An efficient blind decision feedback equalizer
1
An efficient blind decision feedback equalizer Alban Goupil and Jacques Palicot
Abstract Efficient and fully blind Decision Feedback Equalizer (DFE) remains an open issue, mainly because of the potential errors in the decision loop. Based on the Weighted Decision Feedback Equalizer (WDFE), our previous work aiming at decreasing the error propagation phenomena, we propose a new blind DFE called Blind Weighted Decision Feedback Equalizer. The main idea of the WDFE was to commute between Linear Recursive Equalizer (LRE) and DFE for the error computation at the decision device. In this paper, we extend this idea in order to commute also both the algorithms and the filtering structure. This commutation is performed softly and blindly.
I. I NTRODUCTION At first sight, the drawbacks of the DFE, for example error propagation, make it a poor candidate for blind equalization, although its performance when optimal is particularly attractive. Extending our previous work Weighted Decision Feedback Equalizer [1] we propose here a new fully blind DFE: the Blind-Weigthed Decision Feedback Equalizer (B-WDFE). The main idea is to commute softly and blindly the algorithms, the filtering structure and the decision device between the two extremes cases: the LRE and the DFE. This should be performed under the following constraints, in order to achieve our objective: •
No structure change—i.e. all the parts of the equalizer are used equally for the convergence of the algorithm and for the equalization of the channel once converged. This avoids having one equalizer that opens the eye in the first stage and then another that derives from it once the decisions are sufficiently certain, as in [3].
•
No sudden change in the cost functions—i.e. the equalizer must adapt itself to achieve the best possible correspondence between the convergence phase and the channel variation tracking phase, without changing the cost functions to be optimized.
•
The equalizer must be as simple as possible in its architecture. Thus sample-by-sample algorithms and structures based on filtering are considered positively.
In Section II we first describe the filtering structure of our equalizer then in Section III we explain the proposed algorithm, which softly commutes between a blind algorithm and the Decision Directed (DD) algorithm. We prove the efficiency of our B-WDFE thanks to simulation presented in Section IV and finally we conclude the paper. A. Goupil is with the University of Reims Champagne-Ardenne, France J. Palicot is with SUPELEC/IETR, France
January 22, 2010
DRAFT
2
II. F ILTERING LRE and DFE are classical equalizers whose structure are well-studied. LRE is composed of an automatic gain control (AGC), a feedforward filter (FFF) and a feedback filter (FBF). DFE shares the same components as LRE but it also includes a decision device. This device is classically put in front of the FBF in order to improve the equalizer output when the decisions are correct. Under the hypothesis of correct decision and for the minimum mean square error (MMSE) criterion, the optimal coefficients of the filters of the DFE and the LRE are equal [6]. This fact is the bases of the blind equalizer of Labat et al. [3] (SA-DFE): once the convergence of the filter of the LRE is achieved, the structure of the LRE switches into the structure of the DFE while the filters’ coefficients are kept. In this way, LRE structure is used during the convergence phase of the adaptation and DFE is used during the tracking phase providing good performances and fast convergence. The equalizer presented in this paper, named B-WDFE, deals with the same idea of switching between the LRE and the DFE. But this process must be done smoothly. In this way, there is no need to compute a threshold as in the case of SA-DFE, and no hysteresis phenomenon would appear. This objective is accomplished by using the DFE structure of Belfiore and Park [4] which is shown in figure 1. This way to address the structure switching problem of the SA-DFE was also proposed in [7].
R
×
U
+
g
F(z)
X
+
Z
1 − B(z)
φ (·) 1 − B(z)
Z˜ E˜
+
+ −
Fig. 1.
Blind LRE/DFE structure.
