# Performance analysis of the weighted decision feedback equalizer.

**ABSTRACT** In this paper, we analyze the behavior of the weighted decision feedback equalizer (WDFE), mainly from filtering properties aspects. This equalizer offers the advantage of limiting the error propagation phenomenon. It is well known that this problem is the main drawback of decision feedback equalizers (DFEs), and due to this drawback DFEs are not used very often in practice in severe channels (like wireless channels). The WDFE uses a device that computes a reliability value for making the right decision and decreasing the error propagation phenomenon. We illustrate the WDFE convergence through its error function. Moreover regarding the filtering analysis, we propose a Markov model of the error process involved in the WDFE. We also propose a way to reduce the number of states of the model. Our model associated with the reduction method permits to obtain several characteristic parameters such as, error propagation probability (appropriate to qualify the error propagation phenomenon), time recovery and error burst distribution. Since the classical DFE is a particular case of the WDFE (where the reliability is always equal to one); our model can be applied directly to DFE. As a result of the analysis of this process, we show that the error propagation probability of the WDFE is less than that of the classical DFE. Consequently, the length of the burst of errors also decreases with this new WDFE. Our filtering model shows the efficiency of the WDFE.

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**ABSTRACT:**Efficient and fully blind Decision Feedback Equalizer (DFE) remains an open issue, mainly because of the potential errors in the decision loop. Based on the Weighted Decision Feedback Equalizer (WDFE), our previous work aiming at decreasing the error propagation phenomena, we propose a new blind DFE called Blind Weighted Decision Feedback Equalizer. The main idea of the WDFE was to commute between Linear Recursive Equalizer (LRE) and DFE for the error computation at the decision device. In this paper, we extend this idea in order to commute also both the algorithms and the filtering structure. This commutation is performed softly and blindly.IEEE Communications Letters 06/2010; · 1.16 Impact Factor - SourceAvailable from: mcgill.ca
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Page 1

Performance Analysis of the Weighted

Decision Feedback Equalizer

Jacques Palicot and Alban Goupil

Abstract

In this paper, we analyze the behavior of the Weighted Decision Feedback Equalizer (WDFE), mainly from

filtering properties aspects. This equalizer offers the advantage of limiting the error propagation phenomenon.

It is well known that this problem is the main drawback of Decision Feedback Equalizers (DFEs), and due

to this drawback DFEs are not used very often in practice in severe channels (like wireless channels). The

WDFE uses a device that computes a reliability value for making the right decision and decreasing the error

propagation phenomenon. We illustrate the WDFE convergence through its error function. Moreover regarding

the filtering analysis, we propose a Markov model of the error process involved in the WDFE. We also propose

a way to reduce the number of states of the model. Our model associated with the reduction method permits to

obtain several characteristic parameters such as: error propagation probability (appropriate to qualify the error

propagation phenomenon), time recovery and error burst distribution. Since the classical DFE is a particular case

of the WDFE (where the reliability is always equal to one); our model can be applied directly to DFE. As a

result of the analysis of this process, we show that the error propagation probability of the WDFE is less than

that of the classical DFE. Consequently, the length of the burst of errors also decreases with this new WDFE.

Our filtering model shows the efficiency of the WDFE.

I. INTRODUCTION

The number of services on heterogeneous wireless networks such as GSM, IS95, PDC, DECT and the

future 3G standards like the UMTS proposal in Europe is increasing dramatically. Moreover, one of the most

challenging issues is the interactive multimedia services over wireless networks. Consequently, the spectrum

efficiency of the modulation scheme is becoming extremely important. There are many ways of offering this

higher spectrum efficiency. Two methods are very obvious:

1) An increase in the symbol frequency.

2) An increase in the number of the state of the modulation.

Whatever technique is used, the sensitivity of the transmitted signal to multipath effects also increases and as a

result the well-known Inter Symbol Interference (ISI) phenomenon becomes more prominent in disturbing the

useful symbols. Therefore, the multipath effect will have to be tackled carefully.

In order to tackle the multipath problem of wireless networks, some services have chosen multicarrier

modulations such as for instance DAB and DVB-T in Europe since with single carrier modulation, we need

powerful equalization techniques.

The corresponding author is J. Palicot

J. Palicot is with Supelec, avenue de la boulaie, BP 81127, 35511 Cesson-S´ evign´ e, France (email: jacques.palicot@supelec.fr)

A. Goupil is with D´ ecom, Universit´ e de Reims, UFR Sciences, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France (email:

alban.goupil@univ-reims.fr)

June 25, 2007DRAFT

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It is well known that the Maximum Likelihood Sequence Estimate (MLSE) equalizer is the best but its

computational complexity depends exponentially on both: the number of constellation points and the length of

the channel impulse response. Therefore, MLSE is not very practicable for high spectral efficiency modulation

and because of this reason DFEs are gaining importance. In fact, the latter offers the best compromise between

performance and complexity.

