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The Effect of Timing Jitter on the Performance of a Discrete
Multitone System - Communications, IEEE Transactions onIEEE
TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996 799

The Effect of Timing Jitter on the Performance of a Discrete Multitone System

T. Nicholas Zogakis, Member, IEEE, and John M. Cioffi, Fellow, IEEE

Abstract- The transmission of high-speed data over severely band-limited channels may be accomplished through the use of discrete multitone (DMT) modulation, a modulation technique that has been proposed for a number of new applications. While the performance of a DMT system has been analyzed by a number of authors, these analyses ignore the effect of timing jitter on system performance. Timing jitter becomes an increasingly important concern as higher data rates are supported and larger constellations are allowed on the DMT subchannels. Hence, in this paper, we assume that synchronization is maintained by using a digital phase-locked loop to track a pilot carrier. Given this model, we derive error rate expressions for an uncoded DMT system operating in the presence of timing jitter, and we derive an expression for the interchannel distortion that results from a varying timing offset across the DMT symbol. In addition, we investigate the performance of trellis-coded DMT modulation in the presence of timing jitter. Practical examples from the asymmetric digital subscriber line (ADSL) service are used to illustrate various results.

I. INTRODUCTION

ISCRETE multitone (DMT) modulation is a technique in D which a transmission channel is partitioned into a number of independent, parallel subchannels, each of which may be considered as supporting a lower-speed quadrature amplitude modulated (QAM) signal [l]. Performance is maximized by allocating more bits to subchannels with high signal-to-noise ratios (SNR’s) and fewer or no bits to subchannels with low SNR’s. An example of an application for which DMT modulation is well suited is the asymmetric digital subscriber line (ADSL), a service proposed for providing a high-speed downstream channel, ranging from 1.544 Mb/s to 6.4+ Mb/s, from the central office to the customer, along with a lower- speed upstream channel over existing copper twisted pair [2].

Several authors have evaluated the performance of a DMT system for a variety of applications, focusing on maximizing data rate or maximizing margin under a constraint on the available transmit power [l], [3]-[5]. However, in these analy- ses, perfect synchronization is assumed, whereas in an actual system, the practical timing recovery mechanism will result in some degree of timing jitter. The importance of this form

Paper approved by P. H. Wittke, the Editor for Communication Theory of the IEEE Communications Society. Manuscript received August 15, 1994; revised July 15, 1995. This work was supported in part by a National Science Foundation (NSF) Fellowship and in part by Contracts CASIS 2DPD335 and NSF 2DPL133.

T. N. Zogakis was with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA. He is now with Amati Communications Corporation, Mountain View, CA 94040 USA.

J. M. Cioffi is with the Information Systems Laboratory, Stanford Univer- sity, Stanford, CA 94305 USA.

Publisher Item Identifier S 0090-6778(96)05506-7.

of impairment in the determination of error rate performance increases as the available bandwidth, which is determined by the channel SNR function, decreases and as the data rate increases, since both trends result in larger constellations being used on some of the subchannels. When large spectral efficiency is required and constellations supporting on the order of 10 b or more are allowed, then careful attention must be given to the synchronization scheme.

Similar to more traditional single-carrier modulation tech- niques, the performance of a DMT system may be enhanced by the application of coding. For instance, [6] and [7] present methods for applying trellis coding to DMT modulation, while [8] investigates the performance of a concatenated coding scheme consisting of an inner trellis code and outer Reed-Solomon code when applied to a DMT system. Each of these coding schemes requires const ellation expansion over a subset of the carriers and, thus, potentially increases the susceptibility of the system to timing jitter. Furthermore, trellis decoders are based on the assumption of uncorrelated Gaussian noise, whereas, timing jitter introduces correlated noise into the system. For a DMT system employing trellis coding across the tones as described in [7], the correlation between the phase errors caused by timing jitter on consecutive complex symbols at the input to the trellis decoder is quite strong. Hence, it is not clear whether or not the timing jitter requirements are significantly tighter for a trellis-coded DMT system compared to an uncoded DMT system.

In this paper, we investigate the performance of both an uncoded and a trellis-coded DMT system in the presence of timing jitter. For simplicity, we assume that synchronization is maintained by designating one of the carriers as a pilot signal and using a digital phase-locked1 loop in the receiver to track the pilot carrier. This assumption leads to a tractable analysis and corresponds to the technique implemented in DMT modems for ADSL. Throughout the analysis, exam- ples from the ADSL service are used to illustrate various points.

In Section 11, we establish the DMT timing jitter model that serves as a starting point for the analysis. In Section 111, we analyze the performance of an uncoded DMT system in the presence of timing jitter, and we compare the analytical results to simulation results for two ADSL scenarios. In Section IV, we first address the application of trellis coding to DMT modulation and then investigate the performance of a trellis- coded DMT system in the presence of timing jitter. Finally, in Section V, we discuss some of the implications of our results for the ADSI, service.

0090-6778/96$05.00 0 1996 IEEE

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XOO IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

Bit Allocation

Parallcl

Fig. 2. Baseline DMT receiver.

11. DMT TIMING JITTER MODEL

Figs. 1 and 2 present simplified block diagrams of the DMT system that we consider in this paper. At the input to the transmitter, the bit stream is partitioned into blocks of size b = RT bits, where R is the uncoded bit rate, T is the DMT symbol period, and b is the number of bits contained in one DMT symbol. The bits collected during the ith symbol interval are allocated among f l subchannels or tones in a manner determined during system initialization, with bk bits assigned to tone k and C b k = b. On subchannel k , the bk

bits are mapped to a constellation point Xk,i = Ah,? + j B k , z in a constellation of size 2bk with unity distance between constellation points. Next, the constellation point is scaled by a real multiplier, ,9k, and the collection of constellation points {xl;,z = g k X k , i , k = l , . . . , N } serves as the input to an inverse fast Fourier transform (IFFT) block. The constants { g k } are chosen so that E{ lfi;k,z12} = Pk, the power allocated to the kth tone. The time-domain signal that is transmitted over the channel is obtained by performing a length N = 2N IFFT on the complex symbols { x k , i , k = 0 , l : . . IV - I}, where T0.i = 0 and { x k , i = xkT-k,i, k = F + 1. F + 2. . . ! N -

The kth subchannel is associated with the frequency f k = k A f , where A f = l / T . Hence, the DMT symbol transmitted during the ith symbol period is given by 191, 1111

I}.'

' I n practice, a cyclic prefix [9], [lo] would be added to the data block before transmission to eliminate interblock interfercncc and to make the linear convolution with the channel look like a circular convolution. To simplify our notation, we ignorc this complication since it does not change the main results of our jitter analysis.

- where A k . z = g k A k , , , B k , z = g k B k , , , and g z ( t ) is a rectangu- lar window function defined as

t - ZT - T / 2 + T / ( 2 N ) T .9z(t)

In forming the limits of the summation in (l), we have assumed that the Nyquist bin is not used. The transmitted signal s ( t ) , formed by sending a sequence of DMT symbols, is

30 1v/2-1 z s ( t ) = 1 1 [Akzcos(27rkAft)

The signal is sent over the channel where it is convolved with the channel impulse response h ( t ) , yielding a received signal of

To focus solely on the effect of timing jitter in this initial discussion, we ignore the contribution of additive noise; the noise will be included after the final expressions are obtained. Denoting the Fourier transform of the channel impulse re- sponse by F{h( t ) } = H ( f ) e 3 ' b ( f ) , we may simplify (4) by making use of the relationship [ I l l

h(t) * [go( t ) cos(27rkAft)l

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM 801

where P ( f ) = (-1/27r)(ali/(f)/af) is the group delay of the channel. A similar expression is obtained for the sine function. For our purposes, the possibility of interblock interference may be eliminated by assuming the group delay to be equal to a constant, which we conveniently take to be zero. In a well-designed DMT system, this approximation will be valid since interblock interference would otherwise be detrimental to system performance. Hence, under these assumptions, the final expression for the received analog DMT signal is given by

00 N/2-1

- H k : B k , , s i n ( 2 ~ k A f t + $k)].qgz(t) (6)

where Hk = H ( k A f ) and $ k = 4(kA f ) . At the input to the receiver in Fig. 2, the first step of

demodulation is to sample the signal at a nominal rate of f S = NAf. We denote the sampling instances by ( iN + m)Ts + r,,,, m E { O , l , . . . , N - l}, where rm,, is a timing offset that may vary from sample to sample and T, = I / f s .

Hence, the received sequence of samples is given by

ith symbol period is given by

At this point, assumptions regarding the dependence of rm,% on the block index i and the intrablock index m must be made to allow for further analysis. Since the DPLL is updated at the DMT symbol rate, the simplest approach would be to assume that r,,% is constant over each block and thus independent of r r ~ A more complicated analysis that more closely approximates reality models the change in timing error between consecutive updates as a ramp L samples long followed by a constant for N - L samples, where L depends upon the control voltage bandwidth of the voltage controlled oscillator (VCO) that is part of the phase-locked loop or L = N for a frequency offset. Both cases are considered in the next section.

111. UNCODED DMT JITTER PERFORMANCE

The statistics of the timing error depend upon the timing recovery mechanism used in the DMT system. For simplicity, we assume that synchronization is maintained by using a second-order digital phase-locked loop (DPLL) to track a pilot carrier located at frequency f, = p A f . The pilot signal is generated by sending a fixed constellation point on the pth tone, and the DPLL is updated at the DMT symbol rate. With the DPLL model, a good approximation is that the phase error on the pilot is Gaussian distributed [12], and we define the phase jitter, 06, as the standard deviation of this Gaussian process.