The coefficient g represents the AGC, and F (z) is the FFF. B(z) is a monic polynomial whose coefficients are used twice in the structure and represent the FBF part of the equalizer. The function φ(·) concerns the soft decision. Its choice permits to switch between the LRE and the DFE. Indeed, if φ(z) = z, the complete equalizer performs as a LRE whose FFF is F (z) and the FBF is B(z). When φ(z) is the decision device of the constellation, the equalizer becomes the DFE. Choosing the function φ(·) between this two extreme cases results in an equalizer which behaves between LRE and DFE. The structure of our equalizer is thus static and the adaptation will concern not only the filters’ coefficients but also the soft decision device φ(·) depending on the input and of the convergence state. In order to keep the equalizer simple, the chosen soft decision device is a convex combination of the decision device and the identity: φ(z) = (1 − α) z + α dec(z),
(1)
where dec(·) is the decision and α is a proportion parameter that allows to choose between the LRE and the DFE when α = 0 and α = 1 respectively. This parameter will be subject to adaptation also. January 22, 2010
DRAFT
3
III. A DAPTATION As the extreme cases of the equalizer’s structure are the DFE and the LRE, the adaptation will be also inbetween these two cases. By choosing the soft decision (1), the adaptation will be also a convex combination of the adaptation for the LRE and for the DFE. Thus we describe first the adaption algorithm in the LRE case and then the adaptation in the DFE case followed by the complete algorithm, that is the convex combination of the previous cases. For convenience, the algorithms used thereafter are based on the LMS algorithm. All step sizes needed by the algorithms are noted by the Greek letter µ with subscripts and superscripts that indicate the adapted component. The algorithms are described using the notation of the figure 1. The time index is noted k. Thus, the k-th sample of the signal U is given by uk . If several sample of this signal are needed, we note Uk the column vector [uk , uk−1 , . . .]T . The length of the vectors are given by the context. The coefficients of the FFF F (z) is given by F which is a column vector. Transposition of F is noted by FT . The conjugate of a complex number z whose real and imaginary parts are Re z and − Im z respectively, is noted z ∗ . A. LRE Case In the LRE case, the adaptation concerns the AGC, the FFF and the FBF. Each component is adapted according to its own criterion. Indeed, the equalizer performs like the LRE if the steady-state is not reached. The purpose of the AGC is to adjust the power of the signal to unity. The algorithm used is classical: p 2 G ← G + µG 1 − |uk | and g ← |G|,
(2)
where µG is the adaptation step of this algorithm. The FBF is adapted in order to converge to the whitening filter of the input signal. The criterion concerns the maximization of the power of its output. But this filter is recursive and thus the LMS algorithm is not easily derived. Macchi in [5] proposed and studied the stability of the following algorithm. ∗ BLRE ← BLRE + µLRE B uk Uk−1 .
(3)
The FFF is adapted using the CQA, a variant of the Constant Modulus Algorithm (CMA), presented in [2] which permits to recover the phase of the constellation: FLRE ← FLRE − µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k , F
(4)
where kxk∞ = max(|Re x|, |Im x|) and p(x) is the gradient of kxk∞ . The constant γ depends only on the constellation. B. DFE Case If the convergence is almost done, tracking phase begins, the filters are close to the optimal performance and the decisions are mostly correct. Thus the equalizer must be close to the DFE and the criterion used to adapt
January 22, 2010
DRAFT
4
is directed by the decision. That is the criterion is the MMSE and its estimation is based on the estimated error eˆk = dec(zk ) − zk . The FBF and FFF are thus adapted with the LMS algorithms: FDFE ← FDFE − µDFE eˆk U∗k F
(5)
˜ ∗k BDFE ← BDFE − µDFE ˆk Z B e
(6)
C. Soft Transition and complete algorithm As the soft decision device is a convex combination of the identity and the hard decision device, the adaptation of the coefficients of our equalizer is computed by the same convex combination of the algorithms: (8) = (1 − α) × (4) + α × (5), for the FFF and (9) = (1 − α) × (3) + α × (6) for the FBF: 2 G ← G + (1 − α) µG 1 − |uk | ,
(7)
F ← F − (1 − α) µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k F
(8)
− α µDFE eˆk U∗k , F ∗ DFE ˜ ∗k . B ← B + (1 − α) µLRE ˆk Z B uk Uk−1 − α µB e
(9)
If α = 0, the equalizer is completely equivalent to a LRE adapted as described in (2)–(4). But if α = 1, the equalizer is the DD-DFE described by (5)–(6). However, if α is between 0 and 1, the equalizer is a soft combination of the LRE and the DFE as needed. In order to be blind, the combination parameter α must also be driven by an adaptation algorithm. It is not possible to adjust α based on the estimated error decision eˆk because this estimation is mainly wrong during the convergence phase. A criterion which is robust and behaves well in blind equalization is developed in [2]: the Constant Norm Algorithm (CNA). Thus α is directly computed through the estimation of the CNA criterion: α ← λα + (1 − λ) µα kxk k2p − R where kxkp =
p p
p
2
,
(10)
p
|Re x| + |Im x| . The constant R depends only on the constellation. The parameter λ is close
to 1 and permits to filter the noise in the variation of the CNA criterion. µα is chosen in order to constrain α to be between 0 and 1. The complete algorithm is given in figure 2. IV. S IMULATION RESULTS To test the performance of the B-WDFE, we have carried out a series of simulations for various environments and modulations. All the results are presented in a graph form, plotting the variation of the MSE versus time. Three equalizers are tested: •
The continuously-trained DFE, whose correct decisions are fed back into the FBF. It is considered as the reference, the best results it are possible to reach with DD algorithm.
