- Email: [email protected]
Performance analysis of the weighted decision feedback equalizer
ARTICLE IN PRESS
Signal Processing 88 (2008) 284–295 www.elsevier.com/locate/sigpro
Performance analysis of the weighted decision feedback equalizer Jacques Palicota,, Alban Goupilb a
Supelec, avenue de la boulaie, BP 81127, 35511 Cesson-Se´vigne´, France De´com, Universite´ de Reims, UFR Sciences, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France
b
Received 28 October 2006; received in revised form 25 June 2007; accepted 27 July 2007 Available online 15 August 2007
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. r 2007 Elsevier B.V. All rights reserved. Keywords: Equalization; Decision feedback equalizer; Weighted decisions; Error propagation; Markov chain
1. 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 multiCorresponding author. Tel.: +33 2 99 84 45 41;
fax: +33 2 99 84 45 99. E-mail addresses: [email protected] (J. Palicot), [email protected] (A. Goupil).
media 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
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.07.021
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295
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. 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 decision feedback equalizers (DFEs) are gaining importance. In fact, the latter offers the best compromise between performance and complexity. DFEs are well known for their superior performances 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 [1], 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 m s) [2] 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
285
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 [3]. In this work the author proposed a new decision device based on Bayesian analysis. A similar work has been proposed more recently in [4]. At the same time the author of [5] 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 [5] is that, the weighted decisions are also used for the tracking phase of the algorithm which is not the case in [4,6]. 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 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
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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 [5], 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 (LMSDD). 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 (WDFE), offers the advantage of limiting the error propagation phenomenon. The equalizer proposed in [5] is the classical DFE, to which we add two devices as shown in Fig. 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 DFE. 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 [5] 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 [13,14]. To obtain some k
wk sk
H (z)
+
rk
F (z)
xk
+
zk
Decision and ˆzk Reliability 1 − B (z)
Fig. 1. WDFE scheme.
˜zk
Use
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 recovery, 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. 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 [5,12]. 2. WDFE presentation 2.1. General description Overall scheme of the channel with the weighted version of the DFE is shown in Fig. 1. The notations used in this figure and throughout this paper are the following: sk is the source symbol sequence, HðzÞ the channel transfer function, wk the additive white Gaussian noise, rk the received sequence, F ðzÞ the feed forward filter, 1 BðzÞ the feedback filter, zk the WDFE’s output sequence, and gk the reliability of zk and z~k is the feedback symbols. The difference between the WDFE and the classical DFE is simply the addition of two new devices.
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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 gI and gQ be these reliabilities, then the output of the decision device of the WDFE is
tion for which the decision domain of a symbol is shown in Fig. 2. The point z is the input of the device, and z^ represents the hard decision. The distances d x or y give the distance between the border of the domain, and D is the ‘‘radius’’ of the domain. Given this value, the reliability is given by
z~Ik ¼ f ðgIk Þ^zIk þ ð1 f ðgIk ÞÞzIk ,
ð1Þ
g¼1
Q Q Q zk þ ð1 f ðgQ z~Q k ¼ f ðgk Þ^ k ÞÞzk ,
ð2Þ
where f ðÞ 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 DFE appears clearly in Eqs. (1) and (2). Indeed, when f ðgÞ is equal to 1 the WDFE acts as a classical DFE, whereas when f ðgÞ 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 ¼ f ðgIk Þð^zI zI Þ,
ð3Þ
eQ k
ð4Þ
¼
f ðgQ zQ k Þð^
Q
z Þ.
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.
þ minðdþ x ; dx ; dy ; dy Þ
. (5) D 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 Eq. (5) in the following form: g¼
kz z^k . D
(6)
Moreover, we can also decompose the reliability on the axis of the QAM. We then obtain two reliabilities, which are given by minðdþ x ; dx Þ , D minðdþ y ; dy Þ gQ ¼ . D
gI ¼
ð7Þ ð8Þ
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
+
y
x-
+x
z
Δ -
ˆz
y
2.2. 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 modula-
Fig. 2. Notations for the QAM symbol decision region.
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1 z d g (x)
0.5
Δ
0
b=0.5, a=5
zˆ
a=5 a = 50
-0.5 dmin
a = 10
-1 -1
-0.5
0
0.5
1
x
Fig. 3. Rule 1 scheme.
Fig. 4. Sigmoid.
