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Estimation and equalization of multipath Rician fading channels with stochastic tap coefficients
Signal Processing 68 (1998) 43—57
Estimation and equalization of multipath Rician fading channels with stochastic tap coefficients Bor-Sen Chen*, Yih-Jinn Huang, Sin-Chung Chen Department of Electrical Engineering, National Tsing-Hua University, Hsin-Chu 300, Taiwan, ROC Received 21 August 1996; received in revised form 10 March 1998
Abstract This study investigates the estimation and equalization problems of multipath Rician fading channels with stochastic tap coefficients. The time-varying tap coefficients of fading channels under investigation are typically modeled as complex-valued Gaussian processes. First, least squares estimation methods are used to estimate the mean and variance of the stochastic tap coefficients. Then, based on these estimated statistical characteristics, an optimal equalization filter is designed to recover the transmitted sequence. In the proposed design method, a realizable equalization filter is constructed to minimize the mean square error from the perspective of frequency domain. Also, both calculus of variation technique and spectral factorization method are employed in the design procedure. To exhibit the superior performance of the proposed design method, some illustrative simulations are presented. ( 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Die vorliegende Studie untersucht das Scha¨tzproblem und das Entzerrungsproblem fu¨r Rice’sche Schwundkana¨le mit Mehrwegeausbreitung und stochastischen lmpulsantwortkoeffizienten. Die zeitvarianten lmpulsantwortkoeffizienten der untersuchten Schwundkana¨le werden typischer Weise als komplexwertige Gau{prozesse modelliert. Zuerst werden Kleinste-Quadrate-Methoden benutzt, um Mittelwert und Varianz der stochastischen lmpulsantwortkoeffizienten zu scha¨tzen. Danach wird auf der Grundlage dieser gescha¨tzten statistischen Charakteristiken ein optimales Entzerrerfilter zur Ru¨ckgewinnung der u¨bertragenen Folge entworfen. ln dem vorgestellten Entwurfsverfahren wird ein realisierbares Entzerrerfilter konstruiert, das den mittleren quadratischen Fehler im Frequenzbereich minimiert. Auch werden in dem Entwurfsverfahren sowohl die Variationsrechung als auch die Methode der spektralen Faktorisierung eingesetzt. Um die ho¨here Leistung der vorgeschlagenen Entwurfsmethode zu zeigen, werden einige illustrative Simulationen vorgestellt. ( 1998 Elsevier Science B.V. All rights reserved. Re´sume´ Cette e´tude porte sur les proble`mes d’estimation et d’e´galisation de canaux a` e´vanouissement multi-chemins de type Rice avec des coefficients de filtre ale´atoires. Les coefficients de filtre variant dans le temps des canaux a` e´vanouissement analyse´s sont typiquement mode´lise´s par des processus gaussiens a` valeurs complexes. Tout d’abord, les me´thodes
—————— * Corresponding author. E-mail: [email protected]. 0165-1684/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 7 ) 0 0 0 5 6 - 5
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
d’estimation aux moindres carre´s sont utilise´es pour estimer la moyenne et la variance des coefficients de filtre ale´atoires. Ensuite, sur la base des estime´es de ces caracte´ristiques statistiques, un filtre d’e´galisation optimal est conc7 u pour recouvrer la se´quence transmise. Dans la me´thode de conception propose´e, un filtre d’e´galisation re´alisable est construit pour minimiser l’erreur quadratique moyenne dans le domaine fre´quentiel. De plus, la technique du calcul des variations et la me´thode de factorisation spectrale sont employe´es dans la proce´dure de conception. Quelques exemples de simulation illustratifs sont pre´sente´s pour mettre en lumie`re les performances supe´rieures de la me´thode de conception propose´e. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Multipath Rician fading channel; Wide sense stationary and uncorrelated scattering
1. Introduction Signal reconstruction of the transmitted sequence in the presence of intersymbol interference (ISI) and channel noise is an important research topic for digital communication systems. The ISI is caused by nonideal channel characteristic and the channel noise comes from quantization, external corruption, etc. Both ISI and channel noise are two major impeding factors for high speed data transmission. The role of the equalization filter is to reconstruct the transmitted sequence by removing the effect of ISI and eliminating the corruption of channel noise. In general, an optimal equalization filter can be designed when the transfer function of the transmission channel is precisely known and the statistical properties of the input signal and channel noise are also exactly given. However, in practical digital communication systems, the transmission channels can not be precisely estimated beforehand. Furthermore, the transmission channels may have randomly time-varying impulse responses corrupted by channel noises. This characterization serves as a model for signal transmission over many communication channels [4,6,7,16]. The time-varying impulse responses of these channels are a consequence of the constantly changing physical characteristics of the media or the time variations in the structure of the medium. As a result of such variations, the nature of the transmission channel varies with time. Moreover, the time variations appear to be unpredictable to the user of the channel. Therefore it is reasonable to characterize the time-varying transmission channel statistically.
