890 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Improved Bounds for Error RecoveryTimes of Decision Feedback Equalization

Wendy W. Choy and Norman C. Beaulieu,Senior Member, IEEE

Abstract—New upper and lower bounds on the mean recoverytime of decision feedback equalization (DFE) are derived. Therecovery time is defined as the time it takes the decision feedbackequalizer (DFEQ) to reach the error-free state after an error hascorrupted an error-free DFEQ. The derivations of the boundsassume a causal channel response, independent data symbols, andindependent noise samples. The bounds are found to be tighter,especially at large SNR, than previous bounds in a numericalexample.

Index Terms—Decision feedback equalization, equalizer errorstatistics.

I. INTRODUCTION

I NTERSYMBOL interference (ISI) in a communicationsystem has a deleterious effect on system performance. The

ISI arises because insufficient channel bandwidth causes thepulses to spread into adjacent pulse intervals at the receiverend. This spreading may increase or decrease the noise marginof the received signal depending on the relative polarities ofthe pulses. On the average, however, ISI increases the bit errorprobability.

One of the methods often used to combat the effects ofISI is to use DFE. A DFEQ operates by reconstructing theportion of the ISI due to previously transmitted symbols andthen subtracting out this portion from the received signal.The reconstruction is based on estimating the previouslytransmitted symbols and the channel characteristics.

A DFEQ consists of a feedforward linear filter whichattempts to adjust the response so that there is no precursorcontribution from subsequent symbols, and a decision deviceto estimate the received symbol. The output of the decisiondevice is fed into a feedback filter to estimate the ISI dueto the previously transmitted symbols. This estimate is thensubtracted from the signal at the input of the decision device.Both the feedforward and feedback filters are usually imple-mented using tapped delay lines. The number of taps of thefeedback filter determines the number of previous decisionsthat are used to reconstruct the ISI. The tap coefficients of thefeedback filter are samples of the tail of the system impulseresponse including the channel and the feedforward filter.

Manuscript received September 21, 1994; revised May 12, 1996. Thematerial in this paper was presented in part at the 1995 IEEE InternationalSymposium on Information Theory, Whistler, BC, Canada, September 17–22,1995.

W. W. Choy is with Nortel Technology, Ottawa, Ont., Canada.N. C. Beaulieu is with the Department of Electrical and Computer Engi-

neering, Queen’s University, Kingston, Ont., Canada K7L 3N6.Publisher Item Identifier S 0018-9448(97)02640-0.

Assuming that the past decisions are correct, a DFEQ (withperfect channel identification) can eliminate ISI due to previ-ously transmitted symbols in the span of the feedback filtercompletely. However, decision errors will result in residualISI which may increase the probability of decision error in thefuture detected symbols. This leads to error propagation in theDFEQ. Analysis of a DFEQ is difficult because little is knownabout the distribution of the past decision errors.

It is important to know how fast a DFEQ can recoverfrom an error; that is, how many symbol intervals it takesto clear up an initial error introduced into the feedback filter.Then one knows how many future decisions will be affectedby the error. When the DFEQ has a finite number of tapsin the feedback filter and the system response has a finitetime duration, the communication system can be modeled as afinite-state Markov chain as shown by Monsen [1] and Austin[2]. Austin, in [2], showed how to obtain the mean recoverytime exactly through quasisimulations and discussed boundingthe mean recovery time. However, both of Austin’s approachesrequire computational efforts that grow exponentially with thelength of the DFEQ. The mean recovery time of a DFEQ witherror state transition probabilities of was also computedin [2]. Cantoni and Butler [3] derived an upper bound forthe mean number of symbols required to reach the zero errorstate, starting from an arbitrary initial state and subject tonoise. The bound depends only on the number of taps inthe DFE feedback filter and the number of signal levels.Kennedy and Anderson [4] extended, generalized, and clarifiedthe contributions in [3], and gave a class of channels for whichthe upper bound in [3] is exactly the mean recovery time.

Duttweiler, Mazo, and Messerschmitt in [5] developedan aggregated states model of a DFEQ which was usedto upper-bound the average error probability. Beaulieu [6]modified the model in [5] to compute upper and lower boundsfor the mean recovery time by writing difference equationsfor conditional, state-dependent, mean recovery times. Healso provided analytical proofs of some known results thatpreviously were justified with intuitive arguments. Since thestates of the model in [6] are based on error-free run lengthsof symbol sequences, the model and associated results in [6]will be referred to by the qualifier EFRL. Altekar and Beaulieudeveloped models in [7] that lead to tighter upper bounds onthe average probability of error of a DFEQ than those of [5].

