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A Maximum Rate Loading Algorithm for Discrete Multitone ModulationSystems

Achankeng Leke* and J o h n M . CioffiInformation Systems Laboratory

Stanford University, Stanford, CA 94305-9510, USA

AbstractA novel approach to loading for Discrete Multitone Modu-lation (DMT) systems is proposed. This algorithm assignsenergy to different subchannels in order to maximize th e da tarate for a given margin while previous algorithms proposed[1,2] are aimed mainly at maximizing margin at a target datarate. Two implementations are suggested, both with finitegranularity: the first one leads to the optimal waterfillingsolution, and the second results in a suboptimal but slightlyless complex flat-energy distribution. The algorithm is ex-tended to the case of a ra te adaptive system with both atarget guaranteed fixed data r ate service and a variable one.Simulation results are presented for a rayleigh fading channelwith additive white gaussian noise.

1 IntroductionThe concept of multitone transmission has attracted alot of interest recently as a means to increase the datarate on a channel under given requirements such as fixedtransmitter power budget and equal probability of erroron all subchannels. The two most common forms ofMulticarrier Modulation (MCM) are Orthogonal FrequencyDivision Multiplexing (OFDM) and Discrete Multi-toneModulation (DMT). OFDM has been used in applicationssuch as Digital Audio Broadcasting (DAB) while DMT hasbeen selected by the American National Standards Instituteand the European Telecommunications Standard Instituteas the standard for transmission over Asymmetric DigitalSubscriber Lines (ADSL) [3]. Unlike OFDM which assignsthe same number of bits to each subchannel, DMT systemsassign more bits to subchannels with higher signal-to-noiselevels. The scheme used to assign energy and bits, knownas loading algorithm, is an important aspect of the design ofa DMT system. One well researched method, the waterfillalgorithm, has long been known to generate the optimalenergy distribution and achieve capacity; it has howeverbeen difficult to implement in practice. Other suboptimalalgorithms have been proposed [1,2] which maximize marginfor a target dat a rate. However, transmi tting at the

"This work was supp orte d by a grant from Sumitomo Electric Indus-tries and a Stanford University Graduate Fellowship. The first authorcan be reached via email at alekeQisl.stanford.edu.

maximum achievable da ta rate is crucial for ce rtain classesof channels such as slowly varying rayleigh fading channelsas this will ensure tha t the channel remains constant duringthe transmission of at least one DMT symbol. In this paper,we propose two versions of a new finite granularity loadingalgorithm which maximizes the data rat e at a given margin:a suboptimal flat-energy distribution and a slightly morecomplex capacity achieving waterfilling distribution.

First, the structure of a DMT system is presented andthe existing loading algorithms (Hughes-Hartogs, Chow,and Fischer) are discussed. The algorithm we propose isthen developed and applied to a typical rayleigh fadingchannel with additive white gaussian noise. Finally, thisalgorithm is extended to rate adaptive systems with both afixed guaranteed rate as well as a variable one.

2 DMT StructureA typical DM T structure in shown in Figure 1. The entirebandwidth is divided into N parallel subchannels. An inputbit stream of data rate R bits /sec. is buffered into blocksof b = RT bits, where T is the symbol period. The load-ing algorithm assigns a certain number of bits, bi, to eachsubchannel where

and No , is the number of subchannels turned on.

Figure 1: DMT Transmit/Receive Structure

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The encoder then t ranslates the bits, bi , into symbols, X i ,chosen from the appropriate constellation. The time sym-bols, zi , are obtained by Inverse Fourier Transform, and acyclic prefix of length v - ideally the length of the chan-nel - is added to the beginning of the DMT symbol. Thiscyclic prefix is used to combat intersymbol interference andmake the transmitted sequence look periodic. To eliminateintercarrier interference, the first v received samples in eachDMT block are discarded. Taking the Fourier Transform ofthe other received samples, we obtain independent par-allel subchannels which can be individually decoded using asimple memoryless decoder for each subchannel.

