Multioarrier Modulation for Data
`Transmission: An Idea Whose Time
`Has Come
`
`
`John A. C. Bingham
`
`HE PRINCIPLE OF TRANSMITTING DATA BY
`dividing it into several interleaved bit streams. and using these
`to modulate several carriers. was used more than 30 years ago
`in the Collins Kjneplex system [1], and has been ofcontinuing,
`albeit peripheral. interest ever since. Now, however, interest is
`increasing because modems based on the principle are being
`uscd——or being considered for use—for transmission of data
`and facsimile on the following:
`- General Switched Telephone Network (GSTN)
`0 60-103 kl-I2 Frequency-Division Multiplexed (FDM}
`group-band ‘
`0 Cellular radio
`
`In addition, high-speed data is being considered for transmis-
`sion on the High-rate Digital Subscriber Line {ll-IDSL).
`The technique has been called by many names—
`orthogonally multiplexed Quadrature Amplitude Modulation
`(QAMl [2]. orthogonal FDM [3], and dynamically assigned
`multiple QAM [4}—but we will refer to it by a generic name:
`Multicarrier Modulation (MCM). A more general form of the
`technique, which uses more complex signals as carriers [5], has
`been developed recently as vector coding [61 and structured
`channel signalling ['.-’] [8]. Unless otherwise stated, 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. but
`the two most irnportant ones are first. that an MCM signal can
`be processed in a receiver without the enhancement (by as
`much as 8 dB in some media) of noise or interference that is
`caused by linear equalization ofa single-carrier signal. and sec-
`ond. that the long symbol time used in MCM produces a much
`greater immunity to impulse noise and fast fades.
`The first seven sections of this article will discuss the follow-
`ing: the general technique of parallel transmission on many
`carriers;
`the performance that can be achieved on an
`undistorted channel; algorithms for achieving that perfonn-
`ance: dealing with channel impairments; improving the per-
`formance through coding; and methods of implementation.
`The last two sections discuss duplex operation of MCM and
`the possible use of this on the GSTN.
`
`Multiplexing
`MCM is a form of FDM: the basic principle is shown in Fig-
`ure 1. Input data at MI, bis are grouped into blocks of M hits at
`
`
`
`Serial-
`to-
`ll’. his
`Parallel
`Convener
`
`
`'Gl‘H"I _r“=nr
`
`Fig. I. Basic mtdticarrier troltsmttter.
`
`a block (“symbol”) rate off5. The M bits are used. mu hits‘ for
`the carrier at j;._” to modulate N‘. carriers, which are spaced Af
`apart across any usable frequency band; that is,
`
`fcIn=rtAf‘forn=nl to F12
`
`ill
`
`H
`
`and
`
`where
`
`The modulated carriers are summed for transmission. and
`must be separated in the receiver before demodulation. Three
`methods have been used for this separation:
`0 First. the earliest MCM modems borrowed from conven-
`tional FDM technology. and used filters to corn letely sepa-
`rate thc bands. The transntttted power spectra _or_tt_.tst three
`sub-bands of a rnulticamer system are shown In Figure 2a.
`
`‘Each of the ms typically = 2 to 3-
`
`0153-5804-I90/0005-0005 $01.00 ‘ 1990 IEEE
`
`May I990 . IEEE Communications Magazine
`
`- 5
`
`APPLE 1011
`
`APPLE 1011
`
`1
`
`

`
`dam
`-50
`
`/
`Fleceiued Signal
`Power per
`
`Sub-band rl
`
`F
`
` Received Noise per
`5ub~hancl r.-
`\
`
`10
`
`«to
`an 2'0
`20
`1‘°
`[El Received signal and noise power.
`
`
`
`
`
`6 :-—r—-'-—5--'-4'—3-2-
`
`-1-"0
`
`UN
`
`0
`
`L0
`
`0.5
`
`1.0
`
`0.5
`
`1.0
`
`0.5
`
`fc.n- 1
`
`fl':.n+ ‘I
`‘Fan
`{at FDM filtering
`
`{bl
`
`SDRM spectra
`
`~
`
`to cast Isencuw - nu? functions
`Fig. 2. MGM trurumlr power spectra.