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. 2.3. Reliability use (1) Rule 1 scheme: This technique is fully described in [5], 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 d min . In this case, the function f is 1 if xod min ; (9) f ðxÞ ¼ 0 otherwise: The graphical representation of this rule is given in Fig. 3 for a QAM. If the input of the WDFE device lies inside the square of ‘‘radius’’ d min 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, Eqs. (1)–(4) become: z~k ¼ gk z^k þ ð1 gk Þzk ,
ð10Þ
ek ¼ gk ð^z zÞ.
ð11Þ
Consequently, if g ¼ 1 we obtain a classical DFE and if g ¼ 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 g 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 Eq. (12) is shown in Fig. 4. To meet our requirements, we modify the basic sigmoid function with a compression law. The result is given in Eq. (13) and is shown in Fig. 5 for the compression parameters b ¼ 0:5 and a ¼ 5: 1 expðaxÞ , 1 þ expðaxÞ 1 1 expðaððx=bÞ 1ÞÞ þ1 . gðxÞ ¼ 2 1 þ expðaððx=bÞ 1ÞÞ
gðxÞ ¼
ð12Þ ð13Þ
In these conditions, by substituting f ðgÞ by gðgÞ in Eqs. (1)–(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 Eq. (13), with a ¼ 1 and d min ¼ 1 b, the present rule becomes equivalent to rule-1 and this is also evident from Figs. 4 and 6, for b ¼ 0:5. In Fig. 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
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1
1
DFE
0.8
0.8
0.6
0.6
b = 0.2
J (e)
g (x)
Rule 1(dmin = 0.8)
b = 0.5
0.4
0.4
0.2
0.2
Rule 1 (dmin = 0.5)
Rule 2
Sigmoid (b = 0.5)
0
0 0
0.2
0.6
0.4
0.8
1
0.2
0
0.4
0.6
0.8
1
e/Δ
x
Fig. 7. Error functions Jð^eÞ.
Fig. 5. Compressed sigmoid.
Keeping in mind that the reliability g is connected to the error by relation
1 Sigmoid (b = 0.5)
gðpÞ k ¼ 1
0.8 Rule 2 g (x)
Sigmoid (b = 0.2)
0.4
Jð^eÞ ¼ jg^ej.
Rule 1
0 0
0.2
0.4
0.6
0.8
1
x Fig. 6. f functions comparison.
compromise between the two rules previously defined in [5,12]. 2.4. Study of the error function Jðek Þ Let ½Fk ; Bk be the coefficients of the filters F ðzÞ and BðzÞ at time k. With respect to the use of reliability for the LMS-DD algorithm given in Eqs. (3) and (4), the adaptation is done using ~ k , (14) ½Fkþ1 ; Bkþ1 ¼ ½Fk ; Bk mf ðgk Þ^ek ½Rk ; Z ~ k being symbols in the memory of the with Rk and Z forward and backward filters, and m the step of the algorithm. The error e^k is the classical DD error given by e^k ¼ z^k zk .
(16)
where ðpÞ 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
0.6
0.2
je^ðpÞ k j , D
(15)
(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 g ¼ 1 whatever the error is, we can compare the WDFE algorithm with the classical DFE. In Fig. 7, Jð^eÞ 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 toward 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 toward the same limit when the error tends toward zero, or, equivalently, when the reliability tends toward 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 toward 1 the WDFE error function decreases, whereas the classical DFE error function increases, consequently the tracking behavior is performed in a wiser way.
