Multi Carrier Modulation for Data

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Multicarrier Modulation for DataTransmission: An Idea Whose TimeHas Come

John A . C. Bingham

HE PRINCIPLE O F TRANSMITTING DATA BYdividing it into several interleaved bit streams, and using theseto modulate several carriers, was used more than 30 years agoin the Collins Kineplex system [ 11,and has been of continuing,albeit peripheral, interest ever since. Now, however, interest isincreasing because modems based on the principle are beingused-or being considered for use-for transmission of dataand facsimile on the following:

General Switched Telephone Network (GSTN)60- 108 kH z Frequency-Division Multiplexed (FDM)group-band .Cellular radio

In addit ion, high-speed data is being considered for transmis -sion on the High-rate Digital Subscriber Line (HDSL).

The technique has been called by many names-orthogonally multiplexed Quadra ture Amplitude Modulation(QAM) [2], orthogonal FDM [3], and dynamically assignedmultiple QAM [4]-but we will refer to it by a generic name:Multicarrier Modulation (MCM). A more general form of thetechnique, which uses more complex signals as carriers [5], hasbeen developed recently as vector coding [6 ] and structuredchannel signalling [7] [SI. Unless otherwise s tated, the discus-sion here will concentrate on the special MCM form.

The reasons for the interest in MCM depend upon the trans-mission medium, and have also changed over the years as sig-nal processing techniques (mainly digital) have improved, butthe two most importan t ones are first, that an MCM signal canbe processed in a receiver without the enhancement (by asmuch as 8 dB in some media) of noise or interference that iscaused by linear equal ization of a single-camer signal, and sec-ond, that the long symbol time used in MCM produces a muchgreater immunity to impulse noise and fast fades.

The first seven sections of thi s article will discuss the follow-ing: the general technique of parallel transmission on manycarriers; the performance that can be achieved on anundistorted channel; algorithms for achieving that perform-ance; dealing with channel impairments; improving the per-

formance through coding; and methods of implementation.The last two sections discuss duplex operation of MCM andthe possible use of this on the GSTN.

MultiplexingMCM is a form of FDM; the basic principle is shown in Fig-

ure 1. Input da ta at M f , bfs are grouped into blocks ofM bi ts at

01 63-6804/90/0005-0005 $01 OO 1990 IEEE

Fig. 1. Basic mu lticarrier transmitter.

a block (“symbol”) rate off,. The M bits are used, rn, bits’ forthe camer at fc,, o modulate N , came rs, which are spaced Afapart across any usable frequency band; that is,

fC.“ = n b r f o r n = n l to n2 (1 1

and

“2

M = mn

I= n

where

N c = n 2 - n l + 1

The modulated came rs are summed for transmission, andmust be separated in the receiver before demodulation. Three

methods have been used for this separation:First, the earliest MCM modems borrowed from conven-tional F DM technology, and used filters to completely sepa-rate the bands. The trans mitted power spectra for just threesub-bands of a multi carrier system are shown in Figure 2a.

‘Each of th e ms typically = 2 to 8 .

Ma y 1990 - IEEE Communicat ions Magazine 5



fc,n % n+ 1

(a) FDM filtering

(b) Overlapping SQAM spectra

(c) QASK [sinc(OM - n)12 functions

Fig. 2. MCM transmit power spectra.

Because of the difficul ty of implementing very sharp filters,each of the signals must use a bandwidth, ( I + a%,,which isgreater than the Nyquist minimum, f,; the efficiency ofband usage is fJAf = 1/(1 + a).Second [9- 131 the efficiency of band usage was increased toalmost 100% by using Staggered Quadrature AmplitudeModulation (SQAM); the individual transmit spectra of themodulated carriers still use an excess bandwidth of a, butthey overlap at the - dB frequencies (as shown in Figure2b), and the composite spectrum is flat. If a 5 I , each sub-band overlaps only its immediate neighbors, andorthogonal ity of the sub-bands-with resultant separabi lityin the receiver-is achieved by staggering the data (that is,offsetting it by half a symbol period) on alternate in-phaseand quadrature sub-channels. The amount of filtering re-quired is less than for complete separation, but it is still con-siderable, and the total number of camers must be small(typical ly less than 20).Third [2 ] [4] [14- 161, the carriers are “keyed” by the data,using Quadrature Ampli tude Shift Keying (QASK). The in-dividual spectra are now sinc functions, as shown in Figure2c; they are not bandlimited but, as we shall see, the signalscan still be separated in the receiver; he frequency-divisionis achieved, not by bandpass filtering, but by baseband pro-cessing. The big advantage of this approach is that bothtransmitte r and receiver can be implemented using efficientFast Fourier Transform ( F R ) echniques.