`
`Because ofthe difficulty of implementing very sharp filters,
`each of the signals must use a bandwidth. ( I + a1)’, which is
`greater than the Nyquist minimum, L; the efficiency of
`band usage is j;}A_,f = ll( 1 + a).
`0 Second [9— [3] the efficiency ofband usage was increased to
`almost
`IDCPX by using Sta
`cred Quadrature Amplitude
`Modulation (SQAM); the individual transmit s mi ofthe
`modulated carriers still use an excess bandwi
`th of u. but
`they overlap at the — 3 dB frequencies (as shown in Figure
`2b). and the composite spectrum is flat. If e S 1, each sub-
`band overlaps only its
`immediate neighbors, and
`orthogonality ofthc sub-bands—with resultant separability
`in the receiver—is achieved by staggering the data (that is.
`offsetting it by half a symbol period) on alternate in-phase
`and 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 carriers must be small
`(typically less than 20).
`I Third 2] [4} [l4—l6]. the carriers are “keyed” by the data.
`using uadrature Amplitude Shift Keying (QASK). The in-
`dividual spectra are now sinc functions. as shown in Figure
`2c; they are not bandlitnited but. as we shall see. the signals
`can still be separated in the receiver, the frequency—division
`is achieved. not by bandpass filtering. but by baseband pro-
`cessing. The big advantage of this approach is that both
`transmitter and receiver can be implemented using efficient
`Fast Fourier Transform (FFT) techniques.
`
`Maximum Achievable Bit Rate:
`Seeking the Shannongri-la of Data
`Transmission
`
`The performance of it data transmission system is usually
`analyzed and measured in tennis of the probability of error at a
`given bit rate and Signal-to-Noise Ratio (SNR). It is. however,
`more useful for our purpose-—and, indeed, more appropriate
`for modern data communication systems that use any combi-
`nation of compression. error correction, and flow control—to
`consider the attainable bit rate at a given error rate and SNR.
`For single-carrier signals that are equalized with either :1 Lin-
`
`6 ‘ May I990 - IEEE Communications Magazine
`
`
`
`Totalbnrata : l,"l.tt4+2iI5 + 2B>rE+ ?x5 + 4:-(4-lv 3x3 4- ‘K2!
`.: 52.5 = 15.625 his
`
`
`10
`20
`30
`40
`sell:
`ttll Bit and power nssignrrionts.
`
`Fig. 3. Adaptive loading fitr ct oddity distorted G57‘N channel.
`
`enr Equalizer (LE) or a Decision-Feedback Equalizer (DEE)
`this can be done by inverting the well-known error rate formu-
`las (e.g., those for LE3 [I7] [IS] and DFE-:5 [3]).
`The variables for a multicarrier signal are the number ofbits
`per symbol, run. and the proportion. 'r,,. of the total transmitted
`power, P, that are allotted to each sub-hand. The aggregate bit
`rate is approximately maximized if these variables are chosen
`so that the bit error rates in all the sub-bands are equal. This
`has not been proved rigorously. but it is intuitively reasonable;
`the dependence of error rates on the m,, and 7,, is such that if
`the error rates are unbalanced. the rate in one band will in-
`crease much more than it will decrease in another band.
`In order to calculate the attainable bit rate for a channel
`
`with transfer function liq} and noise power spectrum at the
`input to the receiver U(,O, we can approximate Hff) and U{_,O
`by segments H" and U" centered about carrier frequencies f;._,,,
`as defined in Equation (I). This is illustrated in Figure 3:1 for a
`badly distorted and noisy voiceband channel with f -_- 62.5
`Hz;3 the signal power received in each sub-band is calculated
`assuming that the total transmit power of — 9 d Bm is distribut-
`ed equally across the sub-bands {i.e.. if all the 1!” were equal};
`the total noise powcrin the 0.3 to 3.4 kHz band is — 57 dBm.
`The probability of bit error. 93. in the symbol-by-symbol de-
`tection (i.c.. without the benefit of any coding across symbols)
`
`‘The possible non-whiteness ofthe “noise” is irnpomnt for HDSL,
`where the principal impairment is strongly correlated Near-End Cross-
`Tallt {NEXT}.