•
Our blind B-WDFE.
January 22, 2010
DRAFT
5
Parameters
LRE DFE DFE µα , µG , µLRE F , µB , µF , µB
Input sequence
rk for k = 1, 2, . . .
Output sequence
zk for k = 1, 2, . . .
F ← F0
B ← B0
α ← 0 G ← 1 // Initialization
For each received symbol rk perform // Calculation of the filter outputs p uk ← |G| rk − BT Uk−1 ˜ k−1 zk ← xk − BT E
xk ← FT Uk eˆk ← dec(zk ) − zk
// Combination proportion α ← λα + (1 − λ) µα kxk k2p − R
2
e˜k ← α dec(zk ) + (1 − α)zk − xk // Coefficient updating F ← F − (1 − α) µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k F − α µDFE eˆk U∗k , F ∗ DFE ˆ Z ˜∗ B ← B + (1 − α) µLRE k k B uk Uk−1− α µB e 2 G ← G + (1 − α) µG 1 − |uk |
Fig. 2.
•
Filtering and adaption algorithm of the B-WDFE.
The classical blind LRE, corresponding to the B-WDFE where the proportion parameter α is always zero.
The final curve on the graphs also shows the temporal variations of the proportion parameter α. As it is not possible to compare this directly with the MSE, a different scale has been plotted on the right for measuring its values. For all the simulations, the graphs are the result of an average of the MSEs over 200 independent runs. We tested the B-WDFE on the Macchi channel. The coefficients of this channel [3], [7] are H(z) = 0.8264 − 0.1653z −1 + 0.8512z −2 + 0.1636z −3 + 0.81z −4 . The filter B comprises 11 coefficients while the transversal filter has 31, all initialized to zero except the central coefficient, which is set to 1. The order of magnitude of the step sizes is 5 · 10−4 . The simulation is performed for a 64-QAM, with a signal-to-noise ratio of 30 dB. The step size is chosen in such a way that continuously-trained DFE and our B-WDFE achieve similar MSEs in stationary regime. In these conditions, the result of the B-WDFE is very impressive, because it outperforms the reference continuously trained DFE MSE without any algorithm and filtering commutation: which was our objective. The second simulation result presented in this paper, concerns more specifically the reactivity of the B-WDFE to abrupt changes in its environment. Up to the 40,000th symbol, the channel is given by the first four coefficients of the Macchi channel. Then, the channel is the full version of this channel. This variation of the channel represents the arrival of a strong new echo. In order to maintain the signal-to-noise ratio, these two channels are normalized. The results in figure 4 show the B-WDFE that converges rapidly towards the optimal solution, faster still than the trained DFE. As soon as the new echo appears, the proportion parameter α diminishes in order to switch the WDFE over to a mode that is more blind than DD. Once the variation has been taken on board, the B-WDFE returns to normal convergence.
January 22, 2010
DRAFT
6
0
α parameter 0.6
−5
MSE (dB)
−10 0.4 Trained DFE
Blind WDFE
−15
Blind LRE
0.2
−20
−25
0 0
Fig. 3.
10
20 Iterations ×103
30
40
Performance of the B-WDFE on the Macchi channel for 64-QAM.