In this paper, new, tighter bounds on the recovery times ofDFE are derived by modifying the models of [7] used for error

0018–9448/97$10.00 1997 IEEE

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 891

Fig. 1. Block diagram of decision feedback equalizer.

probability upper bounds. The paper is organized as follows.Sections II, III, and IV give derivations of error recovery timebounds for the single-errors model, double consecutive errorsmodel, and the arbitrary double-errors model, respectively.Error recovery times at zero and infinite signal-to-noise ratio(SNR) are computed in Section V, and a numerical exampleis considered in Section VI.

II. DISTINCT SINGLE ERRORSSTATES MODEL

The system model for a DFEQ is given in Fig. 1. Thechannel can be modeled as a linear, shift-invariant, discrete-time filter with response

(1)

where are the filter coefficients. The channel response isa combination of the transmitter filter, transmission medium,receiver filter, and the DFEQ forward filter. The transmissiondelay and time origin are chosen to makethe channel outputsample at the decision instant. It is assumed, without lossof generality, that The channel samples forcontribute ISI due to previously transmitted symbols, and thosefor contribute ISI due to subsequent symbols. For somepractical channels, ISI due to subsequent symbols is negligiblecompared to ISI due to previously transmitted symbols [1], [4],[5], [8]. Therefore, it is reasonable to assume that for

Furthermore, the mathematical analysis cannot be donewithout this assumption. It is also assumed that the channelresponse has a finite support, giving for where

is an appropriate positive integer [8].The channel is driven by a binary data stream The

symbols are independent and take on valueswith equalprobability. The output of the channel is contaminated by sam-ples of an additive-noise process which are independentof For the present analysis, it is assumed thatareindependent and identically distributed random variables. Thisis the case if additive white Gaussian noise is present at thefront end of the receiver and an infinite tap DFEQ forward

filter designed under a zero-forcing constraint is used [5],[9], [10]. If the minimum mean-square error criterion is usedto design the DFEQ forward filter, or if the DFEQ forwardfilter is of finite length, there may be correlation in the noisesamples. However, for moderate to high SNR’s, the effects ofthis correlation should be small [1], [2]. It is assumed that thenoise samples possess a probability density functionthat is symmetric, unimodal, and nonimpulsive (i.e.,

is nonincreasing for positive, and the noisecumulative distribution function is continuous). The case ofGaussian noise will be considered for numerical examples.

The -level quantizer has input/output characteristic

(2)

The output of the quantizer is the decoded data stream,which is applied to the input of a feedback filter with response

(3)

with being the filter coefficients and being thenumber of filter taps. The operation of the decision feedbackequalizer is mathematically described by

(4)

The equalizer attempts to cancel the ISI due to previouslytransmitted signals. Let

(5)

where represents the error in channel identification. Also,define the random variable

(6)

892 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Fig. 2. The distinct single-errors model state assignment.

Combining (4) to (6) gives

(7)

Note that whenever there is a decision error, andwhen a correct decision is made. The second term in

the argument of the function in (7) represents residual ISIdue to previous incorrect decisions. The third term representsISI due to imperfect channel identification. The fourth termrepresents ISI from the unequalized portion of the channelresponse. Thus the combined residual ISIis

(8)

By adding states which represent single bit errors within thespan of the DFEQ feedback filter to the model in [6], tighterbounds to the recovery time can be obtained. Definestates

for

for not all zero

for

(9)

which are mutually exclusive, and can represent the state ofthe DFEQ at time Fig. 2 illustrates these states. Define therandom process which takes on values from the set

(10)

corresponding to the state of the DFEQ at timeThe valueof will be determined by and If and

, then If and, then If and

, then If and ,then Since we are dealing with a transient propertyof the DFEQ, is an absorbing state. The sequence of states

is not a Markov chain but, as explained in [5] and [7],we can define and assume the existence of the limiting stateoccupancy probabilities

(11)

and the limiting transition probabilities

(12a)

(12b)

The state diagram of the single-errors model is shown inFig. 3. The recovery time defined here is the time it takes theDFEQ to reach state after an error has corrupted an error-freeDFEQ. This is the time it takes the DFEQ to reach statefromstate The variable is defined to be the average timeit takes the DFEQ to reach statefrom state Therefore,the mean channel recovery time is just This is not thesame time as the channel error recovery time defined in theEFRL model in [6]. Further discussion of the differences inthese two definitions of recovery time is given in Section VI.Finally, define the variable to be the average time it takesthe DFEQ to reach state from state (as in the EFRLmodel). Inspection of the state diagram of Fig. 3 yields

(13)

(14)

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 893

Fig. 3. The single-errors model state diagram.