DFEs are well known for their superior performances as compared to transverse equalizers. However, due to

their recursive structure (feedback loop), they can suffer from error propagation and this results in overall mean

square error (MSE) degradation. This problem has already been addressed by many authors. In [2], the problem

of the bounds of this degradation has been addressed. The problem is so great that none of the manufacturers,

in particular consumer electronics manufacturers, use DFEs in their modems for severe channels. To the best

of authors’ knowledge, DFEs are only used for modems on less severe channels, e.g. cable channels.

This problem is very often overcome by the transmission of a known training data sequence (the training

period). This training period is used both for the starting period (blind equalization) and for the tracking period.

During this latter period, the channel may change and the DFE in the decision directed (DD) mode may suffer

from the error propagation. As a consequence, this training period should be transmitted regularly, and this

results in overall throughput degradation. For example a regular training period is transmitted each frame (25

ms) [3] for digital terrestrial television broadcasting in the US in order to avoid error propagation. This results

in a loss of bit-rate, and explains why this problem is still open. This error propagation is a major problem and

its exact solution is yet to be determined.

Many techniques have been proposed to reduce error propagation and thus to improve the overall DFE

performances without transmitting a training period. We can classify all these techniques in three main

categories:

1) The first category comprises of techniques which modify the decision rules.

2) The second category consists of techniques which work directly on the output sequence of the DFE after

the decision device.

3) The third category groups all other alternative techniques.

Among the techniques based on the modification of the decision rules of the DFE, we find firstly the work

carried out in [4]. In this work the author proposed a new decision device based on Bayesian analysis. A similar

work has been proposed more recently in [5]. At the same time the author of [1] proposed another example

of soft decision device based on weighted decisions. It is precisely this work which is analyzed in this paper.

The work presented in [6] also belongs to the first category of techniques. In this paper, some intermediate

decisions are used to roughly smooth the decision-device. An important feature of the work of [1] is that, the

weighted decisions are also used for the tracking phase of the algorithm which is not the case in [5] and in [6].

In the second category of techniques we find the well known Delayed Decision Feedback Sequence Estimation

(DDFSE) [7]. The authors proposed to perform a simplified Viterbi Algorithm on the delayed Decision Feedback

Sequence. The resulting equalizer performance is between the classical DFE (when the feedback window length

is equal to zero) and the MLSE (when window length becomes large enough). In this category, we also find

the work of [8]. In the case of Trellis Coded Modulation (TCM) the authors proposed a new decision device

June 25, 2007 DRAFT

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which comprises memory and takes into account the rules of the code. In fact when error correcting code or

TCM modulation is used the effect of error propagation is more significant, since it generates burst of errors.

In [9], the authors proposed a simple algorithm in order to improve the decisions contained in the feedback

filter.

Among the third category of techniques we find an effective technique which has been proposed in [10].

In this work the authors proposed a blind DFE by commuting, in a reversible way, both its structure and its

adaptation algorithm, according to some measure of performance as, for instance, the MSE. So, in this way,

their DFE does not suffer from the error propagation problem. More recently in [11] the authors deal with this

problem by performing some comparison between the filter coefficients obtained simultaneously by a DFE and

by a channel estimator. If a divergence occurs, the DFE is again initialized by the channel estimator.

In [1], we addressed the problem of error propagation as being the result of both the input of errors in the

feedback filter and the divergence of the algorithm due to “false errors,” in the Least Mean Square Decision

Directed algorithm (LMS-DD). In fact, as shown by many simulation results, it is very difficult to distinguish

between the error propagation itself in the feedback filter and the algorithm divergence. Thus there must be an

efficient solution addressing both aspects. This equalizer, known as the Weighted Decision Feedback Equalizer,

offers the advantage of limiting the error propagation phenomenon.

The equalizer proposed in [1] is the classical DFE, to which we add two devices as shown in Figure 1.

The basic idea is to inject into the feedback filter the decisions if and only if its reliability is sufficiently high.

Otherwise, the data injected in the feedback filter would simply be the output of the filter. Therefore the WDFE

can be considered as a soft transition between the classical recursive linear equalizer and the decision feedback

equalizer.

In [12], we proposed an improvement in the WDFE performances by using a non-linear function of the

reliability for computing the weighted decisions and the error of the LMS algorithm. Moreover, we show that

the rule number 1 presented in [1] becomes a particular case of the new rule presented in [12].

In this paper, we analyze the behavior of the WDFE, mainly from the filtering point of view in order to

explain this good performance level. We also explain the convergence property by illustrating the behavior of

a particular error function of the WDFE.

There is a lot of literature available on the analysis of the error process. Most of it based on a Markov chain

model as in [13] and in [14]. To obtain some characteristics such as error recovery, time recovery, these models

should be specialized as in [15].

