Elecrical Engineering (2006) 88: 527–534 DOI 10.1007/s00202-005-0309-z
O R I G I N A L PA P E R
Te-Jen Su · Sheng-Yi Lin · Gwo-Jia Jong
FIR fuzzy equalizer design for nonlinear channels LMI-based fuzzy
approach
Received: 13 December 2004 / Accepted: 25 April 2005 / Published online: 16 December 2005 © Springer-Verlag 2005
Abstract In this paper an LMI based fuzzy methodology for nonlinear channel equalization from an H∞ perspective is
proposed. According to Takagi-Sugeno (T-S) fuzzy model-ing concept, the discrete-time nonlinear channel can be con-structed by the piecewise linear subsystems. The FIR fuzzy equalizer design for nonlinear channel is transformed into a standard linear matrix inequality (LMI) optimization prob-lem, and the coefficients of the equalizer are obtained by solving LMIs. Besides, the stability of T-S fuzzy system has been investigated based on Lyapunov approach. Finally, sim-ulation result is given to demonstrate the effectiveness of the proposed methodology.
Keywords FIR· Equalizer · LMI · T-S fuzzy model 1 Introduction
In modern digital communication systems, there have been several methods proposed to treat the problem of channel equalization. The equalizer design problem via two-block
H∞optimization technique is presented in [1]. The
risk-sen-sitive FIR equalizer has been formulated as the constrained analytic centering problem, which is another type of con-vex problem [2]. The maximum likelihood Viterbi algorithm and the conventional decision feedback equalizer are used for the Bayesian decision feedback equalizer [3]. However, most of them mentioned above are firmly based on linear filter algorithms. In practical communication systems, the channel may be nonlinear due to nonideal characteristics of the lin-ear devices. In orthogonal frequency division multiplexing (OFDM) transmission, the nonlinear power amplifier and the linear multipath environment result in a nonlinear commu-nication channel and a nonlinear equalization is required for removal of the nonlinear distortion in the channel. Moreover, T.-J. Su (
B
)· S.-Y. Lin · G.-J. JongDepartment of Electronic Engineering,
National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan 807, Republic of China E-mail: [email protected]
in the Magnetic Recording channels, nonlinear channel mod-els are more appropriate and perform significantly better than linear models. In the published literature, the extended Kal-man filter (EKF) has been proposed to treat the nonlinear problem [4, 5]. However, it has a drawback regarding error propagation because of the linearization around the present state estimate.
Recently, fuzzy techniques have been applied in various fields such as control systems, communication systems, sig-nal processing, and so on. Among various fuzzy modeling themes, Takagi-Sugeno (T-S) model based fuzzy control ap-proach has been rapidly and successfully developing in non-linear control frameworks [6]. The T-S fuzzy model offers an effective way to represent the nonlinear systems with a set of fuzzy IF-THEN rules, each of the rules can be repre-sented as a local linear state equation and the overall fuzzy system is achieved by fuzzy “blending” of the these linear state equations.
We propose an LMI-fuzzy based approach to design the FIR fuzzy equalizer for nonlinear discrete-time channels from
H∞perspective, and employ the state-space description in conjunction with bounded real lemma [7, 8] to calculate the optimal γ value and the coefficients of the FIR fuzzy equal-izer, where the γ value, a tolerance level, can be regarded as an indication of the quality of the filter. The T-S fuzzy modeling methodology is also applied to construct the fuzzy equalizer with some seasonal fuzzy rules and membership function. Moreover, the effect of the γ value for different equalizer lengths is discussed.
Two primary contributions of this paper are (a) deal-ing with the nonlinear system by T-S fuzzy model and (b) eliminating the effect of the external disturbance as much as possible.
The rest of this paper is organized as follows. In Sect. 2, the system model is described and a T-S fuzzy model is used to construct the nonlinear channel. Furthermore, the state-space representation for the error transfer function is pre-sented. Stability analysis is presented in Sect. 3. Converting the problems to the task of finding an optimal γ value by solving LMIs is described in Sect. 4. In Sect. 5, a numerical
528 T.-J. Su et al.
HNL(z) K(z)
L(z) Nonlinear
channel FIR fuzzy
equalizer Delay v(t) b(t) y(t) e(t) (t) ~ z ) ( ˆ t z
Fig. 1 System model
example is proposed and simulation results are demonstrated as well. Conclusion is drawn in Sect. 6.
2 System model description
The general structure of the error transfer function corre-sponding to the nonlinear channel equalization problem is illustrated in Fig. 1 [9], where b(t) is the transmitted digital information sequence, v(t) is the unknown noise, e(t) is the error between the fuzzy equalizer output and the delay of the desired transmitted sequence, HNL(z) is the discrete
equiva-lent of the nonlinear time-invariant communication channel,
K(z) is the fuzzy equalizer to be designed and L(z) = z−d
is the delay. The discrete data sequence b(t) passes through the nonlinear time-invariant channel HNL(z), the observation
sequence y(t) is then formed by the addition of an unknown measurement disturbance v(t) with the output of the nonlin-ear communication channel HNL(z).