Next, the received sequence given in (7) is partitioned into blocks of N samples, each of which is transformed by the FFT to obtain an estimate of the transmitted constellation points. To ensure there is no contribution from the past or previous blocks into a sample obtained during the current symbol and, thus, to maintain a reasonable error rate, we must have max{lr,,,l/T} < 1/(2N). Another way of stating this is that the peak timing offset should be less than one-half the sampling period. For example, the DMT parameters for an ADSL system that loop times to the central office include a sampling rate of 2.208 MHz, a pilot carrier of 276 ICHz, and an FFT size of 512 [13]. With these values, the criteria becomes max{ Ir,,% I } < 226 ns, corresponding to a peak phase error of 22.5’ on the pilot. Hence, we can safely assume that the group of N samples, {rTn,& , m = 0, . . . , N - l}, obtained during the

A. Fixed Offset Over DMT Symbol

For the case in which the timing error is assumed to be fixed over the DMT symbol, we replace r,,,, with rl. and compute the discrete Fourier transform (DlT) of (8) to obtain

as an expression for the detected complex point in the Ith bin during the ith symbol period. The complex multiplicative factor ,91Hle3q4 in (99 is a constant that depends upon the channel characteristic and may be compensated by a one-tap frequency domain equalizer (FEQ) in the receiver. Hence, the final expression for the received ]point is

where the noise term (n1,l + j n ~ , l ) included in (10) is a com- plex Gaussian random variable with E{TL?,~} = E{n i , , } = o:. The noise variance after the FEQ is independent of fre- quency since the DMT system is designed for equal probability of error across all subchannels and the received constellation on each tone has a normalized minimum distance of 1.0 after FEQ scaling. In obtaining (lo), we have dropped the DMT symbol index i to signify that the statistics of the variables in (10) are time-invariant.

To compute the two-dimensional (2-D) error rate perfor- mance of the DMT system, we make use of the small angle approximation e34 M (1 + J $ ) to obtain

Si %(Ai - Bi2irlAfr+n1.i) + ~ ( B I +Al2ir lAfr+n~,i ) . (11)

Hence, the probability of a correct decision on the Zth tone given the transmitted constellation point Aq,l + jB,,l and the

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802 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996

phase error 81 = 27rlA f r is

P C I A ~ . ~ I

(12)

where q E {0,1, . . . , 2b1 - I} is a constellation label and Q ( . ) represents the Gaussian probability of error function. Since 81 is related to the phase error on the pilot signal by a constant scaling factor, the 2-D error rate on the Zth tone may be written in terms of the phase jitter 04 as

where

The overall 2-D error rate, obtained by averaging index I , is given by

where 1.i denotes the set of indices corresponding to subchan- nels used for transmission and u = IMl.

The uncoded DMT system's error rate performance as predicted by (15) will be determined by the poorest performing subchannels. Moreover, (13) and (14) indicate that for a particular level of timing jitter, two main factors determine the error rate performance on the Zth tone. The first factor is the frequency of the bin, with higher frequencies experiencing greater levels of jitter than lower frequencies. The second is the size of the constellation supported by the lth bin, where larger constellations are more susceptible to jitter. Fortunately for applications such as ADSL, the higher frequencies typi- cally support smaller constellations than the lower frequencies because of the increase in channel attenuation with frequency.

B. DMT Examples

We now investigate the implications of (15) for the two bit distributions presented in Fig. 3. In both cases, 1616 b are contained in each DMT symbol, and a pilot carrier is located at 276.0 kHz, hence, the null in the bit distributions at this frequency. Scenario A corresponds to transmission over a 9 kft (2.7 km), 26 AWG loop in the presence of near-end crosstalk (NEXT) from ten digital subscriber line (DSL) disturbers and 24 high bit-rate DSL (HDSL) disturbers, and NEXT and far- end crosstalk (FEXT) from ten ADSL disturbers [14]. Scenario B also corresponds to transmission over a 9 kft (2.7 km), 26 AWG loop, but in the presence of NEXT from one T1 disturber in an adjacent wire bundle.

To obtain both bit distributions, we used the practical bit and power allocation algorithm provided in [15]. This

0" I I I 200 400 600 800 1000 1200

frequency (kHz)

Fig. 3. Uncoded bit distributions for two ADSL scenarios

algorithm attempts to find for a fixed data rate the integer bit distribution that maximizes system margin under a total power constraint. The algorithm starts with a flat power distribution and iteratively solves the set of equations

where SNRk is the SNR on the kth subchannel, I' is a constant that depends upon the target error rate, ym is the margin, and b,,, is the maximum number of bits allowed on a subchannel. At the completion of the iterative part of the algorithm, the power on each subchannel is adjusted slightly to ensure equal error rate performance. See [15] for further details. We ran the algorithm on Scenarios A and B with b,,, = 14, I? = 9.8 dB, and a power constraint of 20.0 dBm.

Figs. 4 and 5 present plots of the uncoded error rate curves obtained for the bit distributions in Fig. 3 at various levels of jitter2; square and cross constellations were used on the subchannels in obtaining these results. The continuous curves in the graphs have been obtained by evaluating (15), while the asterisks represent Monte Carlo simulation points obtained by simulating DMT modulation with timing recovery. A solid error rate curve is included in each plot to signify the performance of a system with perfect synchronization.

The correspondence between the theoretical error rate curves and the simulation points in Figs. 4 and 5 verifies the accuracy of the analysis for a wide range of jitter levels. In addition, these plots indicate the importance of choosing a narrow enough DPLL bandwidth or large enough pilot SNR to ensure acceptable error rate performance for a given bit distribution. For instance, although Scenario A results in bits being placed at high frequencies where the jitter is worse,

'Error rate curves are plotted versus the normalized SNR, which is proportional to 1/uF.

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM

m . e

803

- _ _

, / _ _ _

L

10'

Uncoded DMT system's error rate performance for Scenario A at

IO'

Uncoded DMT system's error rate performance for Scenario B at

Scenario B is less tolerable to timing jitter because of the very large constellations used on some of the tones.

Further insight into the degradation in performance caused by timing jitter is presented in Fig. 6 where we plot the 2- D error rate versus the jitter level for normalized SNR's of 11.0 dB and 14.0 dB. This figure clearly illustrates the greater intolerance to timing jitter in Scenario B as compared to Scenario A. In addition, we observe that timing jitter is more critical at lower error rates since the importance of this form of impairment relative to additive noise is greater than at higher error rates. By using the results in Fig. 6, we can determine the maximum tolerable phase jitter for both DMT scenarios and both normalized SNR's to ensure less than a factor of two degradation in the overall error rate. These critical levels are listed in Table I along with the jitter levels required to ensure less than a factor of two degradation on the worst tone in the ~ y s t e m . ~ As is evident from the table, the jitter requirements are quite stringent for both scenarios.

l op3) is beyond the range of the plot in Fig 6 3The jitter level for Scenario A at a normalized SNR of 11 0 dB (P .D %

1oz7-i

Fig. 6. normalized SNR's.

Error rate versus phase jitter for tmo DMT scenarios and two

TABLE 1 JITTER LEVELS REQUIRED TO CAlJSE A FACTOR

OF TWO DEGRADATION IN ERROR RATE

nncou'ed trellis coded

scenario u + , ~ , ~ ~ (worst t o n e ) a+.ma,. (Fz!gure G) uO,mar (Fiyure 9)

A , 10-3

A . - I 0.090" 1 0.14" I 0.16"

C. Variable Offset Over DMT Symbol

The assumption of a constant timing offset over each DMT symbol is equivalent to assuming that the output phase of the VCO changes instantaneously when the control voltage is changed. In practice, the VCO will have a control voltage bandwidth that is determined by a single-pole Butterworth filter. Hence, we model the timing ofFset as

where L is the number of samples, coiresponding to one time constant of the filter's impulse response. By allowing L = N , we also have a model for the case in which the transmit and receive clocks are offset in frequency. Substituting (16) into (8), we find that the block of received samples representing the zth DMT symbol is given by (17), see equation at the bottom of the next page, where 7% = A f r,, 7,-1 = A f r,-l,

In the Appendix, we evaluate the DFT of (17) to derive an and ATt = 7, -

expression for R L , ~ , the received poinc in the lth bin

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804 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996

where Ck,l,i and D k , l , i are implicitly defined in (26). The desired term in (18) is approximately

Hl @l,i G , l , i + jBl,LQ.l,L)

. ! 7 1 & J ~ L (1 + j(l/P)4€!ffi)(Al,, + j&.,) (19)

where E{&,,,} % 0;. Hence, (19) is essentially the same as the expression that we would obtain under the assumption of a fixed phase offset across the DMT symbol.

By the central limit theorem, the interchannel interference introduced by the k # 1 terms in (26) may be considered as additive Gaussian noise. Thus, by defining the power allocated to the kth subchannel as Pk = E{ I & L + j B k , L / 2 } , we arrive at

k f l

as an expression for the power of the interference introduced into the lth subchannel. To assess the significance of the in- terference, we compare the signal-to-interchannel interference ratio (SIR), defined by

k f l

to SNRl = PlH,2/2a;,,, for a system operating at a 2-D error rate of without coding. We use 2F:,, to represent the noise variance on the lth tone before FEQ scaling; this variance is not independent of frequency.