Maximum Achievable Bit Rate:Seeking the Shannongri-la of DataTransmission

The performance of a data transmission system is usually

analyzed and measured in terms of the probability of error at agiven bit rate and Signal-to-Noise Ratio (SNR). It is, however,more useful for our purpose-and, indeed, more appropria tefor modem data communication systems that use any combi-nation o f compress ion, error correct ion, and flow control-toconsider the attainable bit rate at a given error rate and SNR.For single-carrier signals that are equalized with either a Lin-

- 0

dBm-1

t Ieceived Signal

Sub-band n

Received Noise perSub-band n

\

I ’L

10

I

l.O 202. 0

30 40

(a) Received signal and noise power.

3.0 fkHz50 n

Totalbitrate = (1x4 + 2 x 5 + 2 8 x 6 + 7x 5 + 4 x 4 + 3x 3 + 4x2)x 62.5 = 15,625 b/s

0 110 2 0 30 40 50 In

(b) Bit and power assignments.

Fig. 3. Adaptive loadingfor a badly distorted GSTN hannel.

ear Equalizer (LE) or a Decision-Feedback Equalizer (DFE)this can be done by inverting the well-known error ra te formu-las (e.g., those for LEs [171 [181 and DFEs [3]).

The variables for a multicarrier signal are the number of bitsper symbol, m,,, and the proportion, y,,, ofthe total transmittedpower, P, hat are allotted to each sub-band. The aggregate bitrate is approximately maximized if these variables are chosenso that the bit error rates in all the sub-bands are equal. Thishas not been proved rigorously, but it is intuitively reasonable;the dependence of error rates on the m,, nd y,, is such that ifthe error rates are unbalanced, the rate in one band will in-crease much more than it will decrease in anot her band.

In order to calculate the attainable bit rate for a channelwith transfer function H(f l and noise power spectrum at theinput to the receiver U(f l , 2we can approximate H(f l and U(flby segments H n and U,, centered about carrier frequenciesf,,,as defined in Equation (1). This is illustrated in Figure 3a for abadly distorted and noisy voiceband channel with f = 62.5Hq 3 the signal power received in each sub-band is calculatedassuming that the total transmit power of - dBm is distribut-ed equally across the sub-bands (i.e., if all the yn were equal);the total noise power in the 0.3 to 3.4 kH z and is - 57 dBm.

The probability of bit error , P, in the symbol-by-symbol de-tection (i.e., without the benefit of any coding across symbols)

2The possible non-whiteness of the “noise” s important for HDSL,where the principal impairment is strongly correlated Near-End Cross-

Talk (NEXT).

3This is one of the ca mer separations used in Telebit’s “Trailblazer”modem ; the reason for such a choice (62.5 = 8,000/128) will becomeclear later.

6 Ma y 1990 - IEEE Comm unications Magazine



of the QAM signal in sub-band n-assuming no interferencefrom & e signals in the other bands-is

made very small. Then the summation in Equation (4) can beapproximated by an integration, and the maximum bit rate

where

L = 2"'n,

fl = 4(1 - / L ) / m n ,

and K s an error-rate multiplier, which is a little less than 6 if,as is most usual, differential phase modulation and a 3-tapscrambler are used. Q is defined, as usual, by

- r m

P s the total transmitted power, and y,, is the proportion ofthat total allotted to sub-band n.

We would like to solve Equation (2) for m,,, ut this cannotbe done explicitly because m, occurs in three places on therighthand side. Kalet [191 developed upper and lower boundsfor the symbol error rate by considering the li mits of 4(1 -UL), but it is adequate for our purpose4 o consider only an av-erage value of B. For a practica l range of m, rom 2 to 8 Bvariesfrom 1 to 15/32, so an average value of 4 for the combinederror-rate multiplier, BK, il l suffice. Then, as shown in [ 191,Equation (2) can be inverted, and the total number of bits thatcan be transmitted in one symbol with error probability CPusing N, ub-bands can be written:

n2 I n

where

2

E Y n = l

1= n

Ideally, the optimum power distribution, y,,, should be cal-culated by a "water-pouring" procedure that is similar t o thatof Gallager [20], but for high SNRs (corresponding o most ac-ceptable error rates), the optimum y,, are approximately equal.The most efficient use is made of the channel if the symbolrate& is made equal to the carrier separation, Al; and both are

where the frequency range, J ; to &, is that for which theintegrand is > 2 (i.e., the range over which QAM transmissionis possible), and W (= f, - ;, is the measure of that range.

As pointed out by Kalet and Zervos [3], Equation (5 ) is verysimilar to the bit rate for a Single-Carrier QAM (SCQAM) sig-nal equalized by a DFE, which was originally shown by Price[ 181. In fact, the only difference is in the frequency range of theintegration; for the single-camer signal with DFE it should beextended to that for which the integrand is greater than zero,but in practice the extra contribution to the integral is usuallyinsignificant.

It should be noted that Equation (5) assumes that the num-ber of bits per carrier is continuously variable but, in practice,each rn, must be integer.5 It was shown in [ 171 tha t the effectsof this quantizing can be mitigated by adjusting the y, to re-equalize the error rates in all the sub-bands, and it has beenfound from numerous simulations that the total bit rateachieved in this way is only slightly less than that given byEquation (5).