`
`3"l'liis is one ofthe can-ier separations used in Telebifs ‘Trailblazer’
`modem; the reason for such a choice (62.5 = 8.0003128) will become
`clear later.
`
`2
`
`

`
`of the QAM signal in sub-band n——assuming no interference
`from the signals in the other bands—is
`
`made very small. Then the summation in Equation (4) can be
`approximated by an integration, and the maximum bit rate
`
`{P _K I
`5 _ BQ
`
`3
`Ln3—t
`
`l',.PlH_,|” "‘
`U“
`
`l
`
`[2]
`
`Hm = Ago mi‘!
`
`{5}
`
`where
`
`Ln.2:2l|'tfl.
`
`|l: 4(l — lfblfmv
`
`and K is an error—rate multiplier, which is a little less than 6 if,
`as is most usual. differential phase modulation and a 3-tap
`scrambler are used. Q is defined, as usual. by
`
`I
`
`:5
`
`9(3) = —-vi.-: L ettp(—y2I2l dy,
`
`(3;
`
`[P is the total transmitted power, and 7,, is the proportion of
`that total allotted to sub-band it.
`We would like to solve Equation (2) for run, but this cannot
`be done explicitly because mu occurs in three places on the
`righthand side. Kalet [19] developed upper and lower bounds
`for the symbol error rate by considering the limits of 4(l —
`UL}, but it is adequate for our purpose‘ to consider only an av-
`erage value offi. For a practical range of mi,‘ from 2 to 8 fivaries
`from 1 to 1582, so an average value of 4 for the combined
`en-or-rate multiplier, ilk, will sufiice. Then. as shown in [19].
`Equation (2) can be inverted. and the total number ofbits that
`can be transmitted in one symbol with error probability 95
`using N, sub-bands can be written:
`
`M:
`
`“E
`
`..,,,l
`
`7',,I°lH,,l2
`3
`log? 1+-T j (41
`IQ
`15°!-tll
`U.
`
`where
`
`‘2
`
`_ 7:1: '
`fl=lI
`
`ldeally, the optimum power distribution, 15,. should be cal-
`culated by a "water-pouring” procedure that is similar to that
`of Gallager [20], but for high SN'Rs (corresponding to most ac-
`ceptable error rates), the optimum 1,, are approximately equal.
`The most efficient use is made of the channel if the symbol
`rate, fl, is made equal to the carrier separation. 4,1’: and both are
`
`=
`
`'2.
`
`r,
`
`Plum“
`3
`log? 1 + Tfi T
`[Q
`£9”-'4}! WW0
`
`is that for which the
`where the frequency range. J} to
`integrand is > 2 {i.e., the range over which QAM transmission
`is possible}, and W (= f" — jg is the measure of that range.
`As pointed out by Kalet and Zervos [3]. Equation (5) is very
`similar to the hit rate for a Single-—Carrier QAM (SCQAMJ sig-
`nal equalized by a DFE. which was originally shown by Price
`[1 B]. In fact, the only difference is in the frequency range ofthe
`integration: for the single-carrier signal with DFE it should be
`extended to that for which the integrand is greater than zero.
`but in practice the extra contribution to the integral is usually
`insignificant.
`It should be noted that Equation (5) assumes that the num-
`ber of bits per carrier is continuously variable but, in practice.
`each m” must be integer.5 II was shown in [17] that the eflbcls
`of this quantizing can be mitigated by adjusting the 1,, to re-
`cqualizc the error rates in all the sub-bands, and it has been
`found from numerous simulations that
`the total bit rate
`achieved in this way is only slightly less than that given by
`Equation (5).
`Thus. the aggregate bit rate for MCM is approximately
`equal to that for SCQAMIDFE; for channels with attenuation
`distortion or non-white noise this may be considerably greater
`than for SCQAM with a linear equalizer.
`
`Adaptive Loading
`It was shown that if the ratio lH{)')|"/U0) varies significantly
`across the band and a fixed loading is used [21], the error rate
`in the too-heavily-loaded sub-bands may be very high, and the
`overall error rate may be greater than for a single—cnrrier signal
`H7]! The or” must be varied in order to keep all the sub~band
`error rates. 93”, equal; the following procedure for calculating
`the y,, and integer or" was described [I6].