Hence this simulation shows that the B-WDFE remains very reactive even when it is virtually being DD. The proportion parameter α effectively enables this equalizer to successfully measure the opening of the eye, and thereby the global situation. When the latter deteriorates, the B-WDFE reacts accordingly. V. C ONCLUSION The WDFE was particularly attractive to mitigate the error propagation phenomenon [1]. By using the same basic ideas, this paper proposes the B-WDFE. It appears that the concept of combination is applicable not only for the decision device but also for both the algorithms and the filtering structure. By carefully chosing the adaptation algorithms the WDFE turns out to be fully blind and notably efficient. These performances are the consequence of the use of an proportion parameter α, driving the combination of algorithms and structures, blindly estimated through the use of a CNA cost-function. R EFERENCES [1] J. Palicot and A. Goupil, “Performance analysis of the weighted decision feedback equalizer” in Signal Process., vol. 88, pp. 284–295, Feb. 2008.
January 22, 2010
DRAFT
7
0
α parameter 0.6
−5
MSE (dB)
−10 0.4 Blind WDFE
−15
Blind LRE
0.2
−20 Trained DFE −25
0 0
Fig. 4.
20
40 60 3 Iterations ×10
80
100
B-WDFE’s reaction to an abrupt variation of the channel for 64-QAM.
[2] A. Goupil and J. Palicot “New Algorithms for Blind Equalization: the Constant Norm Algorithm” in IEEE Trans. Signal Process., vol. 55, pp. 1436–1444, Apr. 2007. [3] J. Labat, O. Macchi and C. Laot, “Adaptive decision feedback equalization: can you skip the training period?” IEEE Trans. Commun., vol. 46, pp. 921–930, July 1998. [4] C. A. Belfiore and J. H. Park, “Decision feedback equalization,” Proc. IEEE, vol. 67, pp. 1143–1158, Aug. 1979. [5] O. Macchi, Adaptive processing: the least mean squares approach with applications in transmission. [6] J. G. Proakis, Digital communications.
Wiley, 1995.
McGraw-Hill, 1989.
[7] G. Ananthaswamy and D. L. Goeckel, “A Fast-Acquiring Blind Predictive DFE” IEEE Trans. Commun., vol. 50, pp. 1557– 1560, Oct. 2002.
January 22, 2010
DRAFT
An efficient blind decision feedback equalizer Alban Goupil and Jacques Palicot
Abstract Efficient and fully blind Decision Feedback Equalizer (DFE) remains an open issue, mainly because of the potential errors in the decision loop. Based on the Weighted Decision Feedback Equalizer (WDFE), our previous work aiming at decreasing the error propagation phenomena, we propose a new blind DFE called Blind Weighted Decision Feedback Equalizer. The main idea of the WDFE was to commute between Linear Recursive Equalizer (LRE) and DFE for the error computation at the decision device. In this paper, we extend this idea in order to commute also both the algorithms and the filtering structure. This commutation is performed softly and blindly.
I. I NTRODUCTION At first sight, the drawbacks of the DFE, for example error propagation, make it a poor candidate for blind equalization, although its performance when optimal is particularly attractive. Extending our previous work Weighted Decision Feedback Equalizer [1] we propose here a new fully blind DFE: the Blind-Weigthed Decision Feedback Equalizer (B-WDFE). The main idea is to commute softly and blindly the algorithms, the filtering structure and the decision device between the two extremes cases: the LRE and the DFE. This should be performed under the following constraints, in order to achieve our objective: •
No structure change—i.e. all the parts of the equalizer are used equally for the convergence of the algorithm and for the equalization of the channel once converged. This avoids having one equalizer that opens the eye in the first stage and then another that derives from it once the decisions are sufficiently certain, as in [3].
•
No sudden change in the cost functions—i.e. the equalizer must adapt itself to achieve the best possible correspondence between the convergence phase and the channel variation tracking phase, without changing the cost functions to be optimized.
•
The equalizer must be as simple as possible in its architecture. Thus sample-by-sample algorithms and structures based on filtering are considered positively.