Thus, the aggregate bit rate for MCM is approximatelyequal to that for SCQAMIDFE; for channels with attenuationdistortion or non-white noi se this may be considerably greaterthan for SCQAM with a linear equalizer.

Adaptive LoadingIt was shown that if the ratio Iff(fl12/U(flanes significantly

across the band and a fixed loading is used [21], the error ratein the too-heavily-loaded sub-bands may be very high, and theoverall error rate may be greater than for a single-carrier signal[17]! The rn , must be varied in order to keep all the sub-band

error rates, Pn , qual; the following procedure for calculatingthe y, and integer m, was described [16].Given a set of signal-t~-"noise"~ atios, measured in the re-

ceiver when the far t ransmitter is transmitting at the maximumpermitted level in all sub-bands, calculate the terms, AP,,, ofan "incremental power" matrix, where AP,,, = P,, -P, - ,, P, = the transmit power needed in sub-band n totransfer rn bits per symbol at some predefined error rate), andclearly, P o , = 0 .

Then assign bits one at a time to cam ers, each time choos-ing the carrier that requires the least incremental power. Thiscan be described algorithmically:

9 Search row 1 for the smallest AP,,,Assign one more bit to sub-band nIncrement M and Plot; hat is,M' = M + 1 and Plol' = Plot +

b d i n g schemes to allow non-integer m,, have been discussed foruse on the DSL , but it is not clear how much they would increase the ca-pacity.

%he eq uivalent noise should be the power sum of Gaussian noise,NEX T, and inter-symbol and inter-channel interferences.

4Equation ( I ) is exact on ly for square constellations (i.e., m,, even)anyway. For m, = 5 and 7, the "cross" const ellations are slightly moreefficient, and P is slightly lower, for m,, 3 all constellations are lessefficient, and P s significantly higher.

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Move all terms of column n up one place; that is, APi,,’ =

Repeat searchMi+ .n

For the preferred mode of operation for multicamer-atthe highest rate achievable with a predefined error rate-theassignment should be stopped when P,,, ust exceeds P, heavailable power. If, however, transmission at a given bit rate (asynchronous “bit pump”) is insisted upon, then the processshould be stopped at the appropriate value of M. PI,, may thenbe less or more than permitted (that is, the specified error ratewas pessimistic or optimistic, respectively); all allotted powersmust be scaled to adjust P,,, to the correct value.

The resulting power distribution for the channel of Figure2a is shown in Figure 2b. The discontinuities occur because ofthe integer constraint on the number of bits; if Af is small, thenthe SNR can change only slightly from one sub-band to thenext, so that if, for example, the SNR is decreasing, and m, =m,- , 1, the nth carrier will require approximately 3 dB lesspower than the (n - 1)th carrier for the same error rate. The al-gorithm is clearly not water-pouring in the classical sense, butsince it puts every increment of transmit power where it will bemost effective, it appears to be optimum for multic amer trans-mission using QAM constell ations and symbol-by-symbol de-tection.

Feedback from Receiver to TransmitterAdaptive loading requires that the receiver measure the

sub-band SNRs, calculate the best power and bit ass ignments,and send this information back to the transmitter. This mayseem like a big increase in complexity, but it should be notedthat all single-carrier systems that make best use of a channelalso require some feedback. This can be used in three differentways:

Many present fixed-symbol-rate systems use a “fall-back”procedure that requires the feedback of error-rate informa-tion.

Fig. 4. Integrate and dump detection for QASK.

tion of a total of M bits, m, at a time, is most easilyaccomplished by calculating N, complex numbers (each se-lected from a constellation with 2mn points), augmenting themwith n, - zeros in front and N,,, - nz zeros behind, and per-forming an N,,,-point IFFT.

Modulation via an IFFT is equivalent to multicarrierQASK in which the fundamental baseband pulse shape is a rec-tangle, g(t). That is,

g n ( t ) = 1 I T or0 5 < T, a n d = Ootherwise. (6)

In the receiver the signal is demodulated by assembling N,,samples into a block, and performing a real-to-complex FFT.

This is equivalent to demodulating each sub-band separately,and then doing an integrate-and-dump on each product, asshown in Figure 4. If the received baseband pulse in sub-band nis defined as g,‘(t), then the output from the demodulator re-

sulting from an input to another sub-band ( n - k) s g,’(t) mul-tiplied by a cosine or sine wave of the difference frequency U jthat is,

t i + l ) T

cn = g n ‘ ( t ) . e x p ( i k 2 n A f i ) dt. (7)h n , n - k IT

Better use of a channel might be made by calculating andfeeding back an optimum symbol rate, and then using Someform of Maximum Likelihood Seauence Estimation in the

If the channel is non-distorting, so that g,(t) = g,’(t) = 1/T,then these integrals over a time 1lAfare zero for all non-zero k.

receiver. That is,Another approach is to combine trellis coding with an adap-tive symbol-rate and a DFE. A conventional DFE cannot beused, however, because of error propagation, and the func-tion of the feedback part of the DFE must be implementedin the transmitter using. a generalization of Tomlinson

hn , - (i ) = 1 fori = k = 0 , n d = 0 otherwise, (8)

precoding; this requires fhe fgedback of much the same de-tailed channel characteristics as are needed for MCM. and orthogonality between the sub-bands is maintained.