In this paper we propose to generalize this kind of analysis to the WDFE. Moreover, we introduce a state

reduction method in order to reach some characteristics in a general way. These characteristics are Error Re-

covery, Time Recovery, Duration of Burst of Errors, Burst Error Distribution and Error Propagation Probability

(Ppe). This last characteristic is of great importance to characterize the error propagation phenomenon. Since

the classical DFE is a particular case of the WDFE, all the previous results can be applied to the DFE. Then

under some assumptions, we obtain equivalent results as obtained recently by Campbell et al [14]. We show

that with respect to these parameters, the WDFE performs better than the classical DFE, e.g. error propagation

probability of the Weighted DFE is less than that of the classical DFE.

June 25, 2007DRAFT

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Rest of the paper is organized as follows. The second section presents the WDFE. The rules for computing

the reliability value and how to use it are also described in this section. Third section gives an illustration of the

WDFE convergence through its error function. Then, in the fourth section, a Markov Model of Error Probability

Density for DFEs is derived, and with a proper state reduction method, we obtain the expression for the error

propagation probability for both classical DFE and the new WDFE. The results obtained are presented in the

fifth section. It presents the results of the Markov model for the filtering part. These results are obtained with

fixed equalizer coefficients. They confirm and prove that the WDFE performs better than the classical DFE,

something already obtained in [1], [12].

II. WDFE PRESENTATION

A. General description

Overall scheme of the channel with the weighted version of the DFE is shown in Figure 1. The notations used

in this figure and throughout this paper are the following: skis the source symbol sequence, H(z) the channel

transfer function, wkthe additive white Gaussian noise, rkthe received sequence, F(z) the feed forward filter,

1−B(z) the feedback filter, zkthe WDFE’s output sequence, and γkthe reliability of zkand ˜ zkis the feed-back

symbols. The difference between the WDFE and the classical DFE is simply the addition of two new devices.

The first device computes a reliability value for each DFE output. Depending on the way this reliability is

computed, it can appear like a belief or a likelihood measurement.

The second device uses this value in such a way as not to decide on errors in the feedback loop and also to

minimize the effect of errors in the LMS-DD algorithm.

The way of computing the reliability is mainly given by the kind of modulation used. Different versions of the

WDFE depend on the use of this reliability. Moreover, for certain constellations, two reliability computations can

occur. For example, if a QAM is used, we can compute reliability for each axis (In-phase and in-Quadrature).

Let γIand γQbe these reliabilities, then the output of the decision device of the WDFE is

˜ zI

k= f(γI

k) ˆ zI

k+?1 − f(γI

k) ˆ zQ

k)?zI

k,

(1)

˜ zQ

k= f(γQ

k+?1 − f(γQ

k)?zQ

k.

(2)

where f(·) is the function that specifies the kind of reliability to be used. As already mentioned in the

introduction, the soft transition between the classical recursive linear equalizer and the decision feedback

equalizer appears clearly in equations (1) and (2). Indeed, when f(γ) is equal to 1 the WDFE acts as a

classical DFE whereas when f(γ) is equal to 0 the WDFE becomes an IIR filter. In this case, theoretically,

some instability can appear. However, empirically, this phenomenon does not happen, and a simple clipping of

the symbols fed back should avoid this risk.

The algorithm is also changed accordingly. The error ekof the LMS-DD algorithm is simply weighted by

the reliability:

eI

k= f(γI

k)?ˆ zI− zI?,

(3)

eQ

k= f(γQ

k)?ˆ zQ− zQ?.

(4)

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Note that the convex combination comes from an intuitive idea. As we are concerned with the error

propagation phenomenon, it seems hard to derive mathematically a reliability function as well as a soft-decision

device which minimize the error propagation. Thus, our method was firstly aiming at an intuitive equalizer and

secondly to analyze this equalizer from the error propagation point of view.

B. Reliability computation for QAM

The computation of the reliability, which is in the core of the WDFE, is mostly given by the constellation.

We focus only on the QAM modulation for which the decision domain of a symbol is shown in Figure 3. The

point z is the input of the device, and ˆ z represents the hard decision. The distances δ±

x or ygive the distance

between the border of the domain, and ∆ is the “radius” of the domain. Given this value, the reliability is

given by:

γ =min?δ+

x,δ−

x,δ+

∆

y,δ−

y

?

(5)

The relation given above has a natural interpretation. If the input of the device is close to the border, then the

reliability is close to 0 and, if it is near the hard decision point, then the reliability is around 1. The “radius”

is then a constant included here in order to normalize the reliability. In fact, we can write equation (5) in the

following form

γ = 1 −?z − ˆ z?

∆

.

(6)

Moreover, we can also decompose the reliability on the axis of the QAM. We then obtain two reliabilities,

which are given by

γI=min(δ+

γQ=min?δ+

x,δ−

∆

y,δ−

∆

x)

,

(7)

y

?

.

(8)

The example given above corresponds to a QAM symbol inside the constellation. The case of the symbol

on the border of the constellation is the same but, before any computation, some projections are carried out in

order to respect the idea of the reliability.

For other constellations, the reliability is determined in the same way, that is, the normalized distance between

the point and the border of the decision domain.