The expectations in (2 1) are quite computationally intensive to compute, so we instead consider some worst case values for

k f l

as upper bounds to SIRl . Through straightforward, though tedious, mathematical manipulations, it can be shown that ( / C k , l , i I 2 + lDk ,1 ,~ ,1~) depends upon the timing error only in terms of the difference between the timing offsets r, and ~ i - 1

in the ith and ( z - 1)th symbols. Hence, the expectations omitted in obtaining (22) are over the probability distribution function (PDF) of A?;, or equivalently over the PDF of A& = 4; - q5-1 = 27rpA7i. As noted earlier, & is normally distributed with variance n;. Furthermore, since Ad, is a filtered version of 4i, it too is normally distributed with variance a i a = 2 0 $ ( 1 - p ( 1 ) ) , where p(1 ) = E{q5;4z-l}/n$. Thus, as an upper bound to SIRl, we consider the evaluation of (22) for phase error differences in the range IAq& 5 3aad.

To illustrate the types of signal-to-interchannel interference levels expected, we consider an example in which 04 = 0.50" and clQ = 0.08'. The latter value has been obtained based on a loop filter

with 01 = 1.98 x l o p 2 and /3 = 2.00 x 10V4. This is in fact the filter that was used in obtaining the results of Fig. 4. The range of values for which A& contributes significantly in the evaluation of (21) is

Fig. 7 presents plots of SNRl and SIRl,; versus 1 for Scenario A, which was described in Section 111-B. In the graph, we have included six plots of S1Rl.i corresponding to the six combinations of A& = 0.1" or 0.25' and L = 35, 70, or 512. It is clear from Fig. 7 that the signal-to-interference level is well above the signal-to-noise level for all 1. Similar results were obtained for Scenario B but are not included here. Furthermore, we showed in Fig. 4 that the error rate obtained for a jitter level of cr4 = 0.50' is very poor, thus confirming the dominance of the phase rotation on each bin rather than the interchannel interference in determining system performance. In order to achieve a more acceptable error rate, either a DPLL with a narrower bandwidth must be used or the pilot SNR increased, both of which tend to decrease the significance of the interference. Finally, we note that for a coded system, a given error rate will be achieved with a lower SNR, so interchannel interference becomes even less important relative to additive Gaussian noise. Hence, these observations fully justify our replacement of rm:i with TL in (8).

< 0.25'.

IV. TRELLIS-CODED DMT JITTER PERFORMANCE

A. Application of Trellis Code

Trellis coding may be applied to DMT modulation by using a single trellis encoder to operate across the tones in the system [6], [7]. In the receiver, the complex points obtained at the output of the FEQ's are decoded by using a single Viterbi decoder across the tones. Equation (9) shows that a phase error on the pilot carrier translates into a phase rotation on each of the tones, with the amount of rotation determined by the ratio of the center frequency of the tone to the pilot frequency. Hence, the errors introduced by timing jitter at the input of a trellis decoder operating across the tones are strongly correlated.

Many of the good trellis codes constructed for the additive white Gaussian noise (AWGN) channel are multidimensional

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM 805

L = 35 . . L = 7 0

7.. \; . ' -.

subchannel 1 0:

Fig. 7. Comparison of SNR and SIR for three values of L and two values of A i l , [email protected] = 0.1' (upper three curves) and A& = 0.25' (second set of three curves), in Scenario A.

codes that require a fractional number of bits to be supported in each 2-D coordinate. In the case of DMT modulation, this complicates the trellis encoder and decoder since many different multidimensional constellations have to be supported. However, a method for accommodating fractional numbers of bits while at the same time maintaining the simplicity of an integer bit distribution is presented in [16]. Basically, if u tones are used to support btot = b bits per DMT symbol in the uncoded case and F is the normalized redundancy of the trellis code in b/2-D symbol, then an integer bit distribution may be computed for the trellis-coded case with btot = b + uF.

The significance from the standpoint of timing jitter of the proposed method for accommodating multidimensional trellis codes is that the constellation expansion is greater than 2' on some of the carriers. For instance, in the case of a trellis code with a normalized redundancy of F = 0.5, the constellation size on about one-half of the tones is doubled, while for the other half, it remains the same. This is different from expanding each constellation by a factor of 2 O . j .

In the next section, we investigate the performance of Wei's four-dimensional (4-D), 16-state trellis code [ 171 when implemented in a DMT system subjected to timing jitter. This code has been adopted for the ADSL standard and has a normalized redundancy of F = 0.5 and a fundamental coding gain of 4.5 dB [18]. We use the integer-based algorithm for accommodating the trellis code redundancy, and we investigate the same two scenarios as in Section 111-B, but with a constraint of b,,, = 15 enforced for the bit distribution computed for the trellis-coded system.

B. Pegormance of 4-0, 16-State Wei Code

Fig. 8 presents simulation results for Scenario A at three different jitter levels: 04 = 0.16", 04 = 0.22", and ~4 = 0.28". To obtain these jitter levels, we used a fixed DPLL and changed the SNR on the pilot carrier. However, we found that in all our trellis code simulations, the error rate depended upon the jitter level and not the DPLL bandwidth, so the results are general. This is not too surprising since the correlation

' " 7 8 9 10 11 12 13 14 15 16 normalized SNR (dB)

Fig. 8. Performance of Wei code in Scenario ,4 for three jitter levels.

among the 2-0 symbols associated with a trellis error event arises primarily from the correlaticln among neighboring tones rather than between DMT symbols. The solid lines in Fig. 8 illustrate the error rate performance for an uncoded system (rightmost solid line) and a trellis-coded system (leftmost solid line), assuming perfect synchronization. The three curves on the far right of the plot present the error rate performance for the uncoded system subjected to the three different jitter levels and were obtained by evaluating (15). Curves marked by asterisks correspond to results obtained from simulations of the trellis-coded DMT system, where the asterisks denote the actual simulation points.

The results in Fig. 8 indicate thoit for jitter levels satisfying 04 5 0.16' and over 2-D error rates of lop7 and higher, the trellis code provides approximately the same gain relative to an uncoded system at the same jitter level as it does under the conditions of perfect synchronization. In fact, a small improvement in gain is observed in the presence of jitter at a jitter level of 04 = 0.16" and a 2-D error rate of lop6. These results are quite surprising since the errors are correlated at the input to the trellis decoder. Even at the large jitter level of 04 = 0.22', which would be unacceptable for a practical system, the trellis code performs within 0.3 dB of its full gain at an error rate of 1 0 P .

Perhaps a more useful statistic for the DMT designer is the degradation in error rate performance that occurs at a particular SNR as the jitter level is increasjed. Results for Scenario A are presented in Fig. 9 for normalized SNR's of 8.5 dB and 10.25 dB corresponding to coded error rates on the order of lop3 and lop6 with perfect synchronization. The former 2-D error rate is close to the level at which the inner code in a concatenated code might be operating, and both error rates are comparable to the error rates examined in Section 111-B for the uncoded system. As can be seen from the figure, the error rate performance is degraded by less than a factor of two over error rates down to lop6 as long as the jitter is kept below 0.16'.

Similar simulations were conducted for Scenario B, and these results are presented in Figs. 0 and 10. For Fig. 10, we chose pilot SNR's to induce jitter levels of 04 = 0.05',

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806 lEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

, ' , ,

0 002 0 0 4 006 008 0 1 0 1 2 0 1 4 I "

phase jitter (degrees)

Fig. 9. Error rate versus phase jitter for Wei code at two normalized SNR's.

ad, = 0.07", and 04 = 0.10". As in Scenario A, we find that if the jitter level does not lead to a severe degradation in the uncoded system's performance, then the gain of the coded system at the same jitter level is maintained or slightly improved. Of course, the tolerable level of jitter is much smaller in this scenario than in Scenario A, as noted in Section 111-B. Fig. 9 demonstrates the rapid degradation in error rate that occurs at a normalized SNR of 10.25 dB when the jitter level is increased beyond 0.06".

Table I compares the uncoded and trellis-coded systems in terms of the maximum tolerable jitter level required to ensure less than a factor of two degradation in performance at 2-D error rates on the order of and lop6. Under the given criteria, the trellis-coded system is slightly more tolerant to timing jitter at a 2-D error rate of than the uncoded system, but less tolerant at lop3. At first glance, the latter observation seems to be in contradiction with our assertion that for reasonable jitter levels the trellis-coded DMT system maintains its gain with respect to an uncoded DMT system at the same jitter level. The apparent anomaly is resolved by realizing that a factor of two degradation in error rate for the uncoded system translates into a larger loss in dB than a factor of two degradation for the coded system at the same error rate. Furthermore, the difference is larger at higher error rates where the uncoded curve is relatively flat. Hence, some care must be exercised in defining a tolerable system jitter level.

V. DISCUSSION At a fixed DPLL bandwidth, the derivations in Section I11

indicate that the jitter level in a DMT system may be improved by transmitting the pilot signal at a higher frequency, since the jitter in each subchannel is proportional to that on the pilot, with the proportionality constant determined by the ratio of the subchannel frequency to the pilot carrier frequency. However, for the ADSL application, higher carriers typically have less SNR than lower carriers because of the sharp increase in channel attenuation with frequency.

Other considerations in the determination of an appropriate location of the pilot carrier include the most probable locations

7 8 9 10 11 12 13 14 15 16 normalized SNR (dB)

Performance of Wei code in Scenario B for three jitter levels. Fig. 10.

of nulls in the channel spectrum due to bridge taps and the effect of crosstalk arising from other services 2141. For instance, Fig. 3 indicates that if the pilot carrier were used for transmission, then 13 b could be supported at 276.0 kHz for Scenario B but only 6 b for Scenario A. Hence, although we have shown that the jitter requirements are more stringent for the former scenario, the SNR at 276.0 kHz is on the order of 21.0 dB better than for the latter scenario. The SNR difference is a result of the much greater influence of HDSL NEXT as compared to T1 NEXT at the pilot frequency.