Adaptive Loading When NEXT is theDominant Impairment Channel Impairments

Correcting for the Effects of

For high-speed transmission on the subscriber loop, NEXTis usually more harmful than noise. If this NEXT is mainlyfrom other MCM transmitters, a unilateral decision to changethe spectral distribution of one transmitted signal wouldchange the conditions under which the other transmittersmake their decisions; clearly some coordinated strategy for as-signing all the sub-band powers is needed. Work is being doneon this but it is too early to predict the results.

, Modulation and DemodulationModulation is performed on M bits (a symbol or block) of

data at a time-preferably using an Inverse FFT (1FFT)-andsamples of the transmit signal are generated at a sampling rate,Lampor greatest efficiencyf,,,, should be equal to Af multi-plied by an integer power of two. Iff,,, = 2NtOt Af; then N,,,carriers are available for modulation, but the channel will usu-ally be such that only N, carriers can be used. If these are at fre-quencies n , Af to n 2 A f ;as defined in Equation ( I ) , modula-

-Linear Distortion

The primary effect of attenuation a n do r delay distortion inthe channel is that each subcamer is received with a differentamplitude andlor phase, so that the channel can be grosslycharacterized by a single complex number for each sub-band.These are learned from a training signal of unmodulated carri-ers (a “comb”), and inverted to generate the complex coeffi-cients of a set of one-tap equalizers. All subsequent receivedsamples are then multiplied by these inverses.

A secondary effect is that g,’(t) is not rectangular, and alsooverlaps into the preceding and following symbol periods.Moreover, even with an undistorted-but necessarily band-limited-channel, the sub-bands near the ends of the band areasymmetrical, and distort their gns. Thus, there is both Inter-Channel Interference (ICI) (h, ,-do) # 0), and Inter-SymbolInterference (ISI) (h , , f ) # b) ,and even the combination ofthe two (h,,,-k(f l j # 0); orthogonality of the sub-bands islost.

8 Ma y 1990 - IEEE Communications Magazine



It can be seen that the impulse response of each sub-banddepends only on the channel, and that the transient at the be-ginning and end of each g ‘(t) is independent of the separationof the camers (that is, ofthe symbol period, T). One way ofdeali’ng with distortion would be to increase Tenough that dis-tortion becomes insignificant, but in general this is notp ~ s s i b l e . ~ our o ther ways have been described; these are dis-cussed below.

Guard-PeriodThe transients in the g,,’(f)can be avoided [ 11 [ 141 [22] by

postponing the integration in Equation (5) for a time Tg andincreasing the total symbol time to T, = T + Tgr while stlll, ofcourse, retaining T = l/AJ One commercial modem for theGSTN [4] uses T = 128 ms and Tg = 7 ms. This limits the MSEfrom IS1 and IC1 on even the worst lines to less than 1%, but itdoes reduce the total bit rate by 5.2%.

Passband C hannel EqualizationThe reduction in bit rate caused by the use of a guard-period

can be avoided by linearly equalizing the received signal. Be-cause of the reduction of MSE achieved by integrating over along symbol period, the equalizer can be much less complexthan that for SCQAM; furthermore, it may be acceptable insome media to adapt it only during training, and freeze it dur-ing data reception.

(It should be noted that although the signal is being linearlyequalized, this approach does not incur the large noise-

enhancement penalty of single-carrier modulation. The load-ing is calculated from, and the performance determined by, thesub-band SNRs, which are reduced only slightly by the ampli-tude equalization across each sub-band; the equalizationacross the full band acts mainly like a delay equalizer plusmany separate Automatic Gain Controls, or AGCs.)

The conclusion that can be drawn from [23] is that for sucha simple equalizer, a Tapped Delay Line (TDL) structure usingtime-domain convolution is the most efficient. The trainingsignal for this should be an unmodulated subset of the carriers,and the t aps could be calculated either iteratively, by a conven-tional Least Mean Square (LMS) algorithm that takes advan-tage of the cyclic nature of the signal, or by performing an FFT

of the signal to calculate the channel characteristics, invertingthese, and performing an IFFI‘ to calculate the taps.8

The optimum lengths of the data symbol and the TD L are asubject for further investigation. Clearly, as the length of thesymbol is reduced, the effects of IS1 and IC1 become relativelymore important, and the complexity of the equalizer must beincreased. The limit of this would be reached when theequalizer had 2N, parameters, and, since it would then equa-lize the channel response to all N , carriers, it could also takeover the role of the one-tap complex baseband equalizers.