N.B.: For all these reliabilities computations we assume that the a priori decision domain is given by that

of the hard decision of the received symbol.

C. Reliability use

1) Rule 1 scheme: This technique is fully described in [1], so here we will simply recall the useful equations

of this rule. The function f is a threshold function, which allows the WDFE to be a DFE on a certain domain

of inputs and a linear recursive equalizer on the complement domain. This sub-domain is given by a parameter

dmin. In this case, the function f is:

f(x) =

1

if x < dmin

0

otherwise.

(9)

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The graphical representation of this rule is given in Figure 2 for a QAM. If the input of the WDFE device

lies inside the square of “radius” dmin then the WDFE acts as a classical DFE, otherwise it acts as a linear

recursive equalizer.

2) Rule 2 scheme: As seen before, the reliability computed above is not really used as it is. It is first

transformed by a simple function f. The simplest way to transform it is to consider f as the identity function.

Thus, equations (1)-(4) become:

˜ zk= γkˆ zk+ (1 − γk) zk,

ek= γk(ˆ z − z).

(10)

(11)

Consequently, if γ = 1 we obtain a classical DFE and if γ = 0 we obtain a linear feedback equalizer. The

WDFE can be considered as a soft transition between the linear feedback equalizer and the DFE.

3) Improvement of the rule 2 scheme: The improvement consists in the use of a non-linear, but continuous

function of the reliability for computing the weighted decisions and the error of the LMS algorithm. Hence the

reliability is modified; thanks to this non-linear function. The aim of this modification is to increase the effect

of γ when it is a high value and, on the other hand, to decrease this effect when it is a low value. With these

requirements in mind, it is natural to use a sigmoid non-linear function such as that used in neural network

applications.

The classical sigmoid function for several values of a, corresponding to equation (12) is shown in Figure 4.

To meet our requirements, we modify the basic sigmoid function with a compression law. The result is given

in equation (13) and is shown in Figure 5 for the compression parameters b = 0.5 and a = 5.

g(x) =1 − exp(−ax)

1 + exp(−ax)

g(x) =1

2

(12)

?

1 − exp?a?x

b− 1??

1 + exp?a?x

b− 1?? + 1

?

(13)

In these conditions, by substituting f(γ) by g(γ) in equations (1) to (4), we obtain the improved rule. The

main drawback of this improvement is that the new rule becomes user-dependent. In fact, the user should define

compression parameter b. In equation (13), with a = ∞ and dmin= 1−b, the present rule becomes equivalent

to rule-1 and this is also evident from Figure 4 and Figure 6, for b = 0.5. In Figure 6 we have drawn the three

f functions (Rule-1, Rule-2 and Rule-2 improved with the sigmoid rule). We can conclude that the Compressed

Sigmoid Function is a good compromise between the two rules previously defined in [1] and [12].

D. Study of the error function J(ek)

Let?Fk,Bk

for the LMS-DD algorithm given in equations (3) and (4), the adaptation is done using

?be the coefficients of the filters F(z) and B(z) at time k. With respect to the use of reliability

?Fk+1,Bk+1

?=?Fk,Bk

?− µ f(γk) ˆ ek

?Rk,˜Zk

?

(14)

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with Rk and˜Zk being symbols in the memory of the forward and backward filters, and µ the step of the

algorithm. The error ˆ ekis the classical decision directed error given by:

ˆ ek= ˆ zk− zk.

(15)

Keeping in mind that the reliability γ is connected to the error by relation

γ(p)

k

= 1 −

???ˆ e(p)

k

∆

???

,

(16)

where(p)denotes the In-phase or the in-Quadrature axis, it might be interesting to have a look at the behavior

of the global function J, defined by:

J(ˆ e) = |γ ˆ e|

(17)

In fact, the weighted LMS-DD algorithm could be seen as the LMS-DD whose step is weighted by the

reliability, or whose error is weighted. The function J represents the second point of view. Since in the case of

the classical DFE we have γ = 1 whatever the error is, we can compare the WDFE algorithm with the classical

DFE.

In Figure 7, J(ˆ e) is drawn for different f functions. In the case of the classical DFE, the error function

is linear and increases with error. In all the others cases, the error function decreases towards zero when the

reliability is very low (or equivalently when the error is high). The most interesting feature of the sigmoid

function is that it increases the error function when the reliability is high. All the curves tend towards the same

limit when the error tends towards zero, or, equivalently, when the reliability tends towards 1. That means that

for high SNR the algorithm noise tends to have the same behavior. In other words the steady-state error will

have roughly the same behavior at high SNR. However the WDFE will be slightly noisy at practical SNR. At

the opposite, when the error tends towards 1 the WDFE error function decreases whereas the classical DFE

error function increases, consequently the tracking behavior is performed in a wiser way.

We can easily note that, for each iteration in equation (14), the equivalent convergence step of the algorithm

becomes an adaptive step. If adaptive steps are classical for the LMS algorithm, it is interesting to see that in

our case the new equivalent step is weighted by the reliability value of the current output of the WDFE.