An appropriate pilot SNR can be determined during system initialization when the channel SNR function is estimated and the bit and power allocations computed. Furthermore, the SNR could be maintained by continuously monitoring the subchannel for degradation that might arise when other services are initiated after the DMT system has begun to transmit data and adjusting the loop parameters or pilot carrier power accordingly. The main difficulty with this adaptive approach is in ensuring stability of the DPLL.

APPENDIX ANALYSIS OF VARIABLE TIMING OFFSET

To evaluate the discrete Fourier transform of (17), we make use of the transform pairs [ ; ; (2~km/N + 2 ~ k T , - ~

x[m] = +2rrkAT,m/L + $k), 0 2 m < L L L m < N

cf

. exp[j(x/N)(L - 1 ) ( k + kNAF,/L - l ) ] sin((nL/N)(k + ICNAT,/L - I ) ) sin((n/N)(k + kNAT, /L - I ) ) + exp[-j($'k: + 27rk~,-1)]

s in( ( rL/N)(k + kNAT,/L + I ) ) sin((7r/N)(k + kNAT,/L + I ) )

. exp[-j(T/N)(L - l ) ( k + IcNAT,/L + l ) ]

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~

L L m < N *

. exp[ j ( r /N) (k - I ) (N + L - I)] s i n ( ( r / N ) ( N - L ) ( k - 1 ) )

s in( (r /N)(k - I ) ) + exp[-j(& + 2rk7,)]

. exp[- j ( r /N)(k + I ) (N + L - I)] s in( (r /N)(N - L ) ( k + I))

sin((T/N)(k + 1 ) )

Thus, by taking the transform of (17) and using (24) and (25), we arrive at the following expression for the received point on the Ith bin:

N / 2 - 1

s in ( (rL/N)(k + kNATi /L - I ) ) N s i n ( ( r / N ) ( b + kNATi /L - 1 ) )

s in ( (rL/N)(k + kNAT%/L + I ) ) N s i n ( ( r / N ) ( k + k N A T , / L ---I + I ) )

. {- exp[-j(& + 2rki7,)l

(26) s in( (r /N)(N - L ) ( k + I ) )

' N s i n ( ( r / N ) ( k +%-}I' Equation (26) shows that a nonlconstant timing offset over

a DMT symbol causes both interchannel interference as well as amplitude and phase distortion on the desired k = 1 term in the summation. With regard to the latter, we note that

exp[j($l+ 2rI7,- L ) ]

exp[.jrlAT; ( L - L)/L]

+ exp[-j(& + 2rI~-~.-~)] . exp[-j(r/N)(L -- 1)(2I + INAT%/L)]

+ exp[-j(& + h l ~ . , ) ]

. exp[ - j ( r /N) (N + L - l ) ( 2 I ) ]

where H L C ~ , ~ , ~ is the coefficient of &, in (26). Since the first two terms dominate and both 7, and AT, are small, (27) may be simplified to

e x p [ j 2 r 1 ~ , - ~ ] exp[jrLAT,(L - 1 ) / ~ ]

N - N "I L - + e x p [ j 2 ~ / ~ % ] -__- N

After further simplification that involves replacing terms of the form eje with (1 + j e ) , we arrive at the final expression

=HleJ i l [I + j ( ~ / p ) [ ( l - a)$, + a&l]] = m J + l [I + 3 ( 1 / P ) 4 e f f , a ] (29)

where p is the index of the pilot bin, 4, = 2 ~ ~ 7 % is the phase error on the pilot at the start of the ith DMT symbol, a = (L+l) / (2N) , and = (1- -a)~ jz+a&~. An identical expression is obtained for the factor multiplying jBl,% in (26).

From (29), we find that in addition to being scaled and rotated by the channel response, the: transmitted point is rotated as a result of the timing jitter by an amount proportional

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808 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

to a weighted average of the timing offsets at the start of the ( i - 1)th and zth symbols. In the case where there is no frequency offset between the transmit and receive clocks, typically L (( AT so the term dominates in (29), thus leading to the same expression that arises when the VCO output is assumed to change instantaneously. For example, practical values include a sampling rate of f s 1 2.208 MHz, an FFT length of N = 512, and a control voltage bandwidth of 10.0 kHz, which corresponds to a time constant of 16 ps or L = 35 samples. With these numbers, the offset in the ith symbol contributes 96.5% to the determination of the amount of rotation, while that in the ( i - 1)th symbol contributes 3.5%. On the other hand, even if L = N , we may still obtain an accurate assessment of the jitter sensitivity by assuming that the sequence of phase errors on the pilot is given by { 4z}. The justification for this statement is that the phase error variance satisfies

since p(1 ) M 1 for small 01 and j3 in (23). Hence, 4 c ~ , z and 4i are Gaussian random variables with the same mean and essentially the same variance, which are the factors that characterize the jitter.

The interchannel interference caused by the k f 1 terms in (26) may be considered as additive Gaussian noise. The significance of this noise is addressed in Section 111-C.

REFERENCES

[ I ] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” lEEE Commun. Mag., vol. 28, no. 5. pp. 5-14, May 1990.

121 W. Y. Cheii and D. L. Waring, “Applicability of ADSL to support video dial tone in the coppcr loop,” IEEE Commun. Mug. , vol 32, no. 5, pp. 102-109. Mav 1994.

[7] J. C. Tu and J. M. Cioffi, “A Loading Algorithm for the Concatenation of Coset Codes with Multichannel Modulation Methods,” in Proc. 1990 Global Telecommun. Con$, San Diego, CA, Dec. 1990, pp. 1183-1187.

[8] T. N. Zogakis, J. T. Aslanis Jr., and J. M. Cioffi, “Analysis of a concatenated coding scheme for a discrete multitone modulation system,“ in Proc. 1994 IEEE Military Commun. Conf, Ft. Monmouth, NJ, Oct. 1994, pp. 433437.

[9] A. Ruiz, “Frequency-designed modulation for channels with intersymbol interference,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Jan. 1989.

[ IO] J. S. Chow, “Finite-length equalization for multi-carrier transmission systems.” Ph.D. dissertation, Stanford Univ., Stanford, CA, June 1992.

[ l l ] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency- division multiplexing using the discrete fourier transform,” ZEEE Trans. Commun.. vol. COM-19, no. 5, pp. 628-634, Oct. 1971.

[I21 W. C. Lindsey and C. M. Chie, “A survey of digital phase-locked loops,” Proc. ZEEE, vol. PROC-69, no. 4, pp. 410-431, Apr. 1981.

[ 131 American National Standard for Telecommunications, Network and Customer Installation Interfaces: Asymmetric Digital Subscriber Line (ADSL) Metallic Interface, ANSI Standard T1.413, 1995.

[I41 J. J. Werner, “The HDSL environment,” IEEE J . Select. Areas Commun., vol. 9. no. 6, pp. 785-800, Aug. 1991.

[I51 P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels.” ZEEE Trans. Commun., vol. 43, no. 3, pp. 713-775, Mar. 1995.

[16] T. N. Zogakis. J. T. Aslanis Jr., and J. M. Cioffi, “A Coded and Shaped Discrete Multitone System,” IEEE Trans. Cornmun., vol. 43, no. 12, pp. 2941-2949, Dec. 1995.

[ 171 L. F. Wei, “Trellis-Coded Modulation with Multidimensional Constel- lations.“ IEEE Trans. Inform. Theory, vol. IT-33, no. 4, pp. 483-501, July 1987.

[I81 G. D. Fomey Jr., “Coset codes I: Introduction and geometrical classi- fication.” IEEE Truns. lnfijrm. Theory, vol. 34, no. 5 , pp. 1123-1151, Sept. 1988.

T. Nicholas Zogakis (S’88-M’89) received the B S degree in electrical engineering from the University of Florida, Gainesville, in 1989, and the M S and Ph D degrees in electrical engineering from Stan- ford University, CA, in 1990 and 1994, respectively.

During the 1990-1991 academic year, he was with Harris Corporation, Palm Bay, FL, where he worked i n the area of signal identification and modulation recognition Since September 1994, he has been with Amati Communications Corporation, Mountain View, CA, where he works on the anal-

, , 131 I. Kalct, “The Multitone Channel,” IEEE Trans. Commun., vol. 37. no.

2, pp. 119-124, Fcb. 1989. 141 J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitone transceiver

system for HDSL applications,” ZEEE J. Select. Areas Commun., vol. 9, no. 6, pp. 895-908, Aug. 1991.

[ 5 ] J. M. Cioffi, “A multicarrier primer,” in ANSI T1E1.4 Committee Contribution, no. 91-157, Boca Raton, FL, Nov. 1991.

161 A. Ruiz and J . M. Cioffi, “A frequency-domain approach to combined spectral shaping and coding,” in Proc. IY87 h t . Con$ Commiin.. Seattle. WA, June 1987, pp. 1711-1715.

ysis, design, and implementation of DMT modems. His research interests include digital communications, modulation, and coding theory.

Dr. Zogakis is a member of Eta Kappa Nu, Phi Kappa Phi, and the IEEE Communications and Information Theory Societies.

John M. Cioffi (S’77-M’78-SM’90-F’96), for a photograph and biography, see p. 64 of the January issue of this TRANSACTIONS.