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We can easily note that, for each iteration in Eq. (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. 3. Filtering analysis of the WDFE 3.1. 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 z~k ¼ Uðzk Þ,
(18)
where U is the soft decision function and because of this function we can restrict our analysis to the softdecision DFE. In fact, in order to compare the WDFE and the DFE, we can simply study the effect of the function U on the performances, because the DFE model is given by considering U to be the hard decision function. Using the notation of Fig. 1, the input/output relation of the WDFE is given by X X zk ¼ f i rki bi z~ki , (19) i
zk ¼
i40
XX i
j
f j hij ski þ
X i
f i wki
X
bi z~ki ,
i40
(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 ¼ zk sk . It serves to compute the MSE of the output. The second is the error of decision e^k ¼ z^k 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 e~k ¼ z~k 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 X X X ti ski þ f i wki bi e~ki , ð21Þ ek ¼ i
¼ nk
X
i
i40
bi e~ki ,
ð22Þ
i40
where ti is ith coefficient of the impulse response of the combined filter TðzÞ ¼ HðzÞF ðzÞ BðzÞ 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 Eq. (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 e~k ¼ V ðek Þ. 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’’ e~k from the real errors ek , considering that the modulation alphabet is infinite. With the introduction of the function V , we can write ! X (23) e~k ¼ V nk bi e~ki . i40
As the length of the feedback filter is finite, the error sequence e~k follows a Markov process whose order is given by the length of B. It is worth noting that the study of e~k is enough to obtain most of the descriptive parameters of the error process. It is possible to obtain the information about ek from e~k because of Eq. (23). Since e~k is more general than e^k , we get this process through the relation: e^k ¼ Decð~ek Þ. 3.2. Markovian model of the error probability density for DFEs The input/output relation of the ‘‘soft errors’’ of the WDFE is given by ! LB X bi e~ki , (24) e~k ¼ V nk j¼1
where LB is the length of the feedback filter BðzÞ. In order to compute the probability density function of ek we should consider the possible value of e~k . Eq. (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
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resolved easily. We propose simplifying the study by making an approximation of the soft error function V ðÞ. The approximation is obtained using a staircase function: V ðxÞ ¼ vi
for i 2 ½ai ; aiþ1 .
(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 ~ k . Of course, the way in which approximavector E tion is done, affects the quality of the analysis. Let us denote Pr½k 2 E~ i as the probability to be in the state E~ i at time k. The Markov aspect of the error process allows these probabilities to be linked easily by X ~j ! E ~ i Pr½k 2 E ~ j , Pr½k þ 1 2 E~ i ¼ Pr½E (26) j
~ i is the probability of the transiwhere Pr½E~ j ! E ~ j to state E~ i . It should be noted tion between state E 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 f2; 0; þ2g. But the error þ2 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 Eq. (26) corresponds to a multiplication between a matrix and a vector. We denote then PðkÞ as the stack composed of the state probability at time k and Q the matrix whose i; j ~j ! E ~ i . The entry is the transition probability Pr½E time relation between the states is then expressed recursively by the simple matrix multiplication Pðkþ1Þ ¼ QPðkÞ .
(27)
Although the relation in Eq. (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 Pð1Þ ¼ QPð1Þ .
(28)
As the Markov chain is regular, the vector Pð1Þ is the eigenvector of Q associated with the eigenvalue 1, which exists because Q is a stochastic matrix.
291
The approximation in Eq. (25) leads us to the steady-state relation of Eq. (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: e~k
X
T~ 2 Pð1Þ i NðB Ei ; sN Þ,
(29)
i
where Nðm; s2 Þ denotes the Gaussian distribution with mean m and variance s2 , and s2N is the variance of nk from Eq. (22). The Markov chain of the WDFE needs a lot of computation. If the function V ðÞ is approximated by a step function with N steps then the number of states is LN B . Then the dimensions of transition N matrix Q are: LN B LB . Computing the relevant eigenvector of Q may be intractable. But the convergence of Eq. (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. 3.3. 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 F i be the classes which form a partition of the set of states E j , and let R denotes the partitioning matrix defined by 1 if E j belongs to the class F i ; (30) Rij ¼ 0 otherwise:
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The Bayes rule gives us the transition matrix P between the classes during the steady state, which is given by (31)
P ¼ RQC,
where C can be viewed as given the ‘‘weight’’ of the state E i in the class F j . It can be defined as: Pr½k 2 E i . Cij ¼ Rij lim P k!1 E 2F j Pr½k 2 E l
(32)
l
The Markov chain reduction can also be used to compute the probability Pr½F i 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 ¼ Pr½Fe .
(33)
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 T N 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 Fig. 8. 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 nonoptimal. 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 ¼
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 Fig. 8 with the transition probability a ¼ 1. Then Pr½X ¼ k ¼ ð1 Ppe ÞPk1 pe , which is a geometric distribution whose mean is given by Eq. (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 1 − Ppe a
O 1−a
Fig. 8. Reduction into two classes.
Ppe
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295
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, reduction in the number of states in the Markov chain. The Markov aspect occurs here once again:
ð36Þ ð37Þ
The last equation allows us to focus on a length lower than LB because, for longer burst, we obtain the following: Pr½DB 4l4LB ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kLB a0lLB Pr½DB 4LB .