`Given a set of signal»to—“noise“5 ratios, measured in the re-
`ceiver when the far transmitter is transmitting at the maximum
`pcnniltcd level in all sub-bands. calculate the tenns, APHM of
`an “incremental power” matrix, where APR” = P ' fl —
`Pm _ _, ,, {P,,,_,, = the transmit power needed in sub-band rt to
`transfer or bits per symbol at some predefined error rate), and
`clearly. PM 2 0 .
`Then assign bits one at a time to carriers. each time choos-
`ing the canier that requires the least incremental power. This
`can be described algorithmically:
`I Search row 1 for the smallest AP,-In
`I Assign one more hit to sub-band H
`0 Increment M and PM: that is.
`M‘ =: M + l and Pm,‘ = PM + nP,._,,
`
`‘Equation (1) is exact only for square constellations (i.e.. or even)
`anyway. For m = 5 and T. the “cross” constellations are sligh more
`efficient. and lllis slightly lower, for m,, --= 3 all constellations tireless
`efficient, and P is significantly higher.
`
`5Coding schemes to allow non-integer on have been discussed for
`use on the DSL. but it is not clear how much tliey would increase the ca-
`pacity.
`
`"The equivalent noise should be the power sum offilunsian noise,
`NEXT, and inter-symbol and inter-channel iotetferenoea
`
`May 1990-11555 Communications Magazine
`
`0
`
`'7
`
`3
`
`

`
`0 lglfiive all terms of column rt up one place; that is, AP”,' =
`i + Lit
`- Repeal search
`For the preferred mode of operation for multicarrier—at
`the highest rate achievable with a predefined error i-ate~—the
`assignment should be stopped when PM just exceeds P, the
`available power. If. however. transmission at agiven bit rate (a
`synchronous “bit pump’) is insisted upon, then the process
`should be stopped at the appropriate value of M. Pm, may then
`be less or more than permitted (that is. the specified error rate
`was pessimistic or optimistic. respectively): all allotted powers
`must be scaled to adjust PM to the correct value.
`The resulting power distribution for the channel of Figure
`2a is shown in Figure 2b. The discontinuities occur because of
`the integer constraint on the number of bits; ifdfis small, then
`the SNR can change only slightly from one sub-band to the
`next, so that if, for example, the SNR is decreasing. and m,, =
`.?fl"_ , — l. the nth carrier will require approximately 3 dB less
`power than the (ri — llth carrier for the same error rate. The al-
`gorithm is clearly not water-pouring in the classical sense, but
`since it puts every increment of transmit power where it will be
`most effective. it appears to be optimum for multicarrier trans-
`mission using QAM constellations and symbol-by-symbol de-
`tcction.
`
`Feedback from Receiver to Transmitter
`Adaptive loading requires that the receiver measure the
`sub-band SNRs, calculate the best power and bit assignments,
`and send this information back to the transmitter. This may
`seem like a big increase in complexity. but it should be noted
`that all single-carrier systems that make best use of a channel
`also require some feedback. This can be used in three diiierent
`ways:
`
`0 Many present fixed7symbol-rate systems use a "‘fall—back"
`procedure that requires the feedback of error-rate inforrna-
`tion.
`
`I Better use of a channel might be made by calculating and
`feeding back an optimum symbol rate. and then using some
`font} of Maximum Likelihood Sequence Estimation in the
`receiver.
`
`- Another approach is to combine trellis coding with an adap-
`tive symbol-rate and a DFE. A conventional DFE cannot be
`used. however, because cl’ error propagation, and the func-
`tion of the feedback pan of the DFE must be implemented
`in the transmitter using a generalization of Tomlinson
`preceding; this requires the feedback of much the same de-
`tailed channel characteristics as are needed for MCM.
`
`Adaptive Loading When NEXT is the
`Dominant Impairment
`For higlbspecd transmission on the subscriber loop, NEXT
`is usually more harmful than noise. If this NEXT is mainly
`from other MCM transmitters, a unilateral decision to change
`the spectral distribution of one transmitted signal would
`change the conditions under which the other transmitters
`make their decisions; clearly some coordinated strategy for as-
`signing all the sub—b:trid powers is needed. Work is being done
`on this but it is too early to predict the results.