The Effect of Timing Jitter on the Performance of a Discrete Multitone System

T. Nicholas Zogakis, Member, IEEE, and John M. Cioffi, Fellow, IEEE

Abstract- The transmission of high-speed data over severely band-limited channels may be accomplished through the use of discrete multitone (DMT) modulation, a modulation technique that has been proposed for a number of new applications. While the performance of a DMT system has been analyzed by a number of authors, these analyses ignore the effect of timing jitter on system performance. Timing jitter becomes an increasingly important concern as higher data rates are supported and larger constellations are allowed on the DMT subchannels. Hence, in this paper, we assume that synchronization is maintained by using a digital phase-locked loop to track a pilot carrier. Given this model, we derive error rate expressions for an uncoded DMT system operating in the presence of timing jitter, and we derive an expression for the interchannel distortion that results from a varying timing offset across the DMT symbol. In addition, we investigate the performance of trellis-coded DMT modulation in the presence of timing jitter. Practical examples from the asymmetric digital subscriber line (ADSL) service are used to illustrate various results.

I. INTRODUCTION

ISCRETE multitone (DMT) modulation is a technique in D which a transmission channel is partitioned into a number of independent, parallel subchannels, each of which may be considered as supporting a lower-speed quadrature amplitude modulated (QAM) signal [l]. Performance is maximized by allocating more bits to subchannels with high signal-to-noise ratios (SNR’s) and fewer or no bits to subchannels with low SNR’s. An example of an application for which DMT modulation is well suited is the asymmetric digital subscriber line (ADSL), a service proposed for providing a high-speed downstream channel, ranging from 1.544 Mb/s to 6.4+ Mb/s, from the central office to the customer, along with a lower- speed upstream channel over existing copper twisted pair [2].

Several authors have evaluated the performance of a DMT system for a variety of applications, focusing on maximizing data rate or maximizing margin under a constraint on the available transmit power [l], [3]-[5]. However, in these analy- ses, perfect synchronization is assumed, whereas in an actual system, the practical timing recovery mechanism will result in some degree of timing jitter. The importance of this form

Paper approved by P. H. Wittke, the Editor for Communication Theory of the IEEE Communications Society. Manuscript received August 15, 1994; revised July 15, 1995. This work was supported in part by a National Science Foundation (NSF) Fellowship and in part by Contracts CASIS 2DPD335 and NSF 2DPL133.

T. N. Zogakis was with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA. He is now with Amati Communications Corporation, Mountain View, CA 94040 USA.

J. M. Cioffi is with the Information Systems Laboratory, Stanford Univer- sity, Stanford, CA 94305 USA.

Publisher Item Identifier S 0090-6778(96)05506-7.

of impairment in the determination of error rate performance increases as the available bandwidth, which is determined by the channel SNR function, decreases and as the data rate increases, since both trends result in larger constellations being used on some of the subchannels. When large spectral efficiency is required and constellations supporting on the order of 10 b or more are allowed, then careful attention must be given to the synchronization scheme.

Similar to more traditional single-carrier modulation tech- niques, the performance of a DMT system may be enhanced by the application of coding. For instance, [6] and [7] present methods for applying trellis coding to DMT modulation, while [8] investigates the performance of a concatenated coding scheme consisting of an inner trellis code and outer Reed-Solomon code when applied to a DMT system. Each of these coding schemes requires const ellation expansion over a subset of the carriers and, thus, potentially increases the susceptibility of the system to timing jitter. Furthermore, trellis decoders are based on the assumption of uncorrelated Gaussian noise, whereas, timing jitter introduces correlated noise into the system. For a DMT system employing trellis coding across the tones as described in [7], the correlation between the phase errors caused by timing jitter on consecutive complex symbols at the input to the trellis decoder is quite strong. Hence, it is not clear whether or not the timing jitter requirements are significantly tighter for a trellis-coded DMT system compared to an uncoded DMT system.

In this paper, we investigate the performance of both an uncoded and a trellis-coded DMT system in the presence of timing jitter. For simplicity, we assume that synchronization is maintained by designating one of the carriers as a pilot signal and using a digital phase-locked1 loop in the receiver to track the pilot carrier. This assumption leads to a tractable analysis and corresponds to the technique implemented in DMT modems for ADSL. Throughout the analysis, exam- ples from the ADSL service are used to illustrate various points.

In Section 11, we establish the DMT timing jitter model that serves as a starting point for the analysis. In Section 111, we analyze the performance of an uncoded DMT system in the presence of timing jitter, and we compare the analytical results to simulation results for two ADSL scenarios. In Section IV, we first address the application of trellis coding to DMT modulation and then investigate the performance of a trellis- coded DMT system in the presence of timing jitter. Finally, in Section V, we discuss some of the implications of our results for the ADSI, service.

0090-6778/96$05.00 0 1996 IEEE

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XOO IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

Bit Allocation

Parallcl

Fig. 2. Baseline DMT receiver.

11. DMT TIMING JITTER MODEL

Figs. 1 and 2 present simplified block diagrams of the DMT system that we consider in this paper. At the input to the transmitter, the bit stream is partitioned into blocks of size b = RT bits, where R is the uncoded bit rate, T is the DMT symbol period, and b is the number of bits contained in one DMT symbol. The bits collected during the ith symbol interval are allocated among f l subchannels or tones in a manner determined during system initialization, with bk bits assigned to tone k and C b k = b. On subchannel k , the bk

bits are mapped to a constellation point Xk,i = Ah,? + j B k , z in a constellation of size 2bk with unity distance between constellation points. Next, the constellation point is scaled by a real multiplier, ,9k, and the collection of constellation points {xl;,z = g k X k , i , k = l , . . . , N } serves as the input to an inverse fast Fourier transform (IFFT) block. The constants { g k } are chosen so that E{ lfi;k,z12} = Pk, the power allocated to the kth tone. The time-domain signal that is transmitted over the channel is obtained by performing a length N = 2N IFFT on the complex symbols { x k , i , k = 0 , l : . . IV - I}, where T0.i = 0 and { x k , i = xkT-k,i, k = F + 1. F + 2. . . ! N -

The kth subchannel is associated with the frequency f k = k A f , where A f = l / T . Hence, the DMT symbol transmitted during the ith symbol period is given by 191, 1111

I}.'

' I n practice, a cyclic prefix [9], [lo] would be added to the data block before transmission to eliminate interblock interfercncc and to make the linear convolution with the channel look like a circular convolution. To simplify our notation, we ignorc this complication since it does not change the main results of our jitter analysis.

- where A k . z = g k A k , , , B k , z = g k B k , , , and g z ( t ) is a rectangu- lar window function defined as

t - ZT - T / 2 + T / ( 2 N ) T .9z(t)

In forming the limits of the summation in (l), we have assumed that the Nyquist bin is not used. The transmitted signal s ( t ) , formed by sending a sequence of DMT symbols, is

30 1v/2-1 z s ( t ) = 1 1 [Akzcos(27rkAft)

The signal is sent over the channel where it is convolved with the channel impulse response h ( t ) , yielding a received signal of

To focus solely on the effect of timing jitter in this initial discussion, we ignore the contribution of additive noise; the noise will be included after the final expressions are obtained. Denoting the Fourier transform of the channel impulse re- sponse by F{h( t ) } = H ( f ) e 3 ' b ( f ) , we may simplify (4) by making use of the relationship [ I l l

h(t) * [go( t ) cos(27rkAft)l

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM 801

where P ( f ) = (-1/27r)(ali/(f)/af) is the group delay of the channel. A similar expression is obtained for the sine function. For our purposes, the possibility of interblock interference may be eliminated by assuming the group delay to be equal to a constant, which we conveniently take to be zero. In a well-designed DMT system, this approximation will be valid since interblock interference would otherwise be detrimental to system performance. Hence, under these assumptions, the final expression for the received analog DMT signal is given by

00 N/2-1

- H k : B k , , s i n ( 2 ~ k A f t + $k)].qgz(t) (6)

where Hk = H ( k A f ) and $ k = 4(kA f ) . At the input to the receiver in Fig. 2, the first step of

demodulation is to sample the signal at a nominal rate of f S = NAf. We denote the sampling instances by ( iN + m)Ts + r,,,, m E { O , l , . . . , N - l}, where rm,, is a timing offset that may vary from sample to sample and T, = I / f s .

Hence, the received sequence of samples is given by

ith symbol period is given by

At this point, assumptions regarding the dependence of rm,% on the block index i and the intrablock index m must be made to allow for further analysis. Since the DPLL is updated at the DMT symbol rate, the simplest approach would be to assume that r,,% is constant over each block and thus independent of r r ~ A more complicated analysis that more closely approximates reality models the change in timing error between consecutive updates as a ramp L samples long followed by a constant for N - L samples, where L depends upon the control voltage bandwidth of the voltage controlled oscillator (VCO) that is part of the phase-locked loop or L = N for a frequency offset. Both cases are considered in the next section.

111. UNCODED DMT JITTER PERFORMANCE

The statistics of the timing error depend upon the timing recovery mechanism used in the DMT system. For simplicity, we assume that synchronization is maintained by using a second-order digital phase-locked loop (DPLL) to track a pilot carrier located at frequency f, = p A f . The pilot signal is generated by sending a fixed constellation point on the pth tone, and the DPLL is updated at the DMT symbol rate. With the DPLL model, a good approximation is that the phase error on the pilot is Gaussian distributed [12], and we define the phase jitter, 06, as the standard deviation of this Gaussian process.