ð38Þ
Of course, the method of computing the probability Pr½DB 4l for loLB uses the reduction of the Markov chain. The class F i for i 2 f0; . . . ; LB g corresponds to the pattern E; . . . ; E ; 0; X . . . . |fflfflfflfflffl{zfflfflfflfflffl}
(39)
i times
For each class F i the possible transitions are F iþ1 if an error is made or F 0 if a correct decision is made. In the latter case, the burst is finished. Moreover the state F LB is 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 Pr½DB 4l ¼ uT0 Qul ,
0.2 0.1 Symbol error rate
Pr½DB 4l 1.
ð35Þ
To validate the model, a comparison between a Monte Carlo simulation and the model has been done as shown in Fig. 9, with a channel set to H ¼ ½1; 0:2; 0:2 and a backward filter set to B ¼ ½0:24; 0:24. The equalizer does not use a feedforward filter. The WDFE uses the rule-2 approximated with 16 stairs for a BPSK. Fig. 10 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 [5,12]. We choose a feedback length of 2 in order to compute the time recovery for both the DFE and
0.05 Simulation
Model
0.02 ISI free bound
0.01 0.005 0
2
4
6
8
SNR (dB) Fig. 9. Model validation for the WDFE rule 2.
(40)
where ui denotes the vector whose elements are 0 but the i þ 1th is 1. We consider that the matrix Q is constructed assuming that the class i position is in the i þ 1 vector position. Eq. (40) gives us all the information about burst length distribution as Pr½DB ¼ l. 4. 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.
0.64 Error propagation probability
Pr½DB 4l ¼ Pr½^ek a0; e^k1 a0; . . . ; e^kl a0, ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kl a0 Pr½^ek1 a0; . . . ; e^kl a0, ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kl a0
293
0.62 0.6 0.58 DFE
0.56 0.54
WDFE
0.52 0
2
4 SNR (dB)
6
8
Fig. 10. Error propagation probability for the DFE and the WDFE.
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0.6
3.5 DFE
3.4
Pr [B = l]
Recovery time
3.6
3.3
0.4
0.2
3.2
WDFE rule 1
DFE
3.1
0 12
10
14
16
WDFE rule1 2
18
SNR
6 Length
8
10
Fig. 13. Error burst distribution.
Fig. 11. Recovery time.
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 ½1; 0:6; 0:1. The results are still very good as shown in Fig. 13.
0.6
Pr [B = l]
4
0.4
0.2
5. Conclusion
DFE 0
WDFE rule1 2
4
6
8
Length Fig. 12. Error burst distribution.
the WDFE. The channel and feedback coefficients were H ¼ B ¼ ½1; 0:6; 0:1. The time recovery plots for the two equalizers are shown in Fig. 11. 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 cannot 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 Fig. 12. In the latter simulation, the channel coefficients were ½1; 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
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] D.L. Duttweiler, J.E. Mazo, D.G. Messerschmitt, An upper bound on the error probability in decision-feedback equalization, IEEE Trans. Inform. Theory IT-20 (4) (July 1974) 490–497. [2] R. Hopkins, Digital terrestrial HDTV for North America: the grand alliance HDTV system, IEEE Trans. Consum. Electron. 40 (3) (August 1994) 185–198.