III. FILTERING ANALYSIS OF THE WDFE

A. Error model of the filtering

We need to develop a satisfactory model for the filtering before analyzing the error process of WDFE. As the

computation of the reliability and also its use is carried out symbol by symbol, and does not use any memory,

the WDFE filtering could be seen as a classical DFE filtering with a soft decision device. Therefore we can

write:

˜ zk= U(zk)

(18)

where U is the soft decision function and because of this function we can restrict our analysis to the soft-

decision DFE. In fact, in order to compare the WDFE and the DFE, we can simply study the effect of the

June 25, 2007DRAFT

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function U on the performances, because the DFE model is given by considering U to be the hard decision

function. Using the notation of Figure 1, the input/output relation of the WDFE is given by:

zk=

?

i

firk−i−

?

fiwk−i−

i>0

bi˜ zk−i,

(19)

zk=

?

i

?

j

fjhi−jsk−i+

?

i

?

i>0

bi˜ zk−i,

(20)

where the lower letters xkdenotes the sample of the sequence X. Before going any further, we should point out

that three kinds of error have to be considered. The first is the output error, which is given by: ek= zk− sk.

It serves to compute the MSE of the output. The second is the error of decision ˆ ek = ˆ zk− zk, which is

the difference between the hard decision and the source. This error is used to compute the error probability.

However, since the WDFE is a soft decision DFE, a third error type is given by ˜ ek= ˜ zk− sk. This error is

responsible for the error propagation, because it represents the error contained in the memory of the feedback

filter. Of course, this error is equal to the decision error for the classical DFE. But, due to its presence, all the

results of the DFE should be reformulated for the WDFE.

Using above error definitions, we obtain a relation followed by the error sequences as:

ek=

?

i

tisk−i+

?

i

fiwk−i−

?

i>0

bi˜ ek−i,

(21)

= nk−

?

i>0

bi˜ ek−i

(22)

where ti is i-th coefficient of the impulse response of the combined filter T(z) = H(z) F(z) − B(z) and

where nkis considered as a noise. This noise is composed of the effect of the residual ISI and the Gaussian

noise filtered by the feedforward filter. The probability density function of nkis then a Gaussian mixture. As

the relation in equation (22) involves two kinds of error, we first make an assumption. Without a great loss

of generality, we assume the existence of a function V derived from U such that ˜ ek = V (ek). Indeed, the

way to compute the error is independent of the input symbol itself, except when it is at the border of the

constellation. In fact, the function V gives the “soft errors” ˜ ek from the real errors ek, considering that the

modulation alphabet is infinite. With the introduction of the function V , we can write

˜ ek= V

?

nk−

?

i>0

bi˜ ek−i

?

(23)

As the length of the feedback filter is finite, the error sequence ˜ ekfollows a Markov process whose order is

given by the length of B. It is worth noting that the study of ˜ ekis enough to obtain most of the descriptive

parameters of the error process. It is possible to obtain the information about ekfrom ˜ ekbecause of equation (23).

Since ˜ ekis more general than ˆ ek, we get this process through the relation: ˆ ek= Dec(˜ ek).

B. Markovian Model of the Error Probability Density for DFE’s

The input/output relation of the “soft errors” of the WDFE is given by:

˜ ek= V

?

nk−

LB

?

j=1

bi˜ ek−i

?

(24)

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where LBis the length of the feedback filter B(z). In order to compute the probability density function of ek

we should consider the possible value of ˜ ek.

Equation (24) can be used in order to obtain a general relation that the probability density function should

satisfy. However, the problem is quite complex to be resolved easily. We propose simplifying the study by

making an approximation of the soft error function V (·). The approximation is obtained using a staircase

function:

V (x) = vi

for i ∈ [ai;ai+1]. (25)

This kind of approximation allows us to consider the error as a Markov process whose time and states are

discrete. The states are given by the memory of the feedback filter and are encapsulated by the vector˜Ek. Of

course, the way in which approximation is done, affects the quality of the analysis. Let us denote Pr[k ∈˜Ei] as

the probability to be in the state˜Eiat time k. The Markov aspect of the error process allows these probabilities

to be linked easily by:

Pr[k + 1 ∈˜Ei] =

?

j

Pr[˜Ej→˜Ei] Pr[k ∈˜Ej],

(26)

where Pr[˜Ej→˜Ei] is the probability of the transition between state˜Ej to state˜Ei. It should be noted that

this transition probability depends only on the states involved and not on the time. This probability should be

computed carefully. For example consider a BPSK modulation and a classical DFE. The possible error values

are in the set {−2,0,+2}. But the error +2 is associated only with the source symbol −1. That is not the case

of the error 0, which can occur for both source symbols. So, the transition probability computation should take

this fact into account.

