IEEE 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
`
`Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 02,2010 at 10:06:04 EDT from IEEE Xplore. Restrictions apply.
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`Dish
`Exhibit 1010, Page 1
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`

`
`XOO
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996
`
`Bit Allocation
`
`Fig. I . Baseline DMT transmitter.
`
`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 X k , i = Ah,? + j B k , z
`in a constellation of size 2bk with unity distance between
`constellation points. Next, the constellation point is scaled
`points {xl;,z = g k X k , i , k = l , . . . , N } serves as the input to
`by a real multiplier, ,9k, and the collection of constellation
`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 -
`I}.'
`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
`
`'In 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.
`
`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
`H(kAf).90(t - P(kaf))
`' cos(27TkAft + l i , ( k A f ) )
`
`(5)
`
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`
`-
`
`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
`
`s ( t ) = 1 1 [Akzcos(27rkAft)
`
`z
`
`30
`
`1v/2-1
`
`Dish
`Exhibit 1010, Page 2
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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
`
`(6)
`
`- H k : B k , , s i n ( 2 ~ k A f t + $k)].qgz(t)
`where Hk = H ( k A f ) and $ k = 4 ( k A 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
`
`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
`
`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
`, m = 0, . . . , N - l}, obtained during the
`of N samples, {rTn,&
`
`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 +Al2irlAfr+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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`Dish
`Exhibit 1010, Page 3
`
`

`
`802
`
`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
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 7, JULY 1996
`
`0"
`
`I
`
`200
`
`400
`
`600
`frequency (kHz)
`
`I
`800
`
`1000
`
`I
`1200
`
`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
`= 14, I? = 9.8 dB,
`algorithm on Scenarios A and B with b,,,
`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,
`
`rate curves are plotted versus the normalized SNR, which is
`'Error
`proportional to 1/uF.
`
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`Dish
`Exhibit 1010, Page 4
`
`

`
`ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM
`
`~
`
`803
`
`IO'
`
`IO2
`
`i o 3
`
`1oz7-i
`
`i o 4
`
`P i o 5
`g
`g IOd
`
`L
`
`10'
`
`1 o8
`
`m
`e
`
`.
`io4.
`:
`
`a,
`
`2
`
`
`
`
`
`SNR = 11 0 dB, scenario B
`-
`SNR = 11 0 dB, scenario A
`_ _
`SNR = 14 0 dB. scenano B
`_ _ _
`SNR = 14 0 dB, scenario A
`
`, /
`
`Fig. 4.
`Uncoded DMT system's error rate performance for Scenario A at
`various jitter levels.
`
`Fig. 6.
`Error rate versus phase jitter for tmo DMT scenarios and two
`normalized SNR's.
`
`phase iitler (degrees)
`
`IO'
`
`l o 2
`
`i o 3
`
`1 o4
`
`TABLE 1
`JITTER LEVELS REQUIRED TO CAlJSE A FACTOR
`IN ERROR RATE
`OF TWO DEGRADATION
`
`nncou'ed
`(worst tone) a+.ma,. (Fz!gure G)
`scenario u + , ~ , ~ ~
`
`trellis coded
`uO,mar (Fiyure 9)
`
`A ,
`
`10-3
`
`B, - 10-6
`A . -
`
`I
`
`0.036"
`
`0.090"
`
`1
`
`O.OG(i"
`
`0.14"
`
`I
`
`0.067"
`
`0.16"
`
`Fig. 5.
`Uncoded DMT system's error rate performance for Scenario B at
`various jitter levels.
`
`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.
`3The jitter level for Scenario A at a normalized SNR of 11 0 dB ( P . D %
`lop3) is beyond the range of the plot in Fig 6
`
`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,
`and ATt = 7, -
`In the Appendix, we evaluate the DFT of (17) to derive an
`
`expression for R L , ~ , the received poinc in the lth bin
`
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`Exhibit 1010, Page 5
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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)
`(1 + j(l/P)4€!ffi)(Al,, + j&.,)
`(19)
`.!71&J~L
`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+jBk,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
`without coding. We use 2F:,, to represent the
`rate of
`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
`
`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
`< 0.25'.
`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).
`
`IV. TRELLIS-CODED DMT JITTER PERFORMANCE
`
`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 + l D k , 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.
`
`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
`
`Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 02,2010 at 10:06:04 EDT from IEEE Xplore. Restrictions apply.
`
`Dish
`Exhibit 1010, Page 6
`
`

`
`ZOGAKIS AND CIOFFI: THE EFFECT OF TIMING JITTER ON THE PERFORMANCE OF A DISCRETE MULTITONE SYSTEM
`
`805
`
`7..
`\;
`
`.
`' -.
`'.
`' '
`
`L = 35
`. L = 7 0
`
`.
`
`_ _
`
`[r
`
`
`
`- SNR - SNR
`
`1 0:
`
`50
`
`100
`
`200
`
`250
`
`150
`subchannel
`Fig. 7. Comparison of SNR and SIR for three values of L and two values
`of A i l , A@2 = 0.1' (upper three curves) and A& = 0.25' (second set of
`three curves), in Scenario A.
`
`I
`300
`
`'" 7
`
`8
`
`9
`
`10
`
`12
`11
`normalized SNR (dB)
`Fig. 8. Performance of Wei code in Scenario ,4 for three jitter levels.
`
`13
`
`14
`
`15
`
`16
`
`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',
`
`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
`
`Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 02,2010 at 10:06:04 EDT from IEEE Xplore. Restrictions apply.
`
`Dish
`Exhibit 1010, Page 7
`
`

`
`806
`
`lEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. I, JULY 1996
`
`7
`
`8
`
`9
`
`10
`
`11
`12
`normalized SNR (dB)
`
`13
`
`14
`
`15
`
`16
`
`Fig. 10.
`
`Performance of Wei code in Scenario B for three jitter levels.
`
`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 a