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295 [3] D.P. Taylor, The estimate feedback equalizer: a suboptimum nonlinear receiver, IEEE Trans. Commun. 21 (9) (September 1973) 979–990. [4] J. Balakrishna, H. Viswanathan, 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, March 2000. [5] J. Palicot, A weighted decision feedback equalizer with limited error propagation, in: Proceedings of ICC’00, New Orleans, USA, Mexico, June 2000. [6] M. Reuter, J.C. Allen, J.R. Zeidler, R.C. North, Mitigating error propagation effects in a decision feedback equalizer, IEEE Trans. Commun. 49 (11) (November 2001) 2028–2041. [7] A. Duel-hallen, C. Heegard, Delayed decision-feedback sequence estimation, IEEE Trans. Commun. 37 (5) (May 1989) 428–436. [8] H. Besbes, M. Jaidane-Saidane, J. Ezzine, On exact convergence results of adaptive filters: the finite alphabet case, Signal Processing 80 (7) (July 2000) 1373–1384. [9] A. Goupil, J. Palicot, Maximum likelihood-like improvement of the output sequence SIMO decision feedback equalizer, in: SUMMIT’04, June 2004. [10] J. Labat, O. Macchi, C. Laot, Adaptive decision feedback equalization: can you skip the training period?, IEEE Trans. Commun. 46 (7) (July 1998) 921–930. [11] K. Berberidis, A. Marava, P. Karaivazoglou, J. Palicot, Robust and fast converging decision feedback equalizer
[12]
[13]
[14]
[15]
[16]
[17]
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based on a new adaptive semi-blind channel estimation algorithm, in: Proceedings of GLOBECOM’01, San Antonio, Texas, November 2001. J. Palicot, C. Roland, Improvement of the WDFE performance thanks to a non linear amplification of the reliability function, in: Proceedings of ICT’01, Bucarest, Roumania, June 2001. C. Lu¨tkemeyer, 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. T.J. Willink, P.H. Wittke, L.L. Campbell, Evaluation of the effect of intersymbol interference in decision-feedback equalizer, IEEE Trans. Commun. 48 (4) (April 2000) 629–635. N.C. Beaulieu, Bounds on recovery times of decision feedback equalizers, IEEE Trans. Commun. 42 (10) (October 1994) 2786–2794. A. Goupil, 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. M. Jin, B. Farhang-Boroujeny, G. Mathew, K.C. Indukumar, A novel fast approach for estimating error propagation in decision feedback detectors, IEEE J. Select. Areas Commun. 19 (4) (April 2001) 668–676.
Signal Processing 88 (2008) 284–295 www.elsevier.com/locate/sigpro
Performance analysis of the weighted decision feedback equalizer Jacques Palicota,, Alban Goupilb a
Supelec, avenue de la boulaie, BP 81127, 35511 Cesson-Se´vigne´, France De´com, Universite´ de Reims, UFR Sciences, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France
b
Received 28 October 2006; received in revised form 25 June 2007; accepted 27 July 2007 Available online 15 August 2007
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. r 2007 Elsevier B.V. All rights reserved. Keywords: Equalization; Decision feedback equalizer; Weighted decisions; Error propagation; Markov chain
1. 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 multiCorresponding author. Tel.: +33 2 99 84 45 41;
fax: +33 2 99 84 45 99. E-mail addresses: [email protected] (J. Palicot), [email protected] (A. Goupil).
media 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
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.07.021
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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. 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 decision feedback equalizers (DFEs) are gaining importance. In fact, the latter offers the best compromise between performance and complexity. DFEs are well known for their superior performances 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 [1], 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 m s) [2] 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
285
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 [3]. In this work the author proposed a new decision device based on Bayesian analysis. A similar work has been proposed more recently in [4]. At the same time the author of [5] 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 [5] is that, the weighted decisions are also used for the tracking phase of the algorithm which is not the case in [4,6]. 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 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
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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 [5], 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 (LMSDD). 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 (WDFE), offers the advantage of limiting the error propagation phenomenon. The equalizer proposed in [5] is the classical DFE, to which we add two devices as shown in Fig. 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 DFE. 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 [5] 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 [13,14]. To obtain some k
wk sk
H (z)
+
rk
F (z)
xk
+
zk
Decision and ˆzk Reliability 1 − B (z)
Fig. 1. WDFE scheme.
˜zk
Use
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 recovery, 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. 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 [5,12]. 2. WDFE presentation 2.1. General description Overall scheme of the channel with the weighted version of the DFE is shown in Fig. 1. The notations used in this figure and throughout this paper are the following: sk is the source symbol sequence, HðzÞ the channel transfer function, wk the additive white Gaussian noise, rk the received sequence, F ðzÞ the feed forward filter, 1 BðzÞ the feedback filter, zk the WDFE’s output sequence, and gk the reliability of zk and z~k is the feedback symbols. The difference between the WDFE and the classical DFE is simply the addition of two new devices.
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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 gI and gQ be these reliabilities, then the output of the decision device of the WDFE is
tion for which the decision domain of a symbol is shown in Fig. 2. The point z is the input of the device, and z^ represents the hard decision. The distances d x or y give the distance between the border of the domain, and D is the ‘‘radius’’ of the domain. Given this value, the reliability is given by
z~Ik ¼ f ðgIk Þ^zIk þ ð1 f ðgIk ÞÞzIk ,
ð1Þ
g¼1
Q Q Q zk þ ð1 f ðgQ z~Q k ¼ f ðgk Þ^ k ÞÞzk ,
ð2Þ
where f ðÞ 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 DFE appears clearly in Eqs. (1) and (2). Indeed, when f ðgÞ is equal to 1 the WDFE acts as a classical DFE, whereas when f ðgÞ 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 ¼ f ðgIk Þð^zI zI Þ,
ð3Þ
eQ k
ð4Þ
¼
f ðgQ zQ k Þð^
Q
z Þ.