In Section II we first describe the filtering structure of our equalizer then in Section III we explain the proposed algorithm, which softly commutes between a blind algorithm and the Decision Directed (DD) algorithm. We prove the efficiency of our B-WDFE thanks to simulation presented in Section IV and finally we conclude the paper. A. Goupil is with the University of Reims Champagne-Ardenne, France J. Palicot is with SUPELEC/IETR, France
January 22, 2010
DRAFT
2
II. F ILTERING LRE and DFE are classical equalizers whose structure are well-studied. LRE is composed of an automatic gain control (AGC), a feedforward filter (FFF) and a feedback filter (FBF). DFE shares the same components as LRE but it also includes a decision device. This device is classically put in front of the FBF in order to improve the equalizer output when the decisions are correct. Under the hypothesis of correct decision and for the minimum mean square error (MMSE) criterion, the optimal coefficients of the filters of the DFE and the LRE are equal [6]. This fact is the bases of the blind equalizer of Labat et al. [3] (SA-DFE): once the convergence of the filter of the LRE is achieved, the structure of the LRE switches into the structure of the DFE while the filters’ coefficients are kept. In this way, LRE structure is used during the convergence phase of the adaptation and DFE is used during the tracking phase providing good performances and fast convergence. The equalizer presented in this paper, named B-WDFE, deals with the same idea of switching between the LRE and the DFE. But this process must be done smoothly. In this way, there is no need to compute a threshold as in the case of SA-DFE, and no hysteresis phenomenon would appear. This objective is accomplished by using the DFE structure of Belfiore and Park [4] which is shown in figure 1. This way to address the structure switching problem of the SA-DFE was also proposed in [7].
R
×
U
+
g
F(z)
X
+
Z
1 − B(z)
φ (·) 1 − B(z)
Z˜ E˜
+
+ −
Fig. 1.
Blind LRE/DFE structure.
The coefficient g represents the AGC, and F (z) is the FFF. B(z) is a monic polynomial whose coefficients are used twice in the structure and represent the FBF part of the equalizer. The function φ(·) concerns the soft decision. Its choice permits to switch between the LRE and the DFE. Indeed, if φ(z) = z, the complete equalizer performs as a LRE whose FFF is F (z) and the FBF is B(z). When φ(z) is the decision device of the constellation, the equalizer becomes the DFE. Choosing the function φ(·) between this two extreme cases results in an equalizer which behaves between LRE and DFE. The structure of our equalizer is thus static and the adaptation will concern not only the filters’ coefficients but also the soft decision device φ(·) depending on the input and of the convergence state. In order to keep the equalizer simple, the chosen soft decision device is a convex combination of the decision device and the identity: φ(z) = (1 − α) z + α dec(z),
(1)
where dec(·) is the decision and α is a proportion parameter that allows to choose between the LRE and the DFE when α = 0 and α = 1 respectively. This parameter will be subject to adaptation also. January 22, 2010
DRAFT
3
III. A DAPTATION As the extreme cases of the equalizer’s structure are the DFE and the LRE, the adaptation will be also inbetween these two cases. By choosing the soft decision (1), the adaptation will be also a convex combination of the adaptation for the LRE and for the DFE. Thus we describe first the adaption algorithm in the LRE case and then the adaptation in the DFE case followed by the complete algorithm, that is the convex combination of the previous cases. For convenience, the algorithms used thereafter are based on the LMS algorithm. All step sizes needed by the algorithms are noted by the Greek letter µ with subscripts and superscripts that indicate the adapted component. The algorithms are described using the notation of the figure 1. The time index is noted k. Thus, the k-th sample of the signal U is given by uk . If several sample of this signal are needed, we note Uk the column vector [uk , uk−1 , . . .]T . The length of the vectors are given by the context. The coefficients of the FFF F (z) is given by F which is a column vector. Transposition of F is noted by FT . The conjugate of a complex number z whose real and imaginary parts are Re z and − Im z respectively, is noted z ∗ . A. LRE Case In the LRE case, the adaptation concerns the AGC, the FFF and the FBF. Each component is adapted according to its own criterion. Indeed, the equalizer performs like the LRE if the steady-state is not reached. The purpose of the AGC is to adjust the power of the signal to unity. The algorithm used is classical: p 2 G ← G + µG 1 − |uk | and g ← |G|,
(2)
where µG is the adaptation step of this algorithm. The FBF is adapted in order to converge to the whitening filter of the input signal. The criterion concerns the maximization of the power of its output. But this filter is recursive and thus the LMS algorithm is not easily derived. Macchi in [5] proposed and studied the stability of the following algorithm. ∗ BLRE ← BLRE + µLRE B uk Uk−1 .