`
`Modulation and Demodulatioii
`
`Modulation is performed on M bits (a symbol or block} of
`data at a timc—-—preferably using an Inverse FFT (1 FFT}——and
`samples of the transmit signal are generated at a sampling rate.
`J; mp. For greatest efficiency fmmp should be equal to AImulti-
`p ted by an integer power of two. lffmmp = 2N,0, AL then NM.
`carriers are available for modulation, but the channel will usu-
`ally be such that only N‘. carriers can be used. Ifthese are at fre-
`quencies )1 _.Af to ri3.:l_,IC as defined in Equation (ll. module-
`
`8 0 May I990 - IEEE Communications Magazine
`
` '
`
`'5'. .1...
`
`.
`
`-.'_
`
`.
`
`'
`
`."
`
`.
`
`-
`
`Fig. 4. integrate and dump detection for past:
`
`is most easily
`total of M bits, m,, at a time,
`tion of ii
`accomplished by calculating N‘, complex numbers (each so
`lected from a constellation with 2“"n points). augmenting them
`with rt, - I zeros in front and NM — P12 zeros behind, and per-
`forrriing an NW-point IFFT.
`Modulation via an IFFT is equivalent to multicarrier
`QASK in which the fundamental baseband pulse shape is a ret-
`tangle. am. That is.
`
`gall] = ll?‘ foi-ll St 4: T, and = flotherwiae.
`
`(6)
`
`In the receiver the signal is demodulated by assembling N,
`.
`samples into it bloclt. and performing a real-to-complex
`This is equivalent to demodulating each sub-band separately,
`and then doing an integrate-and-dump on each product, as
`shown in Figure 4. Ifthe received baseband pulse in sub-band It
`is defined as g,l’(r,l, then the output from the demodulator re-
`sulting from an input to another sub-band (ii — Jr] is g,,’{r) mul-
`tiplied by a cosine or sine wave ofthe difference frequency km‘,
`that is,
`
`ll+llT
`
`ha n_*(tl = I
`
`'
`
`I?‘
`
`gu‘lll.exp(fk2II-fiflldt.
`
`l7}
`
`Ifthe channel is non-distorting, so that g,,(U = g,,'(l) = UT,
`then these integrals over a time 1l.«Jfare zero for all non-zero it.
`That is.
`
`ha "_*{l') = ll'ori=k= 0,and=Dol.herwise.
`
`l3)
`
`and orthogonality between the sub-bands is maintained.
`
`Correcting for the Effects of
`Channel Impairments
`Linear Distortion
`The primary effect of attenuation and./or delay distortion in
`the channel is that each subcarrier is received with a different
`amplitude anclfor phase, so that the channel can be grossly
`characterized by a single complex number for each sub-band.
`These are learned from a tra:ini.ng signal ofunmodulated carri-
`ers (a "coinb"'}, and inverted to generate the oompleit coeffi-
`cients of a set of one-tap equalizers. All subsequent received
`sampl are then multiplied by these inverses.
`A secondary effect is that g,,'{t) is not rectangular, and also
`overlaps into the preceding and following symbol periods.
`Moreover. even with an I.Indlsl.Dl'l.¢d—l:lI.tl necessarily band-
`limited——chani:icl, the sub-bands near the ends of the band are
`asymmetrical. and distort their gins. Thus, there is both Inter-
`Channel Interference {ICU U1" _,‘(0) at D), and Inter-Symbol
`Interference (lSl)(lt _,,(;|; 1) at ll‘ , and even the combination of
`the two (Ir,,_,,_ kfj: ll it 0); orthogonality of the sub-bands is
`lost.
`
`4
`
`

`
`It can be seen that the impulse response of each sub-band
`depends only on the channel, and that the transient at the be-
`ginning and end of each g ‘{1} is independent of the separation
`of the carriers (that is, of’the symbol period, 77. One way of
`dealing with distortion would be to increase Tenough that dis-
`tortion becomes insignificant, but
`in general
`this is not
`possible} Four other ways have been described; these are dis-
`cussed below.