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.
þ minðdþ x ; dx ; dy ; dy Þ
. (5) D 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 Eq. (5) in the following form: g¼
kz z^k . D
(6)
Moreover, we can also decompose the reliability on the axis of the QAM. We then obtain two reliabilities, which are given by minðdþ x ; dx Þ , D minðdþ y ; dy Þ gQ ¼ . D
gI ¼
ð7Þ ð8Þ
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
+
y
x-
+x
z
Δ -
ˆz
y
2.2. 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 modula-
Fig. 2. Notations for the QAM symbol decision region.
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1 z d g (x)
0.5
Δ
0
b=0.5, a=5
zˆ
a=5 a = 50
-0.5 dmin
a = 10
-1 -1
-0.5
0
0.5
1
x
Fig. 3. Rule 1 scheme.
Fig. 4. Sigmoid.
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. 2.3. Reliability use (1) Rule 1 scheme: This technique is fully described in [5], 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 d min . In this case, the function f is 1 if xod min ; (9) f ðxÞ ¼ 0 otherwise: The graphical representation of this rule is given in Fig. 3 for a QAM. If the input of the WDFE device lies inside the square of ‘‘radius’’ d min 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, Eqs. (1)–(4) become: z~k ¼ gk z^k þ ð1 gk Þzk ,
ð10Þ
ek ¼ gk ð^z zÞ.
ð11Þ
Consequently, if g ¼ 1 we obtain a classical DFE and if g ¼ 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 g 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 Eq. (12) is shown in Fig. 4. To meet our requirements, we modify the basic sigmoid function with a compression law. The result is given in Eq. (13) and is shown in Fig. 5 for the compression parameters b ¼ 0:5 and a ¼ 5: 1 expðaxÞ , 1 þ expðaxÞ 1 1 expðaððx=bÞ 1ÞÞ þ1 . gðxÞ ¼ 2 1 þ expðaððx=bÞ 1ÞÞ
gðxÞ ¼
ð12Þ ð13Þ
In these conditions, by substituting f ðgÞ by gðgÞ in Eqs. (1)–(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 Eq. (13), with a ¼ 1 and d min ¼ 1 b, the present rule becomes equivalent to rule-1 and this is also evident from Figs. 4 and 6, for b ¼ 0:5. In Fig. 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
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1
1
DFE
0.8
0.8
0.6
0.6
b = 0.2
J (e)
g (x)
Rule 1(dmin = 0.8)
b = 0.5
0.4
0.4
0.2
0.2
Rule 1 (dmin = 0.5)
Rule 2
Sigmoid (b = 0.5)
0
0 0
0.2
0.6
0.4
0.8
1
0.2
0
0.4
0.6
0.8
1
e/Δ
x
Fig. 7. Error functions Jð^eÞ.
Fig. 5. Compressed sigmoid.
Keeping in mind that the reliability g is connected to the error by relation
1 Sigmoid (b = 0.5)
gðpÞ k ¼ 1
0.8 Rule 2 g (x)
Sigmoid (b = 0.2)
0.4
Jð^eÞ ¼ jg^ej.
Rule 1
0 0
0.2
0.4
0.6
0.8
1
x Fig. 6. f functions comparison.
compromise between the two rules previously defined in [5,12]. 2.4. Study of the error function Jðek Þ Let ½Fk ; Bk be the coefficients of the filters F ðzÞ and BðzÞ at time k. With respect to the use of reliability for the LMS-DD algorithm given in Eqs. (3) and (4), the adaptation is done using ~ k , (14) ½Fkþ1 ; Bkþ1 ¼ ½Fk ; Bk mf ðgk Þ^ek ½Rk ; Z ~ k being symbols in the memory of the with Rk and Z forward and backward filters, and m the step of the algorithm. The error e^k is the classical DD error given by e^k ¼ z^k zk .