3 Existing Loading AlgorithmsThree multicarrier loading algorithms, all of which aresuboptimal compared to waterfilling, are in use today:Hughes-Hartogs, Chow [l], nd Fischer [2]. They are allmainly fixed data rate algorithms.The Hughes-Hartogs algorithm generates a table of in-cremental energies required to transmit one additionalbit on each of the subchannels. Then at each step, onemore bit is added to the subchannel that requires theleast incremental energy. The extensive sorting neededrenders the algorithm impractical for applications where thenumber of bits per DMT symbol as well as the number ofsubchannels used are large.Chow's algorithm exploits the fact that the differencebetween the optimal waterfilling distribution and theflat-energy distribution is minimal. Thus, the same amountof energy is assigned to the channels turned on:

(2 )EE n =-and the number of bits in each subchannel is given by:Nonbn = log, (1 + -)NRnr . T m (3)

where S N R , is the signal-to-noise ratio on subchanneln, r is the SNR gap representing how far our system isfrom achieving capacity, and ym is the system performancemargin. This iterative algorithm is repeated, with a dif-ferent set of subchannels turned off at each step, until theoptimal margin is achieved and the sum of the bits in eachsubchannel equals our target, Btarge t .The goal of the third algorithm, proposed by Fischer,is to minimize the probability of error on each subchannelgiven by:

P e = K n . Q (E ) (4)for QAM modulated systems. Kn is the number of nearestneighbors, dn is the minimum distance between constella-

tion points, and Q is the variance of the noise in subchanneln. The minimum is achieved when all subchannels have thesame probability of error. Set ting equal the error probabili-ties, Fischer arrives at a set of iterative equations which alsolead to a flat-energy distribution. The results obtained showa slight improvement in signal-to-noise ratio over Chow's al-gorithm [2].

4 Proposed Loading AlgorithmDMT loading algorithms assign higher bit rates to carrierswith higher SNR levels and lower bit rates to those withlower SNR levels while subchannels wi th, signal-to-noiseratios below a certain threshold are completely turned off.The most crucial aspect of loading algorithms is determiningwhich subchannels to turn off and which ones to turn on.Turning on a subchannel which should actually be offincreases the probability of error in that subchannel whichthen dominates the overall probability of error. Once thesubchannels to be turned on have been determined, thetota l energy can be distributed to them in a variety of waysThe first step in the proposed algorithm is to deter-mine which channels should be turned off. From thewaterfilling solution, a subchannel is turned off when theamount of energy to be assigned to it is negative [5]. LetH n and u: represent respectively the gain and the noisevariance on the nth subchannel. Subchannel m will beturned off if:

where E is the total energy budget, once again is the SNRgap representing how far our system is from achieving ca-pacity, and Non is the number of subchannels turned on. (5)can be rewritten as1 1

9m Non n= l

where gn is the SNR on the nth subchannel with unit inputenergy. Let

Subchannel m is then turned off if g m


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The procedure can be summarized as follows:1. Compute subchannel signal-to-noise ratios gn for unitinput energy.2. Sort the gns in descending order: g1 = gmaZ and g N =3. Set N on equal to the total number of channels N.4. Compute NSR.

gmzn.

5 . If gN,,


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1

0 5

0-5 -4 -3 -2 -1 0 i 2 3 4 5x IO 6(b)2

I 5

10 5

0-5 -4 -3 -2 -1 0 1 2 3 4 5x IO6requency [Hz]

Figure 3: Initial energy distributions obtained with unit in-put energy per subchannel: (a) Flat-Energy. (b) Waterfill.

x IO1(b)4-

$ 3

$ 2cL1

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Frequency [Hz] x 10'

Figure 4: Integer bit distributions obtained: (a) Flat-Energy.(b) Waterfill.The main difference of this algorithm compared to the pre-vious ones is in the way it is implemented: the subchannelsto be turned off are first determined which is the most crucialaspect of any algorithm. After which, any desired energy dis-tribution can be achieved. Furthermore, as in [2], logarithmsare only computed once and sorting is also performed once atthe beginning. In our proposed algorithm, only one iterationloop is performed while the previous ones use two. Finally,

this algorithm maximizes throughput at a given margin whilethe others maximize margin for a given data rate.