(15)

(16)

An observation can be made about and Defineas the ISI conditioned on the state of the DFEQ being

Then, from (12) with , we have

(17a)

(17b)

The residual ISI due to previous incorrect decisions, for statesand is Furthermore,

the total ISI due to incorrect decisions, imperfect channelidentification, and ISI from the unequalized portion of thechannel response is from (8)

Total ISI (18)

for both states. Therefore,

(19)

and

(20)

The recovery time can then be found from (13)–(16) and (20)to be

(21)

It is not feasible, in general, to determine the values of thetransition probabilities, because the stationary distribution of

error sequences within each aggregated state is not known [5].We can, however, bound the recovery time by finding boundsfor the transition probabilities. Upper and lower bounds forthe mean recovery time can be found by finding bounds onthe transition probabilities of the form

(22a)

(22b)

Then, we can bound the mean recovery time by

(23)

To find the lower bounds on , note that [5]

(24)

where is the combined residual ISI defined in (8). Usingthe fact that when is known, knowledge of is irrelevant,we get

(25)

where

(26)

894 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Fig. 4. The double consecutive errors model state assignment.

is the probability of a correct decision conditioned on thecurrent residual ISI Conditioning on the value of thetransmitted symbol gives

(27)

For channels which do not have ISI due to symbols transmittedin the future, is independent of Therefore,

(28)

Then

(29)

It can be shown that is symmetric and unimodal [5],[12]. To find the lower bound for , define and to bethe maximum and minimum possible magnitude, respectively,of the combined ISI conditioned on the state of the DFEQbeing Then

for (30)

From (22a), (25), one obtains the lower bound

(31)

and the upper bound

(32)

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 895

Fig. 5. The double consecutive errors model state diagram.

However, it is not easy, in general, to compute the minimumpossible magnitude of the combined ISI Therefore, isupper-bounded by

(33)

The bounds for are obtained using similar arguments, whichgive

(34)

and

(35)

Discussion of the single-errors model bounds is given inSection VI where an example is considered.

III. D OUBLE CONSECUTIVE/DISTINCT

ARBITRARY DOUBLE-ERRORSMODEL BOUNDS

By defining more complicated error state models, tighterbounds to the recovery time can be obtained. Specifically, the

double consecutive errors model consists of states

for

for

for

for

chosen differently

from for

(36)

896 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Fig. 6. The distinct arbitrary double-errors model state assignment.

that are mutually exclusive, and can represent the state ofthe DFEQ at any given time. Fig. 4 illustrates these states.By examining the state diagram of this model shown in Fig.5, equations can be written for the mean times taken to gofrom different states to state Solving these equations forthe recovery time yields

(37)

where

(38a)

(38b)

(38c)

and

(39a)

(39b)

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 897

(39c)

(39d)

(39e)

(39f)

are the limiting transition probabilities. Note that is therandom process which takes a value from the set

(40)

corresponding to the state of the DFEQ at timeBounds to the recovery time can be found by finding

bounds to the transition probabilities, as was done for thesingle-error states model.

More states can be added to the double consecutive errorsmodel to obtain a better bound on the recovery time [7].In particular, all double errors are included in this model,resulting in states

for

and

for and

and

for and

for chosen to make

the states mutually exclusive

for

(41)

that are mutually exclusive, and can represent the state of theDFEQ at any given time. Fig. 6 illustrates these states. Thestates and are the same as in the double consecutiveerrors model of Section III. The states and containtwo errors with error-free bits between them, and the states

are chosen to make the states mutually exclusive.