(3)
The FFF is adapted using the CQA, a variant of the Constant Modulus Algorithm (CMA), presented in [2] which permits to recover the phase of the constellation: FLRE ← FLRE − µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k , F
(4)
where kxk∞ = max(|Re x|, |Im x|) and p(x) is the gradient of kxk∞ . The constant γ depends only on the constellation. B. DFE Case If the convergence is almost done, tracking phase begins, the filters are close to the optimal performance and the decisions are mostly correct. Thus the equalizer must be close to the DFE and the criterion used to adapt
January 22, 2010
DRAFT
4
is directed by the decision. That is the criterion is the MMSE and its estimation is based on the estimated error eˆk = dec(zk ) − zk . The FBF and FFF are thus adapted with the LMS algorithms: FDFE ← FDFE − µDFE eˆk U∗k F
(5)
˜ ∗k BDFE ← BDFE − µDFE ˆk Z B e
(6)
C. Soft Transition and complete algorithm As the soft decision device is a convex combination of the identity and the hard decision device, the adaptation of the coefficients of our equalizer is computed by the same convex combination of the algorithms: (8) = (1 − α) × (4) + α × (5), for the FFF and (9) = (1 − α) × (3) + α × (6) for the FBF: 2 G ← G + (1 − α) µG 1 − |uk | ,
(7)
F ← F − (1 − α) µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k F
(8)
− α µDFE eˆk U∗k , F ∗ DFE ˜ ∗k . B ← B + (1 − α) µLRE ˆk Z B uk Uk−1 − α µB e
(9)
If α = 0, the equalizer is completely equivalent to a LRE adapted as described in (2)–(4). But if α = 1, the equalizer is the DD-DFE described by (5)–(6). However, if α is between 0 and 1, the equalizer is a soft combination of the LRE and the DFE as needed. In order to be blind, the combination parameter α must also be driven by an adaptation algorithm. It is not possible to adjust α based on the estimated error decision eˆk because this estimation is mainly wrong during the convergence phase. A criterion which is robust and behaves well in blind equalization is developed in [2]: the Constant Norm Algorithm (CNA). Thus α is directly computed through the estimation of the CNA criterion: α ← λα + (1 − λ) µα kxk k2p − R where kxkp =
p p
p
2
,
(10)
p
|Re x| + |Im x| . The constant R depends only on the constellation. The parameter λ is close
to 1 and permits to filter the noise in the variation of the CNA criterion. µα is chosen in order to constrain α to be between 0 and 1. The complete algorithm is given in figure 2. IV. S IMULATION RESULTS To test the performance of the B-WDFE, we have carried out a series of simulations for various environments and modulations. All the results are presented in a graph form, plotting the variation of the MSE versus time. Three equalizers are tested: •
The continuously-trained DFE, whose correct decisions are fed back into the FBF. It is considered as the reference, the best results it are possible to reach with DD algorithm.
•
Our blind B-WDFE.
January 22, 2010
DRAFT
5
Parameters
LRE DFE DFE µα , µG , µLRE F , µB , µF , µB
Input sequence
rk for k = 1, 2, . . .
Output sequence
zk for k = 1, 2, . . .
F ← F0
B ← B0
α ← 0 G ← 1 // Initialization
For each received symbol rk perform // Calculation of the filter outputs p uk ← |G| rk − BT Uk−1 ˜ k−1 zk ← xk − BT E
xk ← FT Uk eˆk ← dec(zk ) − zk
// Combination proportion α ← λα + (1 − λ) µα kxk k2p − R
2
e˜k ← α dec(zk ) + (1 − α)zk − xk // Coefficient updating F ← F − (1 − α) µLRE kxk k∞ − γ kxk k∞ p(xk ) U∗k F − α µDFE eˆk U∗k , F ∗ DFE ˆ Z ˜∗ B ← B + (1 − α) µLRE k k B uk Uk−1− α µB e 2 G ← G + (1 − α) µG 1 − |uk |
Fig. 2.