Our purpose is to design a fuzzy equalizer K(z) which estimatesˆz(t) from the observations y(t), that is to minimize the error e(t) between the equalizer output and the delayed signal of the desired transmitted sequence. We derive a state-space representation for the error transfer function, which is mapping from the input b(t) and the disturbances v(t) to the equalization error sequence e(t).
The nonlinear channel HNL(z), contaminated with the
noise at the output, is described by the following piecewise linear T-S fuzzy model:
Model Rule j:
I F z1(t) is Mj 1AND · · · AND zp(t) is Mjp, THEN s(t + 1) = Acjs(t) + Bcju(t),
y(t) = Ccjs(t) + Dcju(t). j = 1, 2, . . . , L.
(1) where Mjp are the fuzzy sets and L the number of model
rules; s(t) is the state vector, u(t) = b(t) v(t)T is the input vector and y(t) is the measured output; Acj, Bcj, Ccj
and Dcj are constant matrices; z1(t), . . . , zp(t) are premise
variables that may be functions of the state variables, external disturbances, and/or time.
The final outputs of the fuzzy systems (1) are inferred as follows: s(t + 1) = L j =1wj(z(t)) Acjs(t) + Bcju(t) L j =1wj(z(t)) ≡ L j =1 hj(z(t)){Acjs(t) + Bcju(t)} = Acs(t) + Bcu(t), (2) y(t) = L j =1wj(z(t)) Ccjs(t) + Dcju(t) L j =1wj(z(t)) ≡ L j =1 hj(z(t)){Ccjs(t) + Dcju(t)} = Ccs(t) + Dcu(t). (3) where z(t) =z1(t) z2(t) . . . zp(t) , wj(z(t)) = p k=1Mj k(zk(t)), hj(z(t)) = Lwj(z(t)) j =1wj(z(t)) , Ac = L j =1wj(z(t))Acj L j =1wj(z(t)) , Bc = L j =1wj(z(t))Bcj L j =1wj(z(t)) , (4) Cc = L j =1wj(z(t))Ccj L j =1wj(z(t)) and Dc = L j =1wj(z(t))Dcj L j =1wj(z(t)) .
The term Mj k(zk(t)) is the grade of membership function
of zk(t) in Mj k. It is easy to find that
wj(z(t)) ≥ 0, j = 1, 2, . . . , L (5) and L j =1 wj(z(t)) > 0. (6) therefore, hj(z(t)) ≥ 0 for j = 1, 2, . . . , L and L j =1 hj(z(t)) = 1. (7)
The delay operator L(z) = z−dis represented by
(t + 1) = Ad(t) + Bdu(t), (8)
FIR fuzzy equalizer design 529
where d > 0, (t) is the state vector, u(t) =b(t) v(t)T is the input vector and Ad=
0 0 I(d−1)·(d−1)0 , Bd= 1 0· · · 0 0 0· · · 0 T , Cd = 01· (d−1) 1
and I denotes the identity matrix with appropriate dimension.
The state space model for the FIR fuzzy equalizer K(z) =
k0+ k1z−1+ · · · + kR−1z−(R−1)with order R − 1 has the
fol-lowing structure: Model Rule j : I F z1(t) is Mj 1AND · · · AND zp(t) is Mjp, THEN ω(t + 1) = Aejω(t) + Bejy(t), (10) ˆz(t) = Cejω(t) + Dejy(t). j = 1, 2, . . . , L. (11) where Mjpare the fuzzy sets and L is the number of model
rules; ω(t) is the state vector, y(t) is the measured output, ˆz(t) is the output of the fuzzy equalizer, z1(t), . . . , zp(t) are
premise variables that may be functions of the state variables, external disturbances, and/or time. Aej =
0 0 I(R−2)·(R−2) 0 , Bej = I 0(R−2)·1 , Cej = k1j, . . . , k(R−1)j , Dej = k0j
and k0j, k1j, ...., k(R−1)j are the coefficients of the equalizer
which to be designed.
The final outputs of the fuzzy equalizer are inferred as follows: ω(t + 1) = L j =1wj(z(t)) Aejω(t) + Bejy(t) L j =1wj(z(t)) ≡ L j =1 hj(z(t)){Aejω(t) + Bejy(t)} = Aeω(t) + Bey(t), (12) ˆz(t) = L j =1wj(z(t)) Cejω(t) + Dejy(t) L j =1wj(z(t)) ≡ L j =1 hj(z(t)){Cejω(t) + Dejy(t)} = Ceω(t) + Dey(t). (13) where z(t)=z1(t) z2(t) · · · zp(t) , wj(z(t)) = p k=1 Mj k(zk(t)), hj(z(t)) = Lwj(z(t)) j =1wj(z(t)) , Ae= L j =1wj(z(t))Aej L j =1wj(z(t)) , Be= L j =1wj(z(t))Bej L j =1wj(z(t)) , (14) Ce= L j =1wj(z(t))Cej L j =1wj(z(t)) and De= L j =1wj(z(t))Dej L j =1wj(z(t)) .