`
`Guard-Period
`
`The transients in the g,,'{.') can be avoided [l] [14] [22] by
`postponing the integration in Equation {'5} for a time T‘, and
`increasing the total symbol time to T, = T + T3. While Still. Of
`course. retaining T = lfdfi One commercial modem for the
`GSTN [4] uses T = l28 msand T1 = 7 ms. This lirnitsthe MSE
`from ISI and ICI on even the worst lines to less than 1'3, but It
`does reduce the total bit rate by 5.2%.
`
`Passband Channel Equalization
`The reduction in hit 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 a
`long symbol period. the equalizer can be much less complex
`than that for SCQAM; furthermore, it may be acceptable in
`some media to adapt it only during training, and freeze it dur-
`ing data reception.
`(It should be noted that although the signal is being linearly
`equalized,
`this approach does not
`incur the large noise-
`enhancement penalty of single~can'ier modulation. The load-
`ing is calculated from. and the performance determined by, the
`sub-band SNR.-i, which are reduced only slightly by the ampli-
`tude equalization across each sub-band:
`the equalization
`across the full band acts mainly like a delay equalizer plus
`many separate Automatic Gain Controls, or AGCs.)
`The conclusion that can be drawn from [23] is that for such
`a simple equalizer, a Tapped Delay Line (TDL} structure using
`time-domain convolution is the most efficient. The training
`signal for this should be an unmodulated subset of the carriers.
`and the taps 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. inverting
`these, and performing an It-‘FT to calculate the taps.‘
`The optimum lengths of the data symbol and the TDI..are a
`subject for further investigation. Clearly. as the length of the
`symbol is reduced, the effects of IS] and ICI become relatively
`more important, and the complexity of the equalizer must be
`increased. The limit of this would be reached when the
`equalizer had 2N: parameters. and, since it would then equa-
`lize the channel response to all N, carriers, it could also take
`over the role of the one-tap complex baseband equalizers.
`
`Baselrartd Equalization
`The ICI terms defined by setting: = 0 in Equation (6) form
`an N: in: NE matrix. with the terms oil‘ the main diagonal de-
`creasing only very slowly (approximately as Mr]. This would
`require an extremely complicated equalizer, and baseband
`equalization is not used for QASK signals. It can be used, how-
`ever, for SQAM signals [I 3], because each sub-band is filtered
`so as to limit interference to the two adjacent bands; the ICI
`matrix then has terms only on the main and two adjacent diag-
`onals.
`
`7The DSP memory, the processing requirements {proportional to T
`and logzi“, respectively), and the delay through the modem all become
`prohibitive.
`3This is typical of the judicious mixture of frequency- and time-
`domain processing that is used in MCM. See [23] for a discussion of
`the trade-offs, and for more references on frequency-domain process-
`trig.
`
`riff.‘ Unmoclulated Centers
`
`
`
`fl” 7' Lower Sideband of I
`in T
`
`, fl‘ \
`
`Upper Sidebend of {*4
`
`I“ 7
`
`Upper Sideband of f*_B
`
`Lower Sidabancl oi Eh B
`
`Fig. 5. Miu'ri'carri'er spectrum with rrdebands resulting from 60 Hz
`pltusejitler.
`
`Vector Coding. Structured Channel Signaling
`Holsinger [5] showed that orthogonality ofthe sub-band sig-
`nals through a distoned channel can be achieved by using, as
`"carriers," the eigenvcctors of the auto-correlation matrix.
`This approach is presently attracting considerable interest
`[6-B]. but it is too soon to know whether it can compete in cont-
`putational efficiency with passband equalization.
`
`Combination of Different Methods
`
`The 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 (T31 T = I
`to 2%} seems to be a very promising combination.
`Phase Jitter
`
`Phasejitter affects MCM and SCQAM quite differently. Ifa
`composite signal of unmodulated carriers is subjected to phase
`jitter of frequency
`and amplitude less than about 10'. then
`each carrier at rrdfwill generate just two significant sidebands
`at ndf + f. The carriers and their sidebands are shown in Fig-
`ure 5 for lhe case where J;-!4f= 7.68”.