•
Filtering and adaption algorithm of the B-WDFE.
The classical blind LRE, corresponding to the B-WDFE where the proportion parameter α is always zero.
The final curve on the graphs also shows the temporal variations of the proportion parameter α. As it is not possible to compare this directly with the MSE, a different scale has been plotted on the right for measuring its values. For all the simulations, the graphs are the result of an average of the MSEs over 200 independent runs. We tested the B-WDFE on the Macchi channel. The coefficients of this channel [3], [7] are H(z) = 0.8264 − 0.1653z −1 + 0.8512z −2 + 0.1636z −3 + 0.81z −4 . The filter B comprises 11 coefficients while the transversal filter has 31, all initialized to zero except the central coefficient, which is set to 1. The order of magnitude of the step sizes is 5 · 10−4 . The simulation is performed for a 64-QAM, with a signal-to-noise ratio of 30 dB. The step size is chosen in such a way that continuously-trained DFE and our B-WDFE achieve similar MSEs in stationary regime. In these conditions, the result of the B-WDFE is very impressive, because it outperforms the reference continuously trained DFE MSE without any algorithm and filtering commutation: which was our objective. The second simulation result presented in this paper, concerns more specifically the reactivity of the B-WDFE to abrupt changes in its environment. Up to the 40,000th symbol, the channel is given by the first four coefficients of the Macchi channel. Then, the channel is the full version of this channel. This variation of the channel represents the arrival of a strong new echo. In order to maintain the signal-to-noise ratio, these two channels are normalized. The results in figure 4 show the B-WDFE that converges rapidly towards the optimal solution, faster still than the trained DFE. As soon as the new echo appears, the proportion parameter α diminishes in order to switch the WDFE over to a mode that is more blind than DD. Once the variation has been taken on board, the B-WDFE returns to normal convergence.
January 22, 2010
DRAFT
6
0
α parameter 0.6
−5
MSE (dB)
−10 0.4 Trained DFE
Blind WDFE
−15
Blind LRE
0.2
−20
−25
0 0
Fig. 3.
10
20 Iterations ×103
30
40
Performance of the B-WDFE on the Macchi channel for 64-QAM.
Hence this simulation shows that the B-WDFE remains very reactive even when it is virtually being DD. The proportion parameter α effectively enables this equalizer to successfully measure the opening of the eye, and thereby the global situation. When the latter deteriorates, the B-WDFE reacts accordingly. V. C ONCLUSION The WDFE was particularly attractive to mitigate the error propagation phenomenon [1]. By using the same basic ideas, this paper proposes the B-WDFE. It appears that the concept of combination is applicable not only for the decision device but also for both the algorithms and the filtering structure. By carefully chosing the adaptation algorithms the WDFE turns out to be fully blind and notably efficient. These performances are the consequence of the use of an proportion parameter α, driving the combination of algorithms and structures, blindly estimated through the use of a CNA cost-function. R EFERENCES [1] J. Palicot and A. Goupil, “Performance analysis of the weighted decision feedback equalizer” in Signal Process., vol. 88, pp. 284–295, Feb. 2008.
January 22, 2010
DRAFT
7
0
α parameter 0.6
−5
MSE (dB)
−10 0.4 Blind WDFE
−15
Blind LRE
0.2
−20 Trained DFE −25
0 0
Fig. 4.
20
40 60 3 Iterations ×10
80
100
B-WDFE’s reaction to an abrupt variation of the channel for 64-QAM.
[2] A. Goupil and J. Palicot “New Algorithms for Blind Equalization: the Constant Norm Algorithm” in IEEE Trans. Signal Process., vol. 55, pp. 1436–1444, Apr. 2007. [3] J. Labat, O. Macchi and C. Laot, “Adaptive decision feedback equalization: can you skip the training period?” IEEE Trans. Commun., vol. 46, pp. 921–930, July 1998. [4] C. A. Belfiore and J. H. Park, “Decision feedback equalization,” Proc. IEEE, vol. 67, pp. 1143–1158, Aug. 1979. [5] O. Macchi, Adaptive processing: the least mean squares approach with applications in transmission. [6] J. G. Proakis, Digital communications.
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January 22, 2010
DRAFT
