped by L.F.Wei and used in ADSL is 3.6 db for upstream 2.4 db for down stream. The coding gain in up streamdirection is 1.2 db below the gain for down stream, which isdue to the shorter trace back length of up stream DMTsymbol. Typical problem of low channel capacity inupstream direction for long loops is faced due to decreasedcoding gain of up stream. So there is a benefit to investigatemodification to the existing trellis code. The size of DMTsymbol in up stream is 13 pairs of tones compared to 112pairs of tones in down stream. We have exploited this shortfall and implemented 8-DMT symbol based decoding, wheremaximum likelihood soft decision viterbi decoding isapplied to up stream after accumulating eight DMT symbols.The decoding length has now increased from 13 to 104. Thecoding gain is .9 db shown in Fig. 4, which brings the coding

Extend trellis diagram and choose the estimated points

Demodulate each pair of estimated points into twobinary words, v and w

Map the 2 LSBs of v and w to (ul, u2, u3). Directly mapthe remaining bits of v and w to the MSBs of u

Fig. 3 Trellis Decoder

V. SIMULATION RESULTS

Since the SNR is different on the each subchannel,an average SNR of the overall system is computed. Theaverage SNR is calculated as

I N

SNR = SNR. (6)N,

where N is the number of subchannels.The simulation results of BER performance of different

systems are shown in Fig. 4. The 16 state, (3, 2, 5) linear 4Dcode developed by L.F.Wei has been implemented for downstream as well as for up stream. The coding gain in downstream and up stream against uncoded QAM system is 3.6

r~~~~~~~

v

No

.

V2

L

db and 2.4 db respectively.The .9 db coding gain of 8-DMT symbol based

decoding is also shown, which bridges up the gap betweenthe down and up stream gains of ADSL. This gain isproposed to resolve the problem of low channel capacity ofup stream transmission in ADSL for long loops.

The 128 state, (3, 2, 8) linear four 4D codepresented in this contribution, achieves an additional codinggain of 1.4db for ADSL over the 16 state, (3, 2, 5) linear 4DWei's code (Fig. 4). The results are tabulated in Table. 1.

Table. 1 Results of Current Research

The spectral efficiency of different systems isshown in Fig. 5.The spectral efficiency of non integral 8/16QAM is 7 = 3.5 bits IHzls. The spectral efficiency achievedby 128 state, (3, 2, 8) linear 4D code in this system is7 = 6.0 as compared to 7 = 5.0 bits/Hz/s for Wei's 16 state

code. The data rate improvement for down stream (.966MHz) and upstream (112.13 KHz) in ADSL can be given as:

* lbit IHZIs * .966 MHz = .966 Mbps in downstream direction

* lbit IHZIs * 112.13 KHz = 112.13 Kbps inupstream direction.

Fig. 4 Coding Gain of 128 State TCM Code

Fig. 5 Spectral Efficiency 77 of 128 State TCM Code

CONCLUSION

With the detailed description of the ADSLmodulation and coding schemes in section I and section II

respectively, section III represents the performance of a

higher state Trellis Coded Modulation (TCM) for ADSLapplications. A 128 state 4D TCM code is simulated in DMTenvironment for ADSL. This code achieves 1.4 db gain over

the 16 state (3, 2, 5) Wei's code in ADSL shows 20%improvement in spectral efficiency, which is = 6.0

bits/Hz/s as compared to Wei' s code. 8-DMTSymbol based decoding is discussed in Section IV,where maximum likelihood soft decision viterbi decoding isapplied to up stream after accumulating eight DMT symbols.An improvement of .9 db is shown to resolve the problemsof up stream in ADSL.

REFERRENCES

[1] L.F. Wei, "Trellis Coded Modulation withMultidimensional Constellations ", IEEE Trans.Inform. Theory, Vol. IT-33, No. 4, July. 1987

[2] .D.J.Costelo J. Hangenauer, H. Imai and S.B.Wicker, "Applications of Error Control Coding",IEEE Trans. Inform. Theory, Vol. IT-44, No. 6,October. 1998.

[3] Jing Wang, "ADSL Coding Schemes and DiscreteMultitone Modulation for Virtual PeripheralEngine", MSc. Thesis, University of Tsukuba,February 2002.

[4] G. Ungerboeck, "Channel Coding withMultilevel/Phase Signal", IEEE Trans. Inform.Theory, Vol. IT-28, No. 1, January. 1982.

[5] G. Ungerboeck, "Trellis-Coded Modulationwith Redundant Signal Sets Part I:Introduction,"IEEE CommunicationsMagazine, vol. 25, No. 2, pp. 5-11, Feb. 1987

[6] G. Ungerboeck, "Trellis-Coded Modulationwith Redundant Signal Sets Part II: State ofthe Art," IEEE Communications Magazine,vol. 25, No. 2, pp. 12-21, Feb. 1987

Trellis Coded Modulation Gain Comparison at BER = lx0O-6

TCM 16 State 128 StateVs Vs

Un-coded(16+8 QAM) 16 State

Up Stream Down DownStream Stream

1-Symbol 8-Symbolbased based

decoding decodin2.4 db 2.4+.9 3.4 db 1.45 db

=3.3 db _I