Baseband EqualizationThe IC1 terms defined by setting i = 0 in Equation (6) form

an N , x N , matrix, with the terms off the main diagonal de-creasing only very slowly (approximately as l/&. This wouldrequire an extremely complicated equalizer, and basebandequalization is not used for QASK signals. It can be used, how-ever, for SQAM signals [131, because each sub-band is filteredso as to limit interference to the two adjacent bands; the IC1matrix then has terms only on the main and two adjacent diag-onals.

7The DSP memory, th e processing requirements (proportional to Tand logzT, respectively), and th e delay through the modem all becomeprohibitive.

*This is typical of the judicious mixture of frequency- and time-domain processing that is used in MCM. See [23] for a discussion ofth e trade-offs, and for more references on frequencydomain process-ing.

A nmodulated Carriers

1 rl1 TI1 rl1rI1 rI1 rl1rl1rl1rl 1 rI1rI1 rl1 rl1rl1rI1 rf - 7 1Lower Sideband of f k + 7 ’/ fk Y U p p e r Sideband of fk-7 fk+ 7

Upper Sideband of fk-8 Lower Sideband of fk+8

Fig. 5. Multicarrier spectrum with sidebands resulting from 60 Hz

phase jitter.

Vector Coding, Structured Channel SignalingHolsinger [51 showed that orthogonality of the sub-band sig-

nals through a distorted channel can be achieved by using, as“carriers,” the eigenvectors of the auto-correlation matrix.This approach is presently attracting considerable interest[6-81, but it is too soon to know whether it can compete in com-putational efficiency with passband equalization.

Combination of Different MethodsThe above methods are not mutually exclusive, and it is

likely that some combination will provide the best compro-mise between amount of computation and total bit rate;passband equalization with a very short guard-space (TdT 1to 2%) seems to be a very promising combination.

Phase JitterPhase jitter affects MCM and SCQAM quite differently. If a

composite signal of unmodulated carriers is subjected to phasejitter of frequency 4 and amplitude less than about 1O’, heneach carrier at nAf will generate just two significant sidebandsat nAf +4.The carriers and their sidebands are shown in Fig-ure 5 for the case where 4 /Af = 7.689.

Both detection methods in the receiver-an FFI’ or de-modulation followed by an integrate and dump-result inequivalent filter shapings of sinc functions centered at the car-rier frequencies.

It can be seen, therefore, that the sidebands of at least twoother carriers’O contribute to the distortion seen by any givencarrier. Since the data modulated onto these other carriers isuncorrelated with that on the camer under consideration, thejitt er is seen as random distortion about each point in the con-

stellation, as shown in Figure 6a. That is, the jitter power (thetotal power in all the sidebands) is spread evenly over all cam -ers and over all data patterns on those carriers, and it can beadded to the noise on a power basis.

In contrast, a single-carrier constellation is rotated by thejitter, as shown in Figure 6b; the outer points are clearly moresusceptible, and the overall effect upon the error rate withadded noise will be greater than for MCM.

Tracking Phase JitterAlthough the effects of phase jitter are less for MCM than

they are for SCQAM, they should not be ignored; identifiable,discrete components of jit ter should be tracked. Identificationis easier in a multicamer receiver because much of the signalprocessing involves FFTs, but tracking is harder because of thelong symbol period.

One method [24] processes one complete symbol to calcu-

late the remanent phase error (the difference between the input

9f= 7. 8 125 Hz s th e preferred camer separation in the Trailblazer,

1°The number of contr ibut ingcarriers reduces to tw o in the special

an d f i = 60 Hz , th e most common jitter frequency in the U.S.

case of.( ldfbeing a n integer.

May 1990 - IEEE Communications Magazine 11



I I

(a) Multicarrier. (b) Single carrier.

Fig. 6 .Effectsofpha se it ter on one quadrant of a 16-point constellation.

phase and the locally generated tracking phase), passes theerror through narrow-band feedback filters as described in[171, and uses the o utputs to u pdate a phase predictor whichgenerates the tracking phase for the next symbol. It has beenfound that discrete jitter components can be tracked almostperfectly.

Non-Linear DistortionA multicamer signal is the sum of many independent mod-

ulated sinewaves, and its sampled amplitude has an almostGaussian distribution. Therefore, its peak-to-average ratio ismuch higher than th at of SCQAM, and it is more susceptible tonon-linear distortion. The most severe component of this isusually a negative cubic term ("saturation"), and it app earsthat if this can be quantifi ed it can be, at least partially, correct-ed in th e receiver by operat ing on the samples with a comple-mentary nonlinearity.

Impulse NoiseBecause a multicamer signal is integrated over a long sym-

bol perio d, the effects of impulse noise are much less than forSCQAM; indeed , this was one of the original motiva tions forMCM [25]. Tests reported to the Consultative Committee forInternational Telephone and Telegraph (CCITT) [26] showedthat the threshold level for noise to cause errors can be as muchas 11 dB higher for MCM.