Similar to the other models, define to be a randomprocess which takes a value from the set

(42)

corresponding to the state of the DFEQ at timeand definethe limiting transition probabilities

(43a)

(43b)

(43c)

(43d)

(43e)

(43f)

The state diagram for the arbitrary double-errors model isshown in Fig. 7.

Solving for the recovery time, one obtains (see (44) at thebottom of the following page) where

(45)

(46)

Bounds to can be found by finding bounds to thetransition probabilities.

Examples and discussions of the double consecutive andarbitrary double-errors model bounds are given in Section V.In the next section, specializations of the bounds previouslyderived for small and large SNR conditions are examined.

IV. ERROR RECOVERY TIMES AT ZERO AND INFINITE SNR

In this section, the performance of the DFEQ under twospecific cases; namely, the vanishingly small signal case andthe noiseless case are examined. At vanishingly small signallevels, the SNR can be considered to approach zero. Theprobability of making a correct decision is , and theprobability of making an incorrect decision is . The meanrecovery time is computed for the three models by setting

(47)

898 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Fig. 7. The arbitrary double-errors model state diagram.

in (21) for the single-errors model

(48)

in (37)–(38c) for the double consecutive errors model, and

(49)

in (44)–(46) for the arbitrary double-errors model. The valueis obtained for all three models. This value of mean

recovery time is also the upper bound of the mean channelrecovery time derived in [6].

The performance of a DFEQ on a noiseless channel is alsoof interest. Under this assumption, one is solely looking at theeffects of error propagation on the performance of the DFEQ.There are two cases to be considered under the noiselessassumption. Recall that is the combined residual ISI asdefined in (8). The DFEQ eye is closed if for all errorand data sequences, and the DFEQ eye is open if forall error and data sequences. When the DFEQ eye is closed,the probability of making a correct decision is the same asthat of making an incorrect decision which is . The meanrecovery time is for all three models. Whenthe DFEQ eye is open, the probability of making a correct

(44)

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 899

Fig. 7 (Continued). The arbitrary double-errors model state diagram.

decision is , and the mean recovery time is computed from(21), (37)–(38), and (44)–(46) with

(50)

in (21) for the single-errors model

(51)

in (37)–(38) for the double consecutive errors model, and

(52)

in (44)–(46) for the arbitrary double-errors model. The result isfor all three models. Note that is the length of the DFEQ;

this agrees with the physical consideration that it takes at leastthe length of the DFEQ to clear an error in the first tap.

V. NUMERICAL EXAMPLE

The bounds derived in Sections II and III for the meanrecovery time were computed for the following example. Thechannel has impulse response [6]

(53)

A -tap DFEQ is used to equalize the channel, and it isassumed that The noise samples are assumed tobe Gaussian, each with density

(54)

900 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

Fig. 8. Mean recovery time bounds of the three models.

Fig. 9. Mean recovery time bounds of the three models.

where is the noise variance. The SNR in decibelsSNRis defined as

SNR (55)

The average recovery time bounds in numbers of symbols forthe three models are plotted versus SNRin Fig. 8. For thisexample, the recovery time changes by six orders of magnitude

as SNR increases from small to large. Clearly, tight recoverybounds must be SNR-dependent. The lower bounds of the threemodels, although different, are graphically coincident. FromFig. 8, one can see that the upper and lower bounds coincidefor small SNR values (SNR less than 10 dB) and largeSNR values SNR greater than 10 dB). Fig. 9 showsthe recovery time bounds for the practical region of SNRfrom 2 to 10 dB. At large SNR where the bounds coincide,the recovery time bounds are computed to bewhich is

CHOY AND BEAULIEU: BOUNDS FOR ERROR RECOVERY TIMES OF DECISION FEEDBACK EQUALIZTION 901

Fig. 10. Comparison of mean recovery time bounds.

the number of taps of the DFEQ. We can also observe fromFig. 9 that although the consecutive double-errors model givesonly a slightly better bound than the single-errors model, thearbitrary double-errors model gives a much tighter bound forSNR between 2 and 10 dB. For example, at SNR 5dB, the upper bound of the double consecutive errors model(70 symbols) is one symbol smaller than that of the single-errors model (71 symbols), but the arbitrary double-errorsmodel gives a much smaller upper bound (55 symbols) thanthe other two models, which is 16 symbols less than that ofthe single-errors model upper bound.