Next, the received sequence given in (7) is partitioned into blocks of N samples, each of which is transformed by the FFT to obtain an estimate of the transmitted constellation points. To ensure there is no contribution from the past or previous blocks into a sample obtained during the current symbol and, thus, to maintain a reasonable error rate, we must have max{lr,,,l/T} < 1/(2N). Another way of stating this is that the peak timing offset should be less than one-half the sampling period. For example, the DMT parameters for an ADSL system that loop times to the central office include a sampling rate of 2.208 MHz, a pilot carrier of 276 ICHz, and an FFT size of 512 [13]. With these values, the criteria becomes max{ Ir,,% I } < 226 ns, corresponding to a peak phase error of 22.5’ on the pilot. Hence, we can safely assume that the group of N samples, {rTn,& , m = 0, . . . , N - l}, obtained during the

A. Fixed Offset Over DMT Symbol

For the case in which the timing error is assumed to be fixed over the DMT symbol, we replace r,,,, with rl. and compute the discrete Fourier transform (DlT) of (8) to obtain

as an expression for the detected complex point in the Ith bin during the ith symbol period. The complex multiplicative factor ,91Hle3q4 in (99 is a constant that depends upon the channel characteristic and may be compensated by a one-tap frequency domain equalizer (FEQ) in the receiver. Hence, the final expression for the received ]point is

where the noise term (n1,l + j n ~ , l ) included in (10) is a com- plex Gaussian random variable with E{TL?,~} = E{n i , , } = o:. The noise variance after the FEQ is independent of fre- quency since the DMT system is designed for equal probability of error across all subchannels and the received constellation on each tone has a normalized minimum distance of 1.0 after FEQ scaling. In obtaining (lo), we have dropped the DMT symbol index i to signify that the statistics of the variables in (10) are time-invariant.

To compute the two-dimensional (2-D) error rate perfor- mance of the DMT system, we make use of the small angle approximation e34 M (1 + J $ ) to obtain

Si %(Ai - Bi2irlAfr+n1.i) + ~ ( B I +Al2ir lAfr+n~,i ) . (11)

Hence, the probability of a correct decision on the Zth tone given the transmitted constellation point Aq,l + jB,,l and the

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802 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996

phase error 81 = 27rlA f r is

P C I A ~ . ~ I

(12)

where q E {0,1, . . . , 2b1 - I} is a constellation label and Q ( . ) represents the Gaussian probability of error function. Since 81 is related to the phase error on the pilot signal by a constant scaling factor, the 2-D error rate on the Zth tone may be written in terms of the phase jitter 04 as

where

The overall 2-D error rate, obtained by averaging index I , is given by

where 1.i denotes the set of indices corresponding to subchan- nels used for transmission and u = IMl.

The uncoded DMT system's error rate performance as predicted by (15) will be determined by the poorest performing subchannels. Moreover, (13) and (14) indicate that for a particular level of timing jitter, two main factors determine the error rate performance on the Zth tone. The first factor is the frequency of the bin, with higher frequencies experiencing greater levels of jitter than lower frequencies. The second is the size of the constellation supported by the lth bin, where larger constellations are more susceptible to jitter. Fortunately for applications such as ADSL, the higher frequencies typi- cally support smaller constellations than the lower frequencies because of the increase in channel attenuation with frequency.

B. DMT Examples

We now investigate the implications of (15) for the two bit distributions presented in Fig. 3. In both cases, 1616 b are contained in each DMT symbol, and a pilot carrier is located at 276.0 kHz, hence, the null in the bit distributions at this frequency. Scenario A corresponds to transmission over a 9 kft (2.7 km), 26 AWG loop in the presence of near-end crosstalk (NEXT) from ten digital subscriber line (DSL) disturbers and 24 high bit-rate DSL (HDSL) disturbers, and NEXT and far- end crosstalk (FEXT) from ten ADSL disturbers [14]. Scenario B also corresponds to transmission over a 9 kft (2.7 km), 26 AWG loop, but in the presence of NEXT from one T1 disturber in an adjacent wire bundle.

To obtain both bit distributions, we used the practical bit and power allocation algorithm provided in [15]. This

0" I I I 200 400 600 800 1000 1200

frequency (kHz)

Fig. 3. Uncoded bit distributions for two ADSL scenarios

algorithm attempts to find for a fixed data rate the integer bit distribution that maximizes system margin under a total power constraint. The algorithm starts with a flat power distribution and iteratively solves the set of equations

where SNRk is the SNR on the kth subchannel, I' is a constant that depends upon the target error rate, ym is the margin, and b,,, is the maximum number of bits allowed on a subchannel. At the completion of the iterative part of the algorithm, the power on each subchannel is adjusted slightly to ensure equal error rate performance. See [15] for further details. We ran the algorithm on Scenarios A and B with b,,, = 14, I? = 9.8 dB, and a power constraint of 20.0 dBm.

Figs. 4 and 5 present plots of the uncoded error rate curves obtained for the bit distributions in Fig. 3 at various levels of jitter2; square and cross constellations were used on the subchannels in obtaining these results. The continuous curves in the graphs have been obtained by evaluating (15), while the asterisks represent Monte Carlo simulation points obtained by simulating DMT modulation with timing recovery. A solid error rate curve is included in each plot to signify the performance of a system with perfect synchronization.

The correspondence between the theoretical error rate curves and the simulation points in Figs. 4 and 5 verifies the accuracy of the analysis for a wide range of jitter levels. In addition, these plots indicate the importance of choosing a narrow enough DPLL bandwidth or large enough pilot SNR to ensure acceptable error rate performance for a given bit distribution. For instance, although Scenario A results in bits being placed at high frequencies where the jitter is worse,

'Error rate curves are plotted versus the normalized SNR, which is proportional to 1/uF.

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM

m . e

803

- _ _

, / _ _ _

L

10'

Uncoded DMT system's error rate performance for Scenario A at

IO'

Uncoded DMT system's error rate performance for Scenario B at

Scenario B is less tolerable to timing jitter because of the very large constellations used on some of the tones.

Further insight into the degradation in performance caused by timing jitter is presented in Fig. 6 where we plot the 2- D error rate versus the jitter level for normalized SNR's of 11.0 dB and 14.0 dB. This figure clearly illustrates the greater intolerance to timing jitter in Scenario B as compared to Scenario A. In addition, we observe that timing jitter is more critical at lower error rates since the importance of this form of impairment relative to additive noise is greater than at higher error rates. By using the results in Fig. 6, we can determine the maximum tolerable phase jitter for both DMT scenarios and both normalized SNR's to ensure less than a factor of two degradation in the overall error rate. These critical levels are listed in Table I along with the jitter levels required to ensure less than a factor of two degradation on the worst tone in the ~ y s t e m . ~ As is evident from the table, the jitter requirements are quite stringent for both scenarios.

l op3) is beyond the range of the plot in Fig 6 3The jitter level for Scenario A at a normalized SNR of 11 0 dB (P .D %

1oz7-i

Fig. 6. normalized SNR's.

Error rate versus phase jitter for tmo DMT scenarios and two

TABLE 1 JITTER LEVELS REQUIRED TO CAlJSE A FACTOR

OF TWO DEGRADATION IN ERROR RATE

nncou'ed trellis coded

scenario u + , ~ , ~ ~ (worst t o n e ) a+.ma,. (Fz!gure G) uO,mar (Fiyure 9)

A , 10-3

A . - I 0.090" 1 0.14" I 0.16"

C. Variable Offset Over DMT Symbol

The assumption of a constant timing offset over each DMT symbol is equivalent to assuming that the output phase of the VCO changes instantaneously when the control voltage is changed. In practice, the VCO will have a control voltage bandwidth that is determined by a single-pole Butterworth filter. Hence, we model the timing ofFset as

where L is the number of samples, coiresponding to one time constant of the filter's impulse response. By allowing L = N , we also have a model for the case in which the transmit and receive clocks are offset in frequency. Substituting (16) into (8), we find that the block of received samples representing the zth DMT symbol is given by (17), see equation at the bottom of the next page, where 7% = A f r,, 7,-1 = A f r,-l,

In the Appendix, we evaluate the DFT of (17) to derive an and ATt = 7, -

expression for R L , ~ , the received poinc in the lth bin

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804 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996

where Ck,l,i and D k , l , i are implicitly defined in (26). The desired term in (18) is approximately

Hl @l,i G , l , i + jBl,LQ.l,L)

. ! 7 1 & J ~ L (1 + j(l/P)4€!ffi)(Al,, + j&.,) (19)

where E{&,,,} % 0;. Hence, (19) is essentially the same as the expression that we would obtain under the assumption of a fixed phase offset across the DMT symbol.

By the central limit theorem, the interchannel interference introduced by the k # 1 terms in (26) may be considered as additive Gaussian noise. Thus, by defining the power allocated to the kth subchannel as Pk = E{ I & L + j B k , L / 2 } , we arrive at

k f l

as an expression for the power of the interference introduced into the lth subchannel. To assess the significance of the in- terference, we compare the signal-to-interchannel interference ratio (SIR), defined by

k f l

to SNRl = PlH,2/2a;,,, for a system operating at a 2-D error rate of without coding. We use 2F:,, to represent the noise variance on the lth tone before FEQ scaling; this variance is not independent of frequency.