(16)
where ðpÞ 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
0.6
0.2
je^ðpÞ k j , D
(15)
(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 g ¼ 1 whatever the error is, we can compare the WDFE algorithm with the classical DFE. In Fig. 7, Jð^eÞ 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 toward 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 toward the same limit when the error tends toward zero, or, equivalently, when the reliability tends toward 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 toward 1 the WDFE error function decreases, whereas the classical DFE error function increases, consequently the tracking behavior is performed in a wiser way.
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290
We can easily note that, for each iteration in Eq. (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. 3. Filtering analysis of the WDFE 3.1. 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 z~k ¼ Uðzk Þ,
(18)
where U is the soft decision function and because of this function we can restrict our analysis to the softdecision DFE. In fact, in order to compare the WDFE and the DFE, we can simply study the effect of the function U on the performances, because the DFE model is given by considering U to be the hard decision function. Using the notation of Fig. 1, the input/output relation of the WDFE is given by X X zk ¼ f i rki bi z~ki , (19) i
zk ¼
i40
XX i
j
f j hij ski þ
X i
f i wki
X
bi z~ki ,
i40
(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 ¼ zk sk . It serves to compute the MSE of the output. The second is the error of decision e^k ¼ z^k 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 e~k ¼ z~k 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 X X X ti ski þ f i wki bi e~ki , ð21Þ ek ¼ i
¼ nk
X
i
i40
bi e~ki ,
ð22Þ
i40
where ti is ith coefficient of the impulse response of the combined filter TðzÞ ¼ HðzÞF ðzÞ BðzÞ 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 Eq. (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 e~k ¼ V ðek Þ. 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’’ e~k from the real errors ek , considering that the modulation alphabet is infinite. With the introduction of the function V , we can write ! X (23) e~k ¼ V nk bi e~ki . i40
As the length of the feedback filter is finite, the error sequence e~k follows a Markov process whose order is given by the length of B. It is worth noting that the study of e~k is enough to obtain most of the descriptive parameters of the error process. It is possible to obtain the information about ek from e~k because of Eq. (23). Since e~k is more general than e^k , we get this process through the relation: e^k ¼ Decð~ek Þ. 3.2. Markovian model of the error probability density for DFEs The input/output relation of the ‘‘soft errors’’ of the WDFE is given by ! LB X bi e~ki , (24) e~k ¼ V nk j¼1
where LB is the length of the feedback filter BðzÞ. In order to compute the probability density function of ek we should consider the possible value of e~k . Eq. (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
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295
resolved easily. We propose simplifying the study by making an approximation of the soft error function V ðÞ. The approximation is obtained using a staircase function: V ðxÞ ¼ vi
for i 2 ½ai ; aiþ1 .
(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 ~ k . Of course, the way in which approximavector E tion is done, affects the quality of the analysis. Let us denote Pr½k 2 E~ i as the probability to be in the state E~ i at time k. The Markov aspect of the error process allows these probabilities to be linked easily by X ~j ! E ~ i Pr½k 2 E ~ j , Pr½k þ 1 2 E~ i ¼ Pr½E (26) j
~ i is the probability of the transiwhere Pr½E~ j ! E ~ j to state E~ i . It should be noted tion between state E 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 f2; 0; þ2g. But the error þ2 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 Eq. (26) corresponds to a multiplication between a matrix and a vector. We denote then PðkÞ as the stack composed of the state probability at time k and Q the matrix whose i; j ~j ! E ~ i . The entry is the transition probability Pr½E time relation between the states is then expressed recursively by the simple matrix multiplication Pðkþ1Þ ¼ QPðkÞ .
(27)
Although the relation in Eq. (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 Pð1Þ ¼ QPð1Þ .
(28)
As the Markov chain is regular, the vector Pð1Þ is the eigenvector of Q associated with the eigenvalue 1, which exists because Q is a stochastic matrix.