These new bounds cannot be compared directly with thebounds in [6] because the recovery times are defined differ-ently. The recovery time defined in [6] is the mean time toreach the error free state after a primary error has corruptedthe DFEQ. A primary error includes both an error corruptingan error-free DFEQ and an error corrupting a DFEQ whichalready contains one or more errors. The recovery time definedhere is the mean time to reach the error free state after anerror has corrupted an error-free DFEQ. Although a directcomparison is not possible, it can nonetheless, be shown thatthe new bounds in this paper are tighter than those of [6].Let be the recovery time as defined in [6]. It can beshown that [12]

(56)

where the quantities are those defined in Section II,and are the probabilities that the DFEQ is instate One cannot find the exact value in (56); however, onecan bound it by examining the partial derivatives of

with respect to and giving

(57)

and

(58)

where and are the lower and upper bounds of ,respectively. These values are given in [7]. Fig. 10 comparesthe two recovery time bounds. At SNR less than 2 dB, thebounds from the two models coincide, but for SNRgreaterthan 2 dB, the bounds obtained from the single-errors modelare much tighter. At SNR greater than 13 dB, the single-errors-model upper bound converges to 20 symbols, whichis the length of the DFEQ, whereas the upper bound in [6]converges to 77 symbols, about four times larger.

VI. CONCLUSIONS

New, tighter bounds for the mean recovery time of a DFEQhave been obtained. In particular, the single-errors model, thedouble consecutive errors model, and the arbitrary double-errors model were used here to derive expressions for themean recovery time. For the numerical example considered,the arbitrary double-errors model gives the tightest bounds forthe mean recovery time. At small SNR values, the new boundsfrom the three models and previous bounds coincide. At largeSNR values, the new bounds are much tighter than previousones. In particular, the new bounds from the three models allapproach , the length of the DFEQ, at large SNR.

902 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997

REFERENCES

[1] P. Monsen, “Adaptive equalization of the slow fading channel,”IEEETrans. Commun., vol. COM-22, pp. 1064–1075, Aug. 1974.

[2] M. E. Austin, “Decision feedback equalization for digital communica-tion over dispersive channels,” Res. Lab. of Electronics. Mass. Inst.Technol., Cambridge, MA, Tech. Rep. 461, Aug. 11, 1967.

[3] A. Cantoni and P. Butler, “Stability of decision feedback inverses,”IEEE Trans. Commun., vol. COM-24, pp. 970–977, Sept. 1976.

[4] R. A. Kennedy and B. D. O. Anderson, “Recovery times of decisionfeedback equalizers on noiseless channels,”IEEE Trans. Commun., vol.COM-35, pp. 1012–1021, Oct. 1987.

[5] D. L. Duttweiler, J. E. Mazo, and D. G. Messerschmitt, “An upper boundon the error probability in decision feedback equalization,”IEEE Trans.Inform. Theory, vol. IT-20, pp. 490–497, July 1974.

[6] N. C. Beaulieu, “Bounds on recovery times of decision feedbackequalizers,”IEEE Trans. Commun., vol. 42, pp. 2786–2796, Oct. 1994.

[7] S. A. Altekar and N. C. Beaulieu, “Upper bounds to the error probabilityof decision feedback equalization,”IEEE Trans. Inform. Theory, vol. 39,pp. 145–156, Jan. 1993.

[8] R. A. Kennedy, B. D. O. Anderson, and R. R. Bitmead, “Tight boundson the error probabilities of decision feedback equalizers,”IEEE Trans.Commun., vol. COM-35, pp. 1022–1028, Oct. 1987.

[9] R. Price, “Nonlinearity feedback-equalized PAM versus capacity fornoisy filter channels,” inInt. Conf. on Communications, Conf. Rec.,1972, sec. 22, pp. 12–16.

[10] P. G. Messerschmitt, “A geometic approach to intersymbol interference.Part I: Zero-forcing and decision-feedback equalization,”Bell Syst. Tech.J., vol 52, pp. 1483–1519, Nov. 1973.

[11] S. A. Altekar, “Upper bounds to the error probability of decision feed-back equalization,” M. Sc. Eng. thesis, Queen’s University, Kingston,Ont., Canada, Sept. 1990.

[12] W. W. Choy, “New results on decision feedback equalization recoverytimes and error rates,” M. Sc. Eng. thesis, Queen’s University, Kingston,Ont., Canada, Apr. 1994.