`Both detection methods in the receiver—an FFT or de-
`modulation followed by an integrate and durnp—result in
`equivalent filter shapings of sinc functions centered at the car-
`rier frequencies.
`11 can be seen, therefore, that the sidebands of at least two
`other carriers"? contribute to the distortion seen by any given
`carrier. Since the data modulated onto these other carriers is
`uncorrelated with that on the carrier under consideration, the
`jitter is seen as random distortion about each point in the con-
`stellation, as shown in Figure 651.. That is, the jitter power (the
`total power in all the sidebands) is spread evenly over all carri-
`ers and over all data patterns on those carriers, and it can be
`added to the noise on a power basis.
`In contrast, a single-carrier constellation is rotated by the
`jitter, as shown in Figure 6b; the outer points are clearly more
`susceptible. and the overall effect upon the error rate with
`added noise will be greater than for MCM.
`
`Tracking Phase Jitter
`Although the effects of phase jitter are less for MCM than
`they are for SCQAM. they should not be ignored; identifiable.
`discrete components ofjitter should be tracked. Identification
`is easier in a multicarrier receiver because much of the signal
`processing involves Fl-Ts, but tracking is harder because of the
`long symbol period.
`One method [24] processes one complete symbol to calcu-
`late the remanent phase error (the difference between the input
`
`9f: ?.8l 2.‘: Hz is the prefefled carrier separation in the Trailblazer.
`and J; = 6|] Hz, the most common jitler Frequency in the U.S.
`'°Tht- number of contributing carriers reduces to two in the special
`case of_l,-ld_l'hcing an integer.
`
`May 1990 ~ IEEE Communications Magazine I
`
`ll
`
`5
`
`

`
`
`
`lei Multiearriar.
`
`[bl Single carrier.
`
`Fig. 6. Effects ofphcsejtrreron ortequadrttnt oft: to-paint constellation.
`
`phase and the locally generated tracking phase), passes the
`error through narrow-band feedback filters as described in
`[I7]. and uses the outputs to update a phase predictor which
`generates the tracking phase for the next symbol. It has been
`found that discrete jitter components can be tracked almost
`perfectly.
`
`Non-Linear Distortion
`A multicarrier signal is the sum of many independent mod-
`ulated sinewavcs, and its sampled amplitude has an almost
`Gaussian distribution. Therefore. its peak-to-average ratio is
`much higher than that ofSCQAM, and it is more susceptible to
`non-linear distortion. The most severe component of this is
`usually a negative cubic tenn (“saturation”). and it appears
`that if this can be quantified it can be. at least partially, correct-
`ed in the receiver by operating on the samples with a comple-
`mentary nonlinearity.
`
`Impulse Noise
`Because a multicarrier signal is integrated over a long synt-
`bol period. the effects of impulse noise are much less than for
`SCQAM: indeed, this was one of the original motivations for
`MCM [25]. Tests reported to the Consultative Committee for
`International Telephone and Telegraph (CCITT) [26] showed
`that the threshold level for noise to cause errors can be as much
`as 11 dB higher for MCM.
`
`Single-Frequency Interference
`There is an interesting tirnelfrequency 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 to
`impulses in the frequency domain (single-tone interference).
`The advantage of MCM lies in the fact that the sources of these
`interferers are discrete.‘ ‘ and their frequencies are usually sta-
`ble (in contrast to the time of occurrence of impulses in the
`time domain}; they can he recognized during training and
`avoided (that is, nearby carriers are not used) by the adaptive
`loading algorithm.
`
`Fades
`in
`Mobile radio channels often suffer widehand fades.
`which the SNR across the whole frequency decreases alarming-
`ly for a short time. A single-can-ier system might have a very
`low error rate between these fades, but would suffer from a
`very high one during a fade; the overall error rate might still be
`intolerable.
`On the other hand, in a multican-ier system both the signal
`and the noise are integrated over the whole symbol period; the
`average SNR and resultant error rate are usually still tolerable.
`
`' ‘A tone at 2.600 H1, which is used in some sinde-freqency signal-
`ling systems. is the most notorious intcrferer in the US.