6 Rate Adaptive SystemThe proposed algorithm is well suited for rate adaptive ap-plications. In such applications, a target data rate and/ormargin are guaranteed. Once the target is reached, any left-

E6 'U)1 5E lF OW

% -4 -3 -2 -1 0 1 2 3 4 5x 10"requency [Hz]

Figure 5: Final energy distributions after bit round-off andrescaling: (a) Flat-Energy. (b) Waterfill.

over channels not turned off can then be used to transmit ad-ditional information. A typical example would be the trans-mission of highly reliable data - high margin and low gap -at a target fixed data rate while the remaining usable sub-channels carry da ta transmi tted at the maximum achievablethroughput, with little or no margin and a higher gap. Thisoptimization problem reduces to:

i , 2

subject to:

i , l

i i

where b i , j and E ~ J epresent respectively the number of bitsand the amount of energy assigned to the i th subchannel ofthe j t h service, RI represents t he fixed data rat e guaranteedby the first service, and E represents the total energy budget.In order to maximize the rate of the second service,the first service should use the minimum number ofsubchannels necessary to achieve the target rate. Thus,subchannels with the best signal-to-noise ratios will carrythe first service. Given the ta rget rat e and margin, Chow'salgorithm [l] can be used to identify this first set ofsubchannels; our proposed algorithm will then maximizethe throughput on the remaining usable subchannels. Anillustration of this application is shown in Figure 6. Thefirst service has a fixed normalized rate of 1 bit/Hz and amargin of 2 dB while the second service transmits at themaximum possible rate with no margin.

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Figure 6: Rate adaptive application. 0: first service. x:second service

7 ConclusionIn this paper, we proposed a new loading algorithm whichmaximizes throughput at a given margin. Two versions ofthe algori thm were presented: the flat-energy method lead-ing to a suboptimal energy distribution and the optimal butslightly more complex waterfilling solution. The proposedalgorithm is well suited for slowly varying rayleigh fadingchannels for which it is advantageous to always transmit atthe maximum achievable da ta r ate in order t o overcome thevariations in the channel. This algorithm was combined withChows in a rate adaptive application to produce two sets ofservices: a highly reliable, fixed rate service along with amaximum rate service with lower reliability in the leftoverusable subchannels.

References[l]P.S. Chow, J.M. Cioffi, and J.A.C. Bingham. A Prac-tical Discrete Multitone Transceiver Loading Algorithmfor Data Transmission over Spectrally Shaped Channels.

IEEE Transactions on Communications, 43(2/3/4) :773-775, February/March/April 1995.[2] R.F.H. Fischer and J.B. Huber. A New Loading Al-

gorithm For Discrete Multitone Transmission. In Proc.IEEE GLOBECOM96, pp. 724-728, London, Novem-ber 1996.[3] American National Standards Institute (ANSI). Asym-metric Digital Subscriber Line (ADSL) Metallic Inter-face Specification. Draft American National Standardfor Telecommunications, December 1995.[4] J.M. Cioffi. A Multicarrier Primer. In ANSI TlEl.4

Committee Contribution 91-157, Boca Raton, Novem-ber 1991.

[5] J.M. Cioffi. Digital Data Transmission. Manuscript preparation.[6] A. Leke and J.M. Cioffi. Transmit Optimization foTime-Invariant Wireless Channels Utilizing a DiscreMultitone Approach. In Proc. IEEE ICC97, pp. 95958, Montreal, June 1997.[7] G . Raleigh, S.N. Diggavi, A.F. Naguib, and A. PaulraCharacterization of Fast fading Vector Channels foMulti- Antenna Communication Systems. In Proceedinof 28 Asilomar Conference on Signals, Systems, an

Computers, v01.2, pp. 853-7, 1994.[8] W.C. Jakes, Microwave Mobile Communications. NeYork: John Wiley, 1974.

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