The expectations in (2 1) are quite computationally intensive to compute, so we instead consider some worst case values for

k f l

as upper bounds to SIRl . Through straightforward, though tedious, mathematical manipulations, it can be shown that ( / C k , l , i I 2 + lDk ,1 ,~ ,1~) depends upon the timing error only in terms of the difference between the timing offsets r, and ~ i - 1

in the ith and ( z - 1)th symbols. Hence, the expectations omitted in obtaining (22) are over the probability distribution function (PDF) of A?;, or equivalently over the PDF of A& = 4; - q5-1 = 27rpA7i. As noted earlier, & is normally distributed with variance n;. Furthermore, since Ad, is a filtered version of 4i, it too is normally distributed with variance a i a = 2 0 $ ( 1 - p ( 1 ) ) , where p(1 ) = E{q5;4z-l}/n$. Thus, as an upper bound to SIRl, we consider the evaluation of (22) for phase error differences in the range IAq& 5 3aad.

To illustrate the types of signal-to-interchannel interference levels expected, we consider an example in which 04 = 0.50" and clQ = 0.08'. The latter value has been obtained based on a loop filter

with 01 = 1.98 x l o p 2 and /3 = 2.00 x 10V4. This is in fact the filter that was used in obtaining the results of Fig. 4. The range of values for which A& contributes significantly in the evaluation of (21) is

Fig. 7 presents plots of SNRl and SIRl,; versus 1 for Scenario A, which was described in Section 111-B. In the graph, we have included six plots of S1Rl.i corresponding to the six combinations of A& = 0.1" or 0.25' and L = 35, 70, or 512. It is clear from Fig. 7 that the signal-to-interference level is well above the signal-to-noise level for all 1. Similar results were obtained for Scenario B but are not included here. Furthermore, we showed in Fig. 4 that the error rate obtained for a jitter level of cr4 = 0.50' is very poor, thus confirming the dominance of the phase rotation on each bin rather than the interchannel interference in determining system performance. In order to achieve a more acceptable error rate, either a DPLL with a narrower bandwidth must be used or the pilot SNR increased, both of which tend to decrease the significance of the interference. Finally, we note that for a coded system, a given error rate will be achieved with a lower SNR, so interchannel interference becomes even less important relative to additive Gaussian noise. Hence, these observations fully justify our replacement of rm:i with TL in (8).

< 0.25'.

IV. TRELLIS-CODED DMT JITTER PERFORMANCE

A. Application of Trellis Code

Trellis coding may be applied to DMT modulation by using a single trellis encoder to operate across the tones in the system [6], [7]. In the receiver, the complex points obtained at the output of the FEQ's are decoded by using a single Viterbi decoder across the tones. Equation (9) shows that a phase error on the pilot carrier translates into a phase rotation on each of the tones, with the amount of rotation determined by the ratio of the center frequency of the tone to the pilot frequency. Hence, the errors introduced by timing jitter at the input of a trellis decoder operating across the tones are strongly correlated.

Many of the good trellis codes constructed for the additive white Gaussian noise (AWGN) channel are multidimensional

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ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM 805

L = 35 . . L = 7 0

7.. \; . ' -.

subchannel 1 0:

Fig. 7. Comparison of SNR and SIR for three values of L and two values of A i l , [email protected] = 0.1' (upper three curves) and A& = 0.25' (second set of three curves), in Scenario A.

codes that require a fractional number of bits to be supported in each 2-D coordinate. In the case of DMT modulation, this complicates the trellis encoder and decoder since many different multidimensional constellations have to be supported. However, a method for accommodating fractional numbers of bits while at the same time maintaining the simplicity of an integer bit distribution is presented in [16]. Basically, if u tones are used to support btot = b bits per DMT symbol in the uncoded case and F is the normalized redundancy of the trellis code in b/2-D symbol, then an integer bit distribution may be computed for the trellis-coded case with btot = b + uF.

The significance from the standpoint of timing jitter of the proposed method for accommodating multidimensional trellis codes is that the constellation expansion is greater than 2' on some of the carriers. For instance, in the case of a trellis code with a normalized redundancy of F = 0.5, the constellation size on about one-half of the tones is doubled, while for the other half, it remains the same. This is different from expanding each constellation by a factor of 2 O . j .

In the next section, we investigate the performance of Wei's four-dimensional (4-D), 16-state trellis code [ 171 when implemented in a DMT system subjected to timing jitter. This code has been adopted for the ADSL standard and has a normalized redundancy of F = 0.5 and a fundamental coding gain of 4.5 dB [18]. We use the integer-based algorithm for accommodating the trellis code redundancy, and we investigate the same two scenarios as in Section 111-B, but with a constraint of b,,, = 15 enforced for the bit distribution computed for the trellis-coded system.

B. Pegormance of 4-0, 16-State Wei Code

Fig. 8 presents simulation results for Scenario A at three different jitter levels: 04 = 0.16", 04 = 0.22", and ~4 = 0.28". To obtain these jitter levels, we used a fixed DPLL and changed the SNR on the pilot carrier. However, we found that in all our trellis code simulations, the error rate depended upon the jitter level and not the DPLL bandwidth, so the results are general. This is not too surprising since the correlation

' " 7 8 9 10 11 12 13 14 15 16 normalized SNR (dB)

Fig. 8. Performance of Wei code in Scenario ,4 for three jitter levels.

among the 2-0 symbols associated with a trellis error event arises primarily from the correlaticln among neighboring tones rather than between DMT symbols. The solid lines in Fig. 8 illustrate the error rate performance for an uncoded system (rightmost solid line) and a trellis-coded system (leftmost solid line), assuming perfect synchronization. The three curves on the far right of the plot present the error rate performance for the uncoded system subjected to the three different jitter levels and were obtained by evaluating (15). Curves marked by asterisks correspond to results obtained from simulations of the trellis-coded DMT system, where the asterisks denote the actual simulation points.

The results in Fig. 8 indicate thoit for jitter levels satisfying 04 5 0.16' and over 2-D error rates of lop7 and higher, the trellis code provides approximately the same gain relative to an uncoded system at the same jitter level as it does under the conditions of perfect synchronization. In fact, a small improvement in gain is observed in the presence of jitter at a jitter level of 04 = 0.16" and a 2-D error rate of lop6. These results are quite surprising since the errors are correlated at the input to the trellis decoder. Even at the large jitter level of 04 = 0.22', which would be unacceptable for a practical system, the trellis code performs within 0.3 dB of its full gain at an error rate of 1 0 P .

Perhaps a more useful statistic for the DMT designer is the degradation in error rate performance that occurs at a particular SNR as the jitter level is increasjed. Results for Scenario A are presented in Fig. 9 for normalized SNR's of 8.5 dB and 10.25 dB corresponding to coded error rates on the order of lop3 and lop6 with perfect synchronization. The former 2-D error rate is close to the level at which the inner code in a concatenated code might be operating, and both error rates are comparable to the error rates examined in Section 111-B for the uncoded system. As can be seen from the figure, the error rate performance is degraded by less than a factor of two over error rates down to lop6 as long as the jitter is kept below 0.16'.

Similar simulations were conducted for Scenario B, and these results are presented in Figs. 0 and 10. For Fig. 10, we chose pilot SNR's to induce jitter levels of 04 = 0.05',

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806 lEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

, ' , ,

0 002 0 0 4 006 008 0 1 0 1 2 0 1 4 I "

phase jitter (degrees)

Fig. 9. Error rate versus phase jitter for Wei code at two normalized SNR's.

ad, = 0.07", and 04 = 0.10". As in Scenario A, we find that if the jitter level does not lead to a severe degradation in the uncoded system's performance, then the gain of the coded system at the same jitter level is maintained or slightly improved. Of course, the tolerable level of jitter is much smaller in this scenario than in Scenario A, as noted in Section 111-B. Fig. 9 demonstrates the rapid degradation in error rate that occurs at a normalized SNR of 10.25 dB when the jitter level is increased beyond 0.06".

Table I compares the uncoded and trellis-coded systems in terms of the maximum tolerable jitter level required to ensure less than a factor of two degradation in performance at 2-D error rates on the order of and lop6. Under the given criteria, the trellis-coded system is slightly more tolerant to timing jitter at a 2-D error rate of than the uncoded system, but less tolerant at lop3. At first glance, the latter observation seems to be in contradiction with our assertion that for reasonable jitter levels the trellis-coded DMT system maintains its gain with respect to an uncoded DMT system at the same jitter level. The apparent anomaly is resolved by realizing that a factor of two degradation in error rate for the uncoded system translates into a larger loss in dB than a factor of two degradation for the coded system at the same error rate. Furthermore, the difference is larger at higher error rates where the uncoded curve is relatively flat. Hence, some care must be exercised in defining a tolerable system jitter level.

V. DISCUSSION At a fixed DPLL bandwidth, the derivations in Section I11

indicate that the jitter level in a DMT system may be improved by transmitting the pilot signal at a higher frequency, since the jitter in each subchannel is proportional to that on the pilot, with the proportionality constant determined by the ratio of the subchannel frequency to the pilot carrier frequency. However, for the ADSL application, higher carriers typically have less SNR than lower carriers because of the sharp increase in channel attenuation with frequency.

Other considerations in the determination of an appropriate location of the pilot carrier include the most probable locations

7 8 9 10 11 12 13 14 15 16 normalized SNR (dB)

Performance of Wei code in Scenario B for three jitter levels. Fig. 10.

of nulls in the channel spectrum due to bridge taps and the effect of crosstalk arising from other services 2141. For instance, Fig. 3 indicates that if the pilot carrier were used for transmission, then 13 b could be supported at 276.0 kHz for Scenario B but only 6 b for Scenario A. Hence, although we have shown that the jitter requirements are more stringent for the former scenario, the SNR at 276.0 kHz is on the order of 21.0 dB better than for the latter scenario. The SNR difference is a result of the much greater influence of HDSL NEXT as compared to T1 NEXT at the pilot frequency.