Single-Frequency InterferenceThere is an interesting time/frequency duality involved

here. An SCQAM signal is sensitive to impulses in the time do-main in the same way that an MCM signal might be sensitive toimpulses in the frequency domain (single-tone interference).The advantage of MCM lies in the fact that the sources of theseinterferers are discrete,' and their frequencies are usually sta-ble (in contrast to the time of occurrence of impulses in thetime domain); they can be recognized during training andavoided (that is, nearby camers are not used) by the adaptiveloading algorithm.

FadesMobile radio channels often suffer wideband fades, in

which the SNR across the whole frequency decreases alarming-ly for a short time. A single-carrier system might have a verylow error rate between these fades, but would suffer from avery high one dur ing a fade; the overall error rate might still beintolerable.

On the other hand, in a multicamer system both the signaland the noise are integrated over the whole symbol perio d theaverage SNR and resultan t error rate are usually still tolerable.

Trellis Code ModulationThe advantages of TCM-about 3.5 dB of coding gain with

present-generation codes and perhaps up to 5 dB with futurecodes-are now widely recognized. Early applications of trel-lis coding to MCM [25] [27] used encoding in the con ventionalway; that is, from symbol to symbol. Only a few cam ers wereused, and the delay through the Viterbi decoder was just tolera-ble because the symbols were fairly short. However, whenMCM was first introduced to the mainstream of modem tech-nology, it was clear that the proposed symbol period of 138 ms

would be so long as to make MCM and conventional trelliscoding incompatible.

The justificat ion for trellis coding of SCQAM in general anddecoding by the Viterbi algorithm in particu lar is that the noiseis white (or almost so); that is, samples of it are almostuncorrelated from symbol to symbol. The time/frequency du-ality of single-/multicamer can be exploited here by recogniz-ing that samples of the noise, averaged over one symbol, arealso uncorrelated from one frequency sub-band to the next,and th at therefore trellis coding can be applied in the same way1281.

Following the terminology of [2 9 , et the m, bits for inputto sub-band n be designated x,', x ) ,... nm . Then x: and Xn 2should be input to the encoder to generate the output set zOz,], zn2, which together with the uncoded bits x,3, ... nm areused to define a point in the appropriate constellation. Thestate of the encoder afte r encoding sub-band n is then used as

the initial state for encoding sub-band ( n + 1).As a result of the adaptive loading, the number of bits, m,,and therefore the size of the nth constellation will probablyvary with n, but this does not matter. The three encoded bitsdefine one of eight sets of points, each containing 2(mn-3)points, and the Viterbi decoding determines these thr ee bitsand, hence, the set; identif ication of a point within the definedset can then be done one sub-band at a time, even though thesize of the set may vary from one sub-band to the next.

Any of the codes that have been developed for single- camercould be used for MCM, but since a decoder will have to dealwith constellations of varying sizes, it would be preferable touse codes and signal mappings that allow constellations togrow smoothly, such as were described in [30].

Block Processing of a Convolutional CodeIt is highly desirable that all of the data in one symbol

(block) be decoded in the same symbol period and from onlythe signals received within th at block. This would not be possi-ble, however, if both conventional encoding and decodingwere used, because, first, a conventional encoder uses its stateafter encoding the last sub-band as the in itial state for encodingthe first sub-band of the next symbol, and second, a conven-tional Viterbi decoder makes a decision about a symbol onlyafter receiving K4 more symbols, where K d , the "look-back"distance or decoding delay, is typically between five and eighttimes the constraint length, I , of the code-about twenty forthe common eight-state codes. Consequently, the last K d sub-bands could not be decoded until the next symbol had been re-ceived and d emodulated.

To achieve full block decoding the look-back distance in thedecoder must be curtailed towards the end of the block. Thiscan be done in two ways:

The encoder can be modified by constraining 1 bits at the

end of the symbol in order to force the 2' state encoder into aknown final state. Then all (M - ) unconstrained bits canbe decoded with no reduction of coding gain. This is easierto do with a feedforward encoder , but it would seem to befeasible even with a non-linear feedback encoder such as isdescribed in CCITT Recommendation V.32.The Viterbi decoder can be modified to decode the last Kdsub-bands by tracing back th e path f rom the smallest final

' A tone at 2,600 Hz , which is u sed in some single-freqency signal-ling systems, is the most notorious interferer in the US .




Fig. 7. Basic multicarrier “mo-dem.

metric, and decoding all of the remaining bits from thenodes on that path. This means that the codin gains for thelast few camers decrease more or less linearly [om the max-imum to about 0 dB for the last cam er. Thi s effect can beanticipated in the original loading of these camers, and willprobably reduce the overall bit rate by about four bits persymbol.