The relation of equation (26) corresponds to a multiplication between a matrix and a vector. We denote then

P(k)as the stack composed of the state probability at time k and Q the matrix whose i,j entry is the transition

probability Pr[˜Ej →˜Ei]. The time relation between the states is then expressed recursively by the simple

matrix multiplication

P(k+1)= Q P(k).

(27)

Although the relation in equation (27) is interesting for observing the transient behavior of the WDFE, its

steady state performances are more valuable for comparing the different versions with the classical DFE. This

steady state exists because the transition diagram of the Markov chain is regular. Thus, the probability of each

state is given by

P(∞)= Q P(∞).

(28)

As the Markov chain is regular, the vector P(∞)is the eigenvector of Q associated with the eigenvalue 1,

which exists because Q is a stochastic matrix.

The approximation in equation (25) leads us to the steady-state relation of equation (28). This relation can

also be seen as an approximation of the probability density function of the errors in the feedback filters be a

Gaussian mixture whose means are given by the states and the feedback filter’s coefficients:

˜ ek∼

?

i

P(∞)

i

N?BT˜Ei, σ2

N

?,

(29)

June 25, 2007DRAFT

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where N(µ,σ2) denotes the Gaussian distribution with mean µ and variance σ2, and σ2

from equation (22).

Nis the variance of nk

The Markov chain of the WDFE needs a lot of computation. If the function V (·) is approximated by a

step function with N steps then the number of states is LN

B. Then the dimensions of transition matrix Q are:

LN

B×LN

Computing the relevant eigenvector of Q may be intractable. But the convergence of equation (27) is fast

B.

enough to reduce the complexity. Moreover the sparse aspect and the numerous symmetries in Q can be used

efficiently to decrease the complexity.

C. Descriptive parameters

The comparison between the WDFE and its classical counterpart cannot be made directly through the

transition matrix because the number of states is not the same for the two equalizers. Hence, one should

compute some parameters that describe the difference between the DFE and the WDFE. We will see that

these parameters can be computed by error modeling using a Markov process. Before giving these descriptive

parameters, we need a tool to compute them. In fact, the states of the WDFE and the DFE should be grouped

together in order to make the Markov process description smaller. For this purpose the states are partitioned

into classes. As we will see below, the different descriptive parameters can be computed from the probability

to be in a specific class during the steady state.

1) Markov chain reduction: We want to compute the probability of transition between the classes of states.

Let Fibe the classes which form a partition of the set of states Ej, and let R denotes the partitioning matrix

defined by:

Rij=

1

if Ejbelongs to the class Fi,

0

otherwise.

(30)

The Bayes rule gives us the transition matrix P between the classes during the steady state, which is given

by

P = RQC,

(31)

where C can be viewed as given the “weight” of the state Eiin the class Fj. It can be defined as:

Cij= Rij lim

k→∞

Pr[k ∈ Ei]

El∈FjPr[k ∈ El].

?

(32)

The Markov chain reduction can also be used to compute the probability Pr[Fi] to be in a class during the

steady state.

2) Error probability: The first descriptive parameter is the error probability. The analytical computation of

this probability for DFE is a very difficult task. However its numerical computation can be carried out through

the chain reduction presented above. As we are interested in the steady state period only, we can define the

error probability as the probability of having a decision error in the first tap of the feedback filter. Therefore,

we can regroup all the states that have a decision error in the first class Feand regroup all the other states in

the other class. Once the reduction has been obtained, the error probability is given by:

Pe= Pr[Fe]

(33)

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Even if this descriptive parameter does not need to compute the class transition matrix explicitly, it shows

some aspects of the method. First the grouping process, which is simple, is performed by a selection of states

that satisfy some criterion. This criterion should only deal with the decision error and not the soft decision

error. This condition is given by the need for comparison between the classical DFE and the WDFE. Therefore

a function should exist, which derives the hard decision error from the soft one. As the reliability is local, this

function always exists.

3) Error recovery time and error propagation: The error recovery time parameter TN measures the mean

time between the appearance of the first error in the feedback filter and the time when there are no longer

any errors in this feedback filter. In others words, it measures the mean time during which the DFE is not

working optimally because of the error propagation phenomenon. The simplest lowest bound is the length of

the feedback filter, because in order to erase the first LBerrors, good decisions should be made.

In order to obtain this time recovery parameter, we should again reduce the Markov chain. However, we will

not compute this parameter in the same way as in [15]. In fact, another parameter, introduced in [16], will be

used and the error recovery time will be computed from it.

Two classes are defined as follow:

• Class E regroups all states that contain at least one decision error, whatever its position is,

• and class O regroups all other states, i.e. the states without any hard decision errors.

The reduced transition diagram is drawn in Figure 13. The loop assigned probability Ppe from E to E is

called “error propagation probability,” and measures the probability of being in a non-optimal state knowing

that the previous state is also non-optimal. This new parameter allows the comparison between the WDFE and

the DFE and this parameter can also be used to compute the error recovery time by using the following relation:

TN=

1

1 − Ppe.