From the above preliminary, we can obtain the state-space model for the error transfer function with mapping input vec-tor u(t) to the error output e(t) = ˜z(t) − ˆz(t) as follows:
x(t + 1) = Ax(t) + Bu(t), (15) e(t) = Cx(t) + Du(t). (16) where d > 0 and A = A0c 0Ad 00 BeCc0 Ae , B =BcBd BeDc T , C =−DeCcCd −Ce , D = [−DeDc] , x(t) =s(t) (t) ω(t)T, u(t) =b(t) v(t)T.
From Eqs. 15 and 16, the equalizer coefficients are only included in the matrices C and D. Therefore, the representa-tion of the error transfer funcrepresenta-tion can be specified as
T (z) = C(zI − A)−1B + D. (17)
Denoting the optimal value γ , the obtained solution guar-antees a disturbance rejection capability, which is optimal in the sense of H∞norm:
T (z)∞< γ (18)
Consequently, we will show how to cast the H∞control
problem into LMI framework in the next section
3 Stability analysis
Stability analysis of T-S fuzzy model systems has been exten-sively studied in literature. It is well known that most stability analysis of nonlinear systems is based on the Lyapunov sta-bility theorem. A single quadratic Lyanopuv function is also employed to prove the stability of a fuzzy system.
Theorem 3.1 The discrete system (15)–(16) is
asymptoti-cally stable if there exist a positive definite symmetric matrix P and a scalar value ε such that the following LMIs are satisfied: ATP A − P APT P A −ε < 0 (19) P > 0 (20) ε > 0 (21)
534 T.-J. Su et al. 1.00e-3 0 2 4 6 8 10 12 14 16 18 SNR in dB 1.00e-1 1.00e-2 0 BER Fuzzy equalizer Extended Kalman filter
Fig. 6 Bit error rate comparison
6 Conclusion
In this paper, both fuzzy control and H∞attenuation
tech-niques are combined to achieve better performance for non-linear systems. The proposed method for nonnon-linear channel equalization has been formulated as a standard LMI opti-mization problem and the FIR fuzzy equalizer coefficients can be easily obtained by the LMI technique. Moreover, the stability of T-S fuzzy system has been investigated based on Lyapunov approach as well.
The given numerical example has demonstrated the effec-tiveness of the proposed methodology for obtaining the opti-mal γ values and coefficients of the fuzzy equalizer. The results show that the longer equalizer outperforms the shorter equalizer. It shows that the performance of the fuzzy equal-izer is clearly superior to that of the extended Kalman fil-ter equalizer. In general, the designed equalizer LMI-based fuzzy approach shows improved BER performance. References
1. Peng SC (1996) An equalizer design for nonminimum phase chan-nel via a two block H∞ optimization technique. Signal Process 51:1–13
2. Erdogan AT, Hassibi B, Kailath T (2000) On linear H∞
equaliza-tion of communicaequaliza-tion channels. Signal Process 48:3227–3231 3. Chen S, Mulgrew B, McLaughlin S (1993) Adaptive Bayesian
equalizer with decision feedback. Signal Process 41(9):2918–2927 4. Qureshi SUH (1985) Adaptive equalization. Proc IEEE 73:1349–
1387
5. Ahmed NU, Radaideh SM (1994) Modified extended Kalman fil-tering. IEEE Trans Automat Control 39:1322–1326
6. Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern SMC 15:116–132
7. Hsiung K -L, Lee L (1999) Lyapunov inequality and bounded real lemma for discrete-time descriptor systems. IEEE Proc Control Theory Appl 146(4):327–331
8. De Souza CE, Xie L (1992) On the discrete-time bounded real lemma with application in the characterization of static state feed-back H-infin controllers. Syst Control Lett 18:61–71
9. Erdogan AT, Hassibi B, Kailath T (2001) FIR H∞ equalization.
Signal Process 81:907–917
10. Chen BS, Tsai CL, Chen DS (2002) Robust H∞and mixed H2/H∞ filter for equalization designs of nonlinear communication system: fuzzy interpolation approach. IEEE Trans Fuzzy Syst 11(3):384– 398
11. Chen S, Gibson GJ, Cowan CFN (1990) Adaptive channel equal-ization using a polynomial-perceptron structure. Proc Inst Elec Eng pt I 137(5):257–264
12. Chen S, Mulgrew B, Grant PM (1993) A clustering technique for digital communications channel equalization using radial basis function networks. IEEE Trans Neural Netw 4:570–579