ImplementationA simplified block diagram of a multicarri er “mo-dem” (the

transmitter of one modem and the receiver of another) isshown in Figure 7. The main processing in the transmitter andreceiver is done with an IFFT and an FFT, respectively. Inorder to compare the amou nts of computation in SCQAM andMultica rrier Quadrat ure Amplitude Shift Keying (MCQASK),which use very different symbol lengths, it is simplest to con-sider a multicarri er symbol as comprising N , equivalent single-carrier (esc) symbols. Then the number of multiplications re-quired to generate one esc symbol-the principal computa-tional “loading”-is approximately proportional to IogpV,,,for MCM and N , for SCQAM, where Ne is the number of tapsin the equalizer. The constants of proportionality are each be-tween 6 and 8, N , is typically 30, and in one widely used imple-mentation of MCM for use on the GSTN, N,,, = 5 12. The re-sultant loadings are about 21 0 for SCQAM and 63 forMCQASK.

In these days of programmable processors that multiply al-most as fast as they d o anything else, the number of multiplica-tions is. however, a simpli stic measure of computational load.These processors were mostly designed to d o sums of products(convolut ions) very efficiently; by cont rast, they a re no t veryefficient at performing FFTs. The net result is that for modula-tion and demodulation the two methods require about thesame number of processor instruction cycles.

It should be noted, however, that in systems using TCM thedifferent amounts of computation needed for demodulationare overshadowed by that needed for the Viterbi decoder,which is common to both modulation techniques.

Echo CancellationAlthough most data communication systems do not need

true full duplex (“maximum” speed in both directions)modems, there are many advantages to be gained from someduplex capability. In order not to reduce the capacity in the pri-mary direction the two signals must occupy the same frequen-cy band, so that if communication is t o be via a two-wire chan-nel, the two signals must be separated in the receivers by echocancellation.

The first Echo Cancelers (ECs) to be developed were signal-driven; a generic one is shown in Figure 8a. The transmi t signalis input to both the four-wire-to-two-wire connector (“hybrid”)and an echo emulator (usually a TDL), which slowly learns thecharacte ristics of the echo path, calculates samples of the esti-mated echo (usually by time-domain convolution), and sub-tracts them from the corrupted receive signal. It was soon rec-ognized, however, that an EC can be greatly simplified if itsinput is the tra nsmit data instead of the modulated an d filteredsignal; a “data-driven’’ EC is shown in Figure 8b.

A straight forward application of data-driven echo cancelingfor QASK MCM would require that, in order t o deal with bothIS1 and ICI, the impulse response of the echo path be modeledas a three-dimensional, N , x N , x 2 matrix. Even with N , =20 (too small for all othe r purposes) the memory a nd process-ing requirements would be prohibitive. More innovative ap-proaches to data-driven ECs are needed, but since they havenot yet appeared, signal-driven ECs are being re-examined.

The Ultimate GSTN Modem:A One-Member Family

Most of the advances in theory and implementation of datatransmission in the last twenty years have been made in coun-tries that have good, and continually improving, transmission

channels, and have been applied to the problem of achievingever higher speeds on those channels; the same ingenuity anddedication have not been applied to the problem of wideningthe range of channels over which a given speed can beachieved. As each advance has been made (be tter timing recov-ery, better tracking of phase jitter, trellis-coding, etc.), it hasseldom been used to improve transmission at the lower speeds,and the older, less efficient techniques have usually sur-vived.

The result of this can be seen in the half-duplex successionof CCITT Recommendations V.26,27,29, and 33. All use dif-ferent coding and modulation schemes, and none is compati-ble with any other; that is, they belong to different families.Finding a supportable speed on a given line (“falling back”) hasbeen likened to climbing down a ladder from which some ofthe rungs are missing.

This has been particularly aggravating for facsimile trans-mission according to Recommendations T.4 and 30. There iscurrently a movement to partially solve the missing rung(“hole”) problem by extending the trellis coding recommendedin V.33 down to 9.6 and 7.2 kbls-speeds previously consid-ered the province of V.29. Nevertheless, ultrareliable transmis-sion at 4.8 kbls will not be ensured, and the erratic, time-consuming stepping-down process will still be necessary.

M ay 1990 - IEEE Communications Magazine 13



Fig. 8. Echo cancelers.

Multicamer modulation solves both these problems bymaking the most advanced techniques usable at any speed-thus achieving the highest possible speed on any line-and byselecting that speed during the modem training, without anyexternal control.

Duplex OperationSimultaneous transmission and reception may be desirable

for any one of many reasons, but the speed and error rate re-quired in the reverse channel will vary greatly with the applica-

tion. The use of MCM would allow the reverse channel to beplaced in the optimum freqency band an d to use the minimumtransmitted power; this could relax the requirements for echocanceling-or, alternatively, extend the dynamic range of themodem-by as much as 18 dB.

AcknowledgmentsI am very grateful to Adam Lender for the original encour-

agement to write this article and for many helpful suggestions,to John Ciofi for many enlightening discussions, and to ananonymous reviewer for the opportunity to see things througha reader's eyes.

