performance analysis of the weighted decision feedback equalizer
TRANSCRIPT
Performance Analysis of the Weighted
Decision Feedback EqualizerJacques 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-Sevigne, France (email: [email protected])
A. Goupil is with Decom, Universite de Reims, UFR Sciences, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France (email:
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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
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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.
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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: sk is the source symbol sequence, H(z) the channel
transfer function, wk the additive white Gaussian noise, rk the received sequence, F (z) the feed forward filter,
1−B(z) the feedback filter, zk the WDFE’s output sequence, and γk the reliability of zk and zk is 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 γI and γQ be these reliabilities, then the output of the decision device of the WDFE is
zIk = f(γI
k) zIk +
(1− f(γI
k))
zIk, (1)
zQk = f(γQ
k ) zQk +
(1− f(γQ
k ))
zQk . (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 ek of the LMS-DD algorithm is simply weighted by
the reliability:
eIk = f(γI
k)(zI − zI
), (3)
eQk = f(γQ
k )(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 z represents the hard decision. The distances δ±x or y give 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 − 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+
x , δ−x )∆
, (7)
γQ =min
(δ+y , δ−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 = γk zk + (1− γk) zk, (10)
ek = γk (z − z) . (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)
(12)
g(x) =12
(1− exp
(a(
xb − 1
))1 + exp
(a(
xb − 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
]be the coefficients of the filters F (z) and B(z) at time k. With respect to the use of reliability
for the LMS-DD algorithm given in equations (3) and (4), the adaptation is done using
[Fk+1,Bk+1
]=[Fk,Bk
]− µ f(γk) 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 ek is the classical decision directed error given by:
ek = zk − zk. (15)
Keeping in mind that the reliability γ is connected to the error by relation
γ(p)k = 1−
∣∣∣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(e) = |γ 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(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
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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
fi rk−i −∑i>0
bizk−i, (19)
zk =∑
i
∑j
fjhi−jsk−i +∑
i
fiwk−i −∑i>0
bizk−i, (20)
where the lower letters xk denotes 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 ek = 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 nk is 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 nk is 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 ek follows a Markov process whose order is
given by the length of B. It is worth noting that the study of ek is enough to obtain most of the descriptive
parameters of the error process. It is possible to obtain the information about ek from ek because of equation (23).
Since ek is more general than ek, we get this process through the relation: 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 LB is 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 Ei at 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)
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where N (µ, σ2) denotes the Gaussian distribution with mean µ and variance σ2, and σ2N is the variance of nk
from equation (22).
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 LNB . Then the dimensions of transition matrix Q are:
LNB×LN
B .
Computing the relevant eigenvector of Q may be intractable. But the convergence of equation (27) is fast
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 Fi be the classes which form a partition of the set of states Ej , and let R denotes the partitioning matrix
defined by:
Rij =
1 if Ej belongs 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 Ei in the class Fj . It can be defined as:
Cij = Rij limk→∞
Pr[k ∈ Ei]∑El∈Fj
Pr[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 Fe and 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 LB errors, 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)P k−1pe , which is a geometric distribution whose mean is given
by equation (34).
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 DB denotes 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
11
reduction in the number of states in the Markov chain. The Markov aspect occurs here once again:
Pr[DB > l] = Pr[ek 6= 0, ek−1 6= 0, . . . , ek−l 6= 0], (35)
= Pr[ek 6= 0|ek−1 6= 0, . . . , ek−l 6= 0]
× Pr[ek−1 6= 0, . . . , ek−l 6= 0], (36)
= Pr[ek 6= 0|ek−1 6= 0, . . . , ek−l 6= 0]
× Pr[DB > l − 1]. (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[ek 6= 0|ek−1 6= 0, . . . , ek−LB6= 0]l−LB Pr[DB > LB ]. (38)
Of course, the method of computing the probability Pr[DB > l] for l < LB uses the reduction of the Markov
chain. The class Fi for i ∈ {0, . . . , LB} corresponds to the pattern
E, . . . , E︸ ︷︷ ︸i times
, 0, X . . . (39)
For each class Fi the possible transitions are Fi+1 if an error is made or F0 if 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] = uT0 Qul (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
12
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
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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,”
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wk
sk H(z) + F (z) +Decision and
ReliabilityUse
1−B(z)
rk xk zk zk
zk
γk
Fig. 1. WDFE scheme.
z
z
d
∆
dmin
Fig. 2. Rule 1 scheme.
June 25, 2007 DRAFT
14
z
z∆
δ−y
δ+yδ−x δ+
x
Fig. 3. Rule 2 scheme.
a = 5
a = 10
a = 50
b = 0.5, a = 5
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
x
g(x
)
Fig. 4. Sigmoid.
June 25, 2007 DRAFT
15
b = 0.2 b = 0.5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
x
g(x
)
Fig. 5. Compressed sigmoid.
Rule 1
Rule 2
Sigmoid (b = 0.5)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
x
g(x
)
Fig. 6. f functions comparison.
June 25, 2007 DRAFT
16
DFE
Rule 1 (dmin = 0.8)
Rule 1 (dmin = 0.5)
Rule 2
Sigmoid (b = 0.2)
Sigmoid (b = 0.5)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
e/∆
J(e
)
Fig. 7. Error functions J(e).
ISI free bound
Simulation Model
0 2 4 6 80.005
0.01
0.02
0.05
0.1
0.2
SNR (dB)
Sym
bol
erro
rra
te
Fig. 8. Model validation for the WDFE rule 2.
June 25, 2007 DRAFT
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DFE
WDFE
0 2 4 6 8
0.52
0.54
0.56
0.58
0.6
0.62
0.64
SNR (dB)
Err
orpr
opag
atio
npr
obab
ility
Fig. 9. Error propagation probability for the DFE and the WDFE.
DFE
WDFE rule 1
10 12 14 16 18
3.1
3.2
3.3
3.4
3.5
3.6
SNR
Rec
over
ytim
e
Fig. 10. Recovery time.
June 25, 2007 DRAFT
18
DFEWDFE rule 1
2 4 6 8
0
0.2
0.4
0.6
Length
Pr[
B=
l]
Fig. 11. Error burst distribution.
DFEWDFE rule 1
2 4 6 8 10
0
0.2
0.4
0.6
Length
Pr[
B=
l]
Fig. 12. Error burst distribution.
June 25, 2007 DRAFT