An appropriate pilot SNR can be determined during system initialization when the channel SNR function is estimated and the bit and power allocations computed. Furthermore, the SNR could be maintained by continuously monitoring the subchannel for degradation that might arise when other services are initiated after the DMT system has begun to transmit data and adjusting the loop parameters or pilot carrier power accordingly. The main difficulty with this adaptive approach is in ensuring stability of the DPLL.

APPENDIX ANALYSIS OF VARIABLE TIMING OFFSET

To evaluate the discrete Fourier transform of (17), we make use of the transform pairs [ ; ; (2~km/N + 2 ~ k T , - ~

x[m] = +2rrkAT,m/L + $k), 0 2 m < L L L m < N

cf

. exp[j(x/N)(L - 1 ) ( k + kNAF,/L - l ) ] sin((nL/N)(k + ICNAT,/L - I ) ) sin((n/N)(k + kNAT, /L - I ) ) + exp[-j($'k: + 27rk~,-1)]

s in( ( rL/N)(k + kNAT,/L + I ) ) sin((7r/N)(k + kNAT,/L + I ) )

. exp[-j(T/N)(L - l ) ( k + IcNAT,/L + l ) ]

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~

L L m < N *

. exp[ j ( r /N) (k - I ) (N + L - I)] s i n ( ( r / N ) ( N - L ) ( k - 1 ) )

s in( (r /N)(k - I ) ) + exp[-j(& + 2rk7,)]

. exp[- j ( r /N)(k + I ) (N + L - I)] s in( (r /N)(N - L ) ( k + I))

sin((T/N)(k + 1 ) )

Thus, by taking the transform of (17) and using (24) and (25), we arrive at the following expression for the received point on the Ith bin:

N / 2 - 1

s in ( (rL/N)(k + kNATi /L - I ) ) N s i n ( ( r / N ) ( b + kNATi /L - 1 ) )

s in ( (rL/N)(k + kNAT%/L + I ) ) N s i n ( ( r / N ) ( k + k N A T , / L ---I + I ) )

. {- exp[-j(& + 2rki7,)l

(26) s in( (r /N)(N - L ) ( k + I ) )

' N s i n ( ( r / N ) ( k +%-}I' Equation (26) shows that a nonlconstant timing offset over

a DMT symbol causes both interchannel interference as well as amplitude and phase distortion on the desired k = 1 term in the summation. With regard to the latter, we note that

exp[j($l+ 2rI7,- L ) ]

exp[.jrlAT; ( L - L)/L]

+ exp[-j(& + 2rI~-~.-~)] . exp[-j(r/N)(L -- 1)(2I + INAT%/L)]

+ exp[-j(& + h l ~ . , ) ]

. exp[ - j ( r /N) (N + L - l ) ( 2 I ) ]

where H L C ~ , ~ , ~ is the coefficient of &, in (26). Since the first two terms dominate and both 7, and AT, are small, (27) may be simplified to

e x p [ j 2 r 1 ~ , - ~ ] exp[jrLAT,(L - 1 ) / ~ ]

N - N "I L - + e x p [ j 2 ~ / ~ % ] -__- N

After further simplification that involves replacing terms of the form eje with (1 + j e ) , we arrive at the final expression

=HleJ i l [I + j ( ~ / p ) [ ( l - a)$, + a&l]] = m J + l [I + 3 ( 1 / P ) 4 e f f , a ] (29)

where p is the index of the pilot bin, 4, = 2 ~ ~ 7 % is the phase error on the pilot at the start of the ith DMT symbol, a = (L+l) / (2N) , and = (1- -a)~ jz+a&~. An identical expression is obtained for the factor multiplying jBl,% in (26).

From (29), we find that in addition to being scaled and rotated by the channel response, the: transmitted point is rotated as a result of the timing jitter by an amount proportional

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808 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996

to a weighted average of the timing offsets at the start of the ( i - 1)th and zth symbols. In the case where there is no frequency offset between the transmit and receive clocks, typically L (( AT so the term dominates in (29), thus leading to the same expression that arises when the VCO output is assumed to change instantaneously. For example, practical values include a sampling rate of f s 1 2.208 MHz, an FFT length of N = 512, and a control voltage bandwidth of 10.0 kHz, which corresponds to a time constant of 16 ps or L = 35 samples. With these numbers, the offset in the ith symbol contributes 96.5% to the determination of the amount of rotation, while that in the ( i - 1)th symbol contributes 3.5%. On the other hand, even if L = N , we may still obtain an accurate assessment of the jitter sensitivity by assuming that the sequence of phase errors on the pilot is given by { 4z}. The justification for this statement is that the phase error variance satisfies

since p(1 ) M 1 for small 01 and j3 in (23). Hence, 4 c ~ , z and 4i are Gaussian random variables with the same mean and essentially the same variance, which are the factors that characterize the jitter.

The interchannel interference caused by the k f 1 terms in (26) may be considered as additive Gaussian noise. The significance of this noise is addressed in Section 111-C.

REFERENCES

[ I ] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” lEEE Commun. Mag., vol. 28, no. 5. pp. 5-14, May 1990.

121 W. Y. Cheii and D. L. Waring, “Applicability of ADSL to support video dial tone in the coppcr loop,” IEEE Commun. Mug. , vol 32, no. 5, pp. 102-109. Mav 1994.

[7] J. C. Tu and J. M. Cioffi, “A Loading Algorithm for the Concatenation of Coset Codes with Multichannel Modulation Methods,” in Proc. 1990 Global Telecommun. Con$, San Diego, CA, Dec. 1990, pp. 1183-1187.

[8] T. N. Zogakis, J. T. Aslanis Jr., and J. M. Cioffi, “Analysis of a concatenated coding scheme for a discrete multitone modulation system,“ in Proc. 1994 IEEE Military Commun. Conf, Ft. Monmouth, NJ, Oct. 1994, pp. 433437.

[9] A. Ruiz, “Frequency-designed modulation for channels with intersymbol interference,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Jan. 1989.

[ IO] J. S. Chow, “Finite-length equalization for multi-carrier transmission systems.” Ph.D. dissertation, Stanford Univ., Stanford, CA, June 1992.

[ l l ] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency- division multiplexing using the discrete fourier transform,” ZEEE Trans. Commun.. vol. COM-19, no. 5, pp. 628-634, Oct. 1971.

[I21 W. C. Lindsey and C. M. Chie, “A survey of digital phase-locked loops,” Proc. ZEEE, vol. PROC-69, no. 4, pp. 410-431, Apr. 1981.

[ 131 American National Standard for Telecommunications, Network and Customer Installation Interfaces: Asymmetric Digital Subscriber Line (ADSL) Metallic Interface, ANSI Standard T1.413, 1995.

[I41 J. J. Werner, “The HDSL environment,” IEEE J . Select. Areas Commun., vol. 9. no. 6, pp. 785-800, Aug. 1991.

[I51 P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels.” ZEEE Trans. Commun., vol. 43, no. 3, pp. 713-775, Mar. 1995.

[16] T. N. Zogakis. J. T. Aslanis Jr., and J. M. Cioffi, “A Coded and Shaped Discrete Multitone System,” IEEE Trans. Cornmun., vol. 43, no. 12, pp. 2941-2949, Dec. 1995.

[ 171 L. F. Wei, “Trellis-Coded Modulation with Multidimensional Constel- lations.“ IEEE Trans. Inform. Theory, vol. IT-33, no. 4, pp. 483-501, July 1987.

[I81 G. D. Fomey Jr., “Coset codes I: Introduction and geometrical classi- fication.” IEEE Truns. lnfijrm. Theory, vol. 34, no. 5 , pp. 1123-1151, Sept. 1988.

T. Nicholas Zogakis (S’88-M’89) received the B S degree in electrical engineering from the University of Florida, Gainesville, in 1989, and the M S and Ph D degrees in electrical engineering from Stan- ford University, CA, in 1990 and 1994, respectively.

During the 1990-1991 academic year, he was with Harris Corporation, Palm Bay, FL, where he worked i n the area of signal identification and modulation recognition Since September 1994, he has been with Amati Communications Corporation, Mountain View, CA, where he works on the anal-

, , 131 I. Kalct, “The Multitone Channel,” IEEE Trans. Commun., vol. 37. no.

2, pp. 119-124, Fcb. 1989. 141 J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitone transceiver

system for HDSL applications,” ZEEE J. Select. Areas Commun., vol. 9, no. 6, pp. 895-908, Aug. 1991.

[ 5 ] J. M. Cioffi, “A multicarrier primer,” in ANSI T1E1.4 Committee Contribution, no. 91-157, Boca Raton, FL, Nov. 1991.

161 A. Ruiz and J . M. Cioffi, “A frequency-domain approach to combined spectral shaping and coding,” in Proc. IY87 h t . Con$ Commiin.. Seattle. WA, June 1987, pp. 1711-1715.

ysis, design, and implementation of DMT modems. His research interests include digital communications, modulation, and coding theory.

Dr. Zogakis is a member of Eta Kappa Nu, Phi Kappa Phi, and the IEEE Communications and Information Theory Societies.

John M. Cioffi (S’77-M’78-SM’90-F’96), for a photograph and biography, see p. 64 of the January issue of this TRANSACTIONS.

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