(34)

To obtain this relation, consider the random variable X measuring the time the DFE is still in the state E

knowing that at time 1 it was in E. Consider also the Markov chain described in Figure 13 with the transition

probability a = 1. Then Pr[X = k] = (1 − Ppe)Pk−1

by equation (34).

pe , which is a geometric distribution whose mean is given

4) Error burst distribution: The error propagation phenomenon results in a continuous burst of errors and

the performances of receivers can thus be affected. In fact, when error correcting codes or TCM modulation is

used, the effect of error propagation is greater, since it generates bursts of errors. This study is very difficult to

carry out (even through simulation), because the computation time can be prohibitive. That is why some authors

have proposed adding specific correlated noise in order to accelerate the simulation time [17]. This error burst

distribution study is based on the Markov chain, as presented above, and simplifies the comparison. Let us

consider that DBdenotes the random variable corresponding to the duration of the consecutive hard decision

errors of the output (i.e. Duration of Bursts of errors). As for time recovery, the error burst computation needs,

June 25, 2007 DRAFT

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reduction in the number of states in the Markov chain. The Markov aspect occurs here once again:

Pr[DB> l] = Pr[ˆ ek?= 0, ˆ ek−1?= 0,..., ˆ ek−l?= 0],

= Pr[ˆ ek?= 0|ˆ ek−1?= 0,..., ˆ ek−l?= 0]

× Pr[ˆ ek−1?= 0,..., ˆ ek−l?= 0],

= Pr[ˆ ek?= 0|ˆ ek−1?= 0,..., ˆ ek−l?= 0]

× Pr[DB> l − 1].

(35)

(36)

(37)

The last equation allows us to focus on a length lower than LB because, for longer burst, we obtain the

following:

Pr[DB

>l>LB]= Pr[ˆ ek

?=0|ˆ ek−1

?=0,..., ˆ ek−LB

?= 0]l−LBPr[DB

>LB].

(38)

Of course, the method of computing the probability Pr[DB> l] for l < LBuses the reduction of the Markov

chain. The class Fifor i ∈ {0,...,LB} corresponds to the pattern

E,...,E

?

???

i times

,0,X ...

(39)

For each class Fithe possible transitions are Fi+1if an error is made or F0if a correct decision is made.

In the latter case, the burst is finished. Moreover the state FLBis special because the transition on error is

directed at itself. If we define Q as the transition matrix of the reduced process, the error burst probability is

given by the relation

Pr[DB> l] = uT

0Qul

(40)

where ui denotes the vector whose elements are 0 but the i + 1-th is 1. We consider that the matrix Q is

constructed assuming that the class i position is in the i + 1 vector position. Equation (40) gives us all the

information about burst length distribution as Pr[DB= l].

IV. RESULTS

The analysis provided above is used to compare DFE and WDFE. The simulation parameters were chosen

to stress the error propagation phenomenon. However, as the computation complexity grows exponentially with

the length of the feedback filter, the channel’s coefficients may seem unrealistically simple.

To validate the model, a comparison between a Monte Carlo simulation and the model has been done as

shown in Figure 8, with a channel set to H = [1,0.2,0.2] and a backward filter set to B = [0.24,0.24].

The equalizer does not use a feedforward filter. The WDFE uses the rule-2 approximated with 16 stairs for a

BPSK. Figure 9 shows the computation of the error propagation probability in the steady state with the same

conditions as the previous simulation for different Signal to Noise Ratios. This result proves that, in exactly

the same conditions, the Error Propagation Probability is lower with the WDFE scheme than with the classical

DFE scheme. This last result is fully in accordance with the overall system performances of [1], [12].

We choose a feedback length of 2 in order to compute the time recovery for both the DFE and the WDFE. The

channel and feedback coefficients were H = B = [1,0.6,0.1]. The time recovery plots for the two equalizers

June 25, 2007 DRAFT

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are shown in Figure 10. It is obvious that the behavior of the WDFE is much better during recovery time. It is

clear that the WDFE decreases the time recovery when the noise is high, but also when the noise is low. The

latter remark can be easily explained by the fact that if there is no noise, the classical DFE can not change its

state, whereas the WDFE, can do it more easily because of soft decision function.

Another very interesting feature of soft function is proved by the results shown in Figure 11. In the latter

simulation, the channel coefficients were [1,0.6] and the feedback filter was perfect with respect to the Zero-

Forcing criterion. The selected modulation was a QPSK and the signal to noise ratio was 19 dB. It is clear

that the length of the error burst is now shorter because of the use of the WDFE and also due to decrease in

the error propagation probability. Another more realistic simulation (longer channel and feedback filters) was

performed with the following channel [1,0.6,0.1]. The results are still very good as shown in Figure 12.