`
`12 0 May 1990 - IEEE Communications Magazine
`
`Trellis Code Modulation
`
`The advantages ofTCM—about 3.5 dB ofcoding gain with
`present-generation codes and perhaps up to 5 dB with future
`codes—are now widely recognized. Early applications of trel-
`lis coding to MCM [25] [27] used encoding in the conventional
`way; that is, from symbol to symbol. Only a few carriers were
`used, and the delay through the Viterbi decoder was just tolera-
`ble because the symbols were fairly short. However, when
`MCM 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 trellis
`coding incompatible.
`The justification for trellis coding of SCQAM in general and
`decoding by the Viterbi algorithm in particular is that the noise
`is white (or almost so); that is, samples of it are almost
`uncorrelated from symbol to symbol. The timeffrequency du-
`ality of single-fmulticarrier can be exploited here by recogniz-
`ing that samples of the noise, averaged over one symbol, are
`also uncorrelated from one frequency sub-band to the next,
`and that therefore trellis coding can be applied in the same way
`[28].
`Following the tenninology of [29 , let the m,, bits for input
`to sub-band it be designated xn’, 1:, .....1r,,’’'. Then x,,,’ and 3:“:
`should be input to the encoder to generate the out ut set 2 0.
`z "
`3 which together with the uncoded bits .7:
`,...r,,"‘ til’:
`uged iio define at point in the appropriate constell tion. The
`state of the encoder after encoding sub-band it is then used as
`the initial state for encoding sub-band (it + 1}.
`As a result of the adaptive loading, the number of bits, in ,
`and therefore the size of the nth constellation will probal:-l'
`vary with It. but this does not matter. The three encoded bits
`define one of eight sets of points, each containing 2""n'3l
`points, and the Viterbi decoding determines these three hits
`and, hence, the set; identification of at point within the defined
`set can then be done one sub-band at a time. even though the
`size of the set may vary front one sub-band to the next.
`Any of the codes that have been developed for single-carrier
`could he used for MGM, but since a decoder will have to deal
`with constellations of varying sizes. it would be preferable to
`use codes and signal mappings that allow constellations to
`grow smoothly, such as were described in [30].
`
`Block Processing ofII Convolution! Code
`It is highly desirable that all of the data in one symbol
`(block) be decoded in the same symbol period and from only
`the signals received within that block. This would not be possi-
`ble, however, if both conventional encoding and decoding
`were used, because, first, a conventional encoder uses its state
`after encoding the last sub-band as the initial state for encoding
`the first sub-band of the next symbol, and second. a conven-
`tional Viterbi decoder makes a decision about a symbol only
`after receiving K more symbols. where K3. the "look-back”
`distance or d
`mg delay, is typically between five and eight
`times the constraint length. I. of the code—about twenty for
`the common eight-state codes. Consequently, the last K‘. sub-
`bands could not be decoded until the next symbol had been re-
`ceivcd and demodulated.
`To achieve full block decoding the look-back distance in the
`decoder must be curtailed towards the end of the block. This
`can be done in two ways:
`- The encoder can be modified by constiraining l bits at the
`end of the symbol in order to force the 2 state encoder into a
`known final state. Then all {M — 1) unconstrained bits can
`be decoded with no reduction of coding gain. This is easier
`to do with a fecdfonavard encoder, but ‘ll. would scent to be
`feasible even with a non-linear feedback encoder such as is
`described in CCITT Recommendation V.32.
`0 The Viterbi decoder can be modified to decode the last Kg
`sub-bands by tracing baclt the path from the smallest final
`
`6
`
`

`
`Lowpass
`Filter
`
`Lowpass
`Filtfil‘
`
`
`
`Fig. ?. Basic mtJl‘tt'eorrier "mo-dem."
`
`bits from the
`metric, and decoding all of the remaini
`nodes on that path. This means that the wow gains for the
`last few carriers decrease more or less linearly tom the max-
`imum to about 0 dB for the last carrier. This effect can be
`anticipated in the original loading of these carriers. and will
`probhtlfily reduce the overall bit rate by about four hits per
`syrn
`.
`
`Implementation
`A simplified block diagram ofa multicarricr "mo—dem" (the
`transmitter of one modem and the receiver of another} is
`shown in Figure T. The main processing in the transmitter and
`receiver is done with an IFFT and an FFT. respec