Multipath fading channels with Rician (or Rayleigh) distribution have been used to characterize the statistical properties of time-varying channels for a long time [9,11,12,22—24], each tap coefficient of multipath fading channels is usually modeled as a complex-valued Gaussian process. To treat the equalization problem for such a timevarying channel, conventional adaptive equalization schemes using recursive least squares (RLS) adaptation algorithm are usually employed to track the channel’s variations and equalize the received signals [3,19]. In [14], Matolak and Wilson develop a maximum likelihood estimator (MLE) to detect the transmitted sequence for a statistically known Rayleigh fading channel. This study proposes a novel design method to treat the estimation and equalization problems for multipath Rician fading channels with stochastic tap coefficients. In the proposed design method, least squares estimation methods are employed to estimate the mean and variance of stochastic tap coefficients. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter which minimizes the mean square error (MSE) is designed from the viewpoint of frequency domain to reconstruct the transmitted sequence. This is the first time to develop a closed form filter to achieve the optimal signal reconstruction under multipath fading channels. Unlike conventional adaptive equalization schemes, the proposed optimal equalization filter is of fixed form which needs not to adapt parameters of filter after the initial training period. Therefore, it is easy to implement and suitable for real time signal reconstruction. Furthermore, calculus of variation technique, spectral factorization method and Cauchy’s residue theorem are employed
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
in the derivation procedure of the optimal equalization filter. Also, an expression for the minimum mean square error (MMSE) is derived. Finally, some illustrative simulations are given to exhibit the superior performance of the proposed design method. The organization of this paper is as follows. In Section 2, the problem description is presented and some modeling aspects of multipath Rician fading channels are discussed. In Section 3, least squares estimation methods are employed to estimate the statistical characteristics of fading channels under investigation. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter minimizing the MSE is derived in Section 4. Two simulation examples are given in Section 5 to exhibit the performance of the proposed design method. Finally, a brief conclusion is given in Section 6.
2. System description and problem formulation Digital signal transmission through a multipath fading channel is usually described by Fig. 1(a). In this figure, the i.i.d., equiprobable input sequence u(n) with zero mean and variance p2u is transmitted every ¹s seconds through a communication system consisting of the transmitter’s pulse shaping filter, the multipath fading channel, the receiver’s matched
45
filter and the sampler. It is well known that the received discrete-time signal, y(n), is given by [8,16,20,24] L y(n)" + hk(n)u(n!k)#w(n), (2.1) k/0 where w(n) denotes the additive white circular complex-valued Gaussian noise with zero mean and variance p2w and hk(n) is the time-varying impulse response of the overall system including transmitter/receiver filters and fading channel (see Fig. 1(b)). In this study, the time-varying impulse response hk(n) under investigation can be expressed in the following form: hk(n)"hk#hI k(n), (2.2) where hk is a non-zero constant mean and hI k(n) is wide-sense stationary and uncorrelated scattering (WSSUS) [6,9,16,24], i.e., E[hI *(n)hI (n!m)]"R I k(m)d(k!l), (2.3) k l h where d[n] is the unit sample sequence defined by
G
d(n)"
1, n"0, 0, otherwise,
and for each k, hI k(n) is a circular complex-valued Gaussian sequence generated by [6,12,20] hI k(n)"okhI k(n!1)#vk(n),
Fig. 1. (a) Signal transmission system through multipath fading channel. (b) An equivalent discrete-time system of (a).
(2.4)
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
where v (n) is an i.i.d., circular complex-valued k Gaussian sequence with zero mean and variance p2k and the parameter o is arbitrarily chosen close v k to unity so that hI (n) represents a generic low-pass k process. From Eq. (2.4), R I k(m) in Eq. (2.3) can be h calculated by
k"0,1,2,¸. Associating Eqs. (2.1) and (2.2), we have
o@m@ k RhI k(m)" p2 . 1!DokD2 vk
h"[h0, h1,2, hL]T,
Remark 1. If the coefficients hk(n) in Eq. (2.2) have non-zero constant means, then the time-varying channel is called the Rician fading channel [4,9]. Otherwise, it is called the Rayleigh fading channel [4,16,24].
and
The following assumptions are imposed on Eqs. (2.1), (2.2), (2.3) and (2.4): AS1: Both w(n) and h (n) are uncorrelated with the k input sequence u(n). AS2: w(n) is uncorrelated with vk(n) for k"0,1,2,¸. Subsequently, w(n) is uncorrelated with hI k(n) and hk(n) for k"0,1,2,¸. AS3: The input sequence u(n) has finite E[Du(n)D4] and E[Du(n)D4]O(E[Du(n)D2])2. For such a time-varying fading channel, the goal of this study is to estimate the channel’s statistical characteristics including each tap coefficient’s mean and variance. Then based on the estimated statistical characteristics, an optimal and realizable equalization filter which minimizes the MSE is designed from the viewpoint of frequency domain to reconstruct the transmitted sequence.
3. Estimation of the channel’s statistical characteristics In this section, a two-step estimation procedure is introduced to determine the channel’s statistical characteristics including each tap coefficient’s mean and variance. This estimation procedure is a generalization of a two-step procedure proposed by Rosenberg [18] to estimate the parameters of a stochastic coefficient regression model. The first step is to estimate each tap coefficient’s mean hk, for
y(n)"hTU(n)#hI T(n)U(n)#w(n),
(3.1)
where the superscript T denotes the transpose,
U(n)"[u(n), u(n!1),2, u(