ReferencesI

I


M. L. Doelz, E. T. Heald, and D. L. Martin , 'Binary Data Transm issionTechniques o r Linear Systems,'Proc. IRE,vol. 45, pp. 656-661, May1957.8. Hirosaki, 'An Orthogonally Multiplexed QAM System Using he Dis-crete Fourier Transform,' IEEE Trans. Comm un., vol. COM-29, pp.

I. Kalet and N. A. Zervos, 'Optimized Decis ion Feedback Equalizationversus Optimized Orthogonal Frequency Division M ultiplexing forHigh-speed Data Transmission Over the Local Cable Network,' IEEEInt'l. Conf Com mun. Rec.. pp. 1.080-1,085, Sept. 1989.J. Fegreus, 'Prestissimo,' Digifal Rev., pp. 82-87, Apr. 1986.J. L. Holsinger, 'Digital Communication Over Fixed Time-ContinuousChannels with Memory-With Special Application to TelephoneChannels," Lincoln Lab. Tech. Rep. No. 366, MIT, Cambridge, MA, Oct.1964S. Kasturia, J. Aslanis, and J. M. Cioffi, 'Vector Coding for Partial-Response Channels,' IEEE Trans. Info. Theory, to be published.J. W. Lechleider, S tructured Channel Signaling,' TIA Tech. Subcom-mittee T 1E 1.4, Contribution No. 89-067, Mar. 1989.J. W . Lechleider, The Optimum Combination of B lock Codes and Re-ceivers or Arbitrary Channels,' IEEE Trans. Commun., o be publishedMay 1990.

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4,731,816 (Mar. 1988). and 4,833,706 (May 1989).J. A . C. Bingham, The Th~ryandPrecticeofModemDesign. ew York:John Wiley and Sons, May 1988.R. Price, Non-Linearly Feedback Equalized PAM vs. Capacity or NoisyNoisy Fi lter Channels,' IEEE Int'l. Conf. Comm un., June 1970.I. Kalet, 'The Multi tone Channel,' IEEE Trans. Commun., vol. 37, pp.119-124, Feb. 1989.R. G. Gallager, Information Theory and Reliable Comm unication, NewYork: John Wiley and Sons, 1968.8. Hirosaki er al. , -A 19.2 kb/s Voice-Band Data Modem Based onOrthogonally Multiplexed QAM Techniques,' IEEE Int'l, ConfCommun. Rec., pp. 661-665. Aug. 1985.W. E. Keasler and D. L. Bitzer, -High Speed Modem Suitable or Opera-ting with a Switched Network,' U. S . Patent No. 4,206,320, June1980.N. Jablon, -Complexity of Frequency Domain Adaptive Filtering forData Modems,' Asilomar Conf. on Circuits and Syst.. 1989.J. A. C. Bingham, 'Method and Apparatus or Correcting or Frequen-cy Offset, Phase Jitter, and Timing Offset in Multicarrier Modems,' U.S. Patent Application filed Mar. 1990.B. Hirosaki, S. Hasegawa, and A. Sabato, -Advanced Group-BandModem Using Orthogonally Multiplexed QAM Technique,' IEEE Trans.Commun., vol. COM-34, pp. 587-592, June 1986.Telebi t Corpora tion, 'Comparative Performance Results for Asym-metrical Duplex, V.32 (extended), and Multicarrier Modems,' CCITTSG XVII, Contribution 056. Sept. 1989.A. Ruiz and J. Cioffi. 'A Frequency Domain Approach to CombinedSpectral Shaping and Coding,' IEEE lnr% Conf. Commun. Rec., pp.1,711-1,715, June 1987.D. Decker eral., -Multi-Channel Trell is Encoder/Decoder,' U. S. atentApplication filed Aug. 1 988.IBM Europe, 'Trellis-Coded Modulation Schemes with 8-State Sys-tematic Encoder and 90' Symmetry or Use in Data Modems Transmit-ting 3-7 Bits per Modulation Interval,' CCITT SG XVII, Contribut ionD180, Oct. 1983.G. Ungerboeck, 'Trellis-Coded Modulation with Redundant SignalSets,- IEEE Commun. Mag., vol. 25, pp. 5-21, Feb. 1987.

BiographyJohn A. C. Bingha m received a B.Sc. degree from Imperial College, Lon-

don in 1956 and an M S.E.E. from S tanford University in 1961.From 1959 to 1963 and 196 6 to 1970, he was with Lenkurt Electric, San

Carlos, CA, working on computational problems of filter design and data trans-mission. From 1972 to 1985, he was the Manager of the Advanced Develop-ment Department a t RacaCVadic, Milpi tas, CA, where, in 1973, he inventedthe VA 34 00, the first full-duplex 1,200 b/s modem. n 1985 , he was a VisitingScholar at ETH, Zurich, Switzerland. He is now a Senior Scientis t at Telebi t Cor-poration. Sunnyvale. CA, working on all types of data transmission.

He is the author of one book and about twen ty papers, and holds five pat-ents

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