291
The approximation in Eq. (25) leads us to the steady-state relation of Eq. (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: e~k
X
T~ 2 Pð1Þ i NðB Ei ; sN Þ,
(29)
i
where Nðm; s2 Þ denotes the Gaussian distribution with mean m and variance s2 , and s2N is the variance of nk from Eq. (22). The Markov chain of the WDFE needs a lot of computation. If the function V ðÞ is approximated by a step function with N steps then the number of states is LN B . Then the dimensions of transition N matrix Q are: LN B LB . Computing the relevant eigenvector of Q may be intractable. But the convergence of Eq. (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. 3.3. 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 F i be the classes which form a partition of the set of states E j , and let R denotes the partitioning matrix defined by 1 if E j belongs to the class F i ; (30) Rij ¼ 0 otherwise:
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The Bayes rule gives us the transition matrix P between the classes during the steady state, which is given by (31)
P ¼ RQC,
where C can be viewed as given the ‘‘weight’’ of the state E i in the class F j . It can be defined as: Pr½k 2 E i . Cij ¼ Rij lim P k!1 E 2F j Pr½k 2 E l
(32)
l
The Markov chain reduction can also be used to compute the probability Pr½F i 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 ¼ Pr½Fe .
(33)
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 T N 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 Fig. 8. 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 nonoptimal. 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 ¼
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 Fig. 8 with the transition probability a ¼ 1. Then Pr½X ¼ k ¼ ð1 Ppe ÞPk1 pe , which is a geometric distribution whose mean is given by Eq. (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 1 − Ppe a
O 1−a
Fig. 8. Reduction into two classes.
Ppe
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295
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, reduction in the number of states in the Markov chain. The Markov aspect occurs here once again:
ð36Þ ð37Þ
The last equation allows us to focus on a length lower than LB because, for longer burst, we obtain the following: Pr½DB 4l4LB ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kLB a0lLB Pr½DB 4LB .
ð38Þ
Of course, the method of computing the probability Pr½DB 4l for loLB uses the reduction of the Markov chain. The class F i for i 2 f0; . . . ; LB g corresponds to the pattern E; . . . ; E ; 0; X . . . . |fflfflfflfflffl{zfflfflfflfflffl}
(39)
i times
For each class F i the possible transitions are F iþ1 if an error is made or F 0 if a correct decision is made. In the latter case, the burst is finished. Moreover the state F LB is 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 Pr½DB 4l ¼ uT0 Qul ,
0.2 0.1 Symbol error rate
Pr½DB 4l 1.
ð35Þ
To validate the model, a comparison between a Monte Carlo simulation and the model has been done as shown in Fig. 9, with a channel set to H ¼ ½1; 0:2; 0:2 and a backward filter set to B ¼ ½0:24; 0:24. The equalizer does not use a feedforward filter. The WDFE uses the rule-2 approximated with 16 stairs for a BPSK. Fig. 10 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 [5,12]. We choose a feedback length of 2 in order to compute the time recovery for both the DFE and
0.05 Simulation
Model
0.02 ISI free bound
0.01 0.005 0
2
4
6
8
SNR (dB) Fig. 9. Model validation for the WDFE rule 2.
(40)
where ui denotes the vector whose elements are 0 but the i þ 1th is 1. We consider that the matrix Q is constructed assuming that the class i position is in the i þ 1 vector position. Eq. (40) gives us all the information about burst length distribution as Pr½DB ¼ l. 4. 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.
0.64 Error propagation probability
Pr½DB 4l ¼ Pr½^ek a0; e^k1 a0; . . . ; e^kl a0, ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kl a0 Pr½^ek1 a0; . . . ; e^kl a0, ¼ Pr½^ek a0j^ek1 a0; . . . ; e^kl a0
293
0.62 0.6 0.58 DFE
0.56 0.54
WDFE
0.52 0
2
4 SNR (dB)
6
8
Fig. 10. Error propagation probability for the DFE and the WDFE.
ARTICLE IN PRESS J. Palicot, A. Goupil / Signal Processing 88 (2008) 284–295
294
0.6
3.5 DFE
3.4
Pr [B = l]
Recovery time
3.6
3.3
0.4
0.2
3.2
WDFE rule 1
DFE
3.1
0 12
10
14
16
WDFE rule1 2
18
SNR
6 Length
8
10
Fig. 13. Error burst distribution.
Fig. 11. Recovery time.
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 ½1; 0:6; 0:1. The results are still very good as shown in Fig. 13.
0.6
Pr [B = l]
4
0.4
0.2
5. Conclusion
DFE 0
WDFE rule1 2
4
6
8
Length Fig. 12. Error burst distribution.
the WDFE. The channel and feedback coefficients were H ¼ B ¼ ½1; 0:6; 0:1. The time recovery plots for the two equalizers are shown in Fig. 11. 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 cannot 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 Fig. 12. In the latter simulation, the channel coefficients were ½1; 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
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.
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