V. CONCLUSION

The WDFE includes the computation and the use of a reliability in order to limit the error propagation

phenomenon. This paper provides an analysis of its behavior in order to quantify the improvement given by

this equalizer. The error probability model for the WDFE proposed in this paper makes it possible to access

several descriptive parameters such as the Error Recovery Time, Burst Error Distribution and Error Propagation

Probability. This model is efficient enough to reach the error propagation probability of both the DFE and the

WDFE. Furthermore, it confirms that this probability is less for WDFE than for classical DFE. Future studies

will investigate the behavior of this model with the adaptive algorithm of the WDFE.

REFERENCES

[1] J. Palicot, “A weighted decision feedback equalizer with limited error propagation,” in Proceedings of ICC’00, New Orleans, USA,

Mexico, June 2000.

[2] D. L. Duttweiler, J. E. Mazo, and D. G. Messerschmitt, “An upper bound on the error probability in decision-feedback equalization,”

IEEE Trans. Inform. Theory, vol. IT-20, no. 4, pp. 490–497, July 1974.

[3] R. Hopkins, “Digital terrestrial HDTV for north america: the grand alliance HDTV system,” IEEE Trans. Consumer Electron., vol. 40,

no. 3, pp. 185–198, Aug. 1994.

[4] D. P. Taylor, “The estimate feedback equalizer: a suboptimum nonlinear receiver,” IEEE Trans. Commun., vol. 21, no. 9, pp. 979–990,

Sept. 1973.

[5] J. Balakrishna, H. Viswanathan, and C. R. Johnson, Jr, “Decision device optimization for soft decision feedback equalization,” in

Proceedings of the 2000 Conference on Information Sciences and Systems, Princeton, NJ, Mar. 2000.

[6] M. Reuter, J. C. Allen, J. R. Zeidler, and R. C. North, “Mitigating error propagation effects in a decision feedback equalizer,” IEEE

Trans. Commun., vol. 49, no. 11, pp. 2028–2041, Nov. 2001.

[7] A. Duel-hallen and C. Heegard, “Delayed decision-feedback sequence estimation,” IEEE Trans. Commun., vol. 37, no. 5, pp. 428–436,

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[8] H. Besbes, M. Jaidane-Saidane, and J. Ezzine, “On exact convergence results of adaptive filters: the finite alphabet case,” Signal

Processing, vol. 80, no. 7, pp. 1373–1384, July 2000.

[9] A. Goupil and J. Palicot, “Maximum likelihood-like improvement of the output sequence SIMO decision feedback equalizer,” in

SUMMIT’04, June 2004.

[10] J. Labat, O. Macchi, and C. Laot, “Adaptive decision feedback equalization: can you skip the training period?” IEEE Trans. Commun.,

vol. 46, no. 7, pp. 921–930, July 1998.

[11] K. Berberidis, A. Marava, P. Karaivazoglou, and J. Palicot, “Robust and fast converging decision feedback equalizer based on a new

adaptive semi-blind channel estimation algorithm,” in Proceedings of GLOBECOM’01, San Antonio, Texas, Nov. 2001.

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[12] J. Palicot and C. Roland, “Improvement of the WDFE performance thanks to a non linear amplification of the reliability function,”

in Proceedings of ICT’01, Bucarest, Roumanie, June 2001.

[13] C. L¨ utkemeyer and T. G. Noll, “A probability state model for the calculation of the BER degradation due to error propagation in

decision feedback equalizers,” in ITC, Porto Carras, Greece, June 1998.

[14] T. J. Willink, P. H. Wittke, and L. L. Campbell, “Evaluation of the effect of intersymbol interference in decision-feedback equalizer,”

IEEE Trans. Commun., vol. 48, no. 4, pp. 629–635, Apr. 2000.

[15] N. C. Beaulieu, “Bounds on recovery times of decision feedback equalizers,” IEEE Trans. Commun., vol. 42, no. 10, pp. 2786–2794,

Oct. 1994.

[16] A. Goupil and J. Palicot, “Markovian model of the error probability density and application to the error propagation probability

computation of the weighted decision feedback equalizer,” in Proceedings of ICASSP’01, Salt Lake city, Utah, May 2001.

[17] M. Jin, B. Farhang-Boroujeny, G. Mathew, and K. C. Indukumar, “A novel fast approach for estimating error propagation in decision

feedback detectors,” IEEE J. Select. Areas Commun., vol. 19, no. 4, pp. 668–676, Apr. 2001.

wk

sk

H(z)+

F(z)+

Decision and

Reliability

Use

1 − B(z)

rk

xk

zk

ˆ zk

˜ zk

γk

Fig. 1.WDFE scheme.

z

ˆ z

d

∆

dmin

Fig. 2.Rule 1 scheme.

June 25, 2007DRAFT

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z

ˆ z

∆

δ−

y

δ+

y

δ−

x

δ+

x

Fig. 3. Rule 2 scheme.

a = 5

a = 10

a = 50

b = 0.5,a = 5

−1

−0.50 0.51

−1

−0.5

0

0.5

1

x

g(x)

Fig. 4.Sigmoid.

June 25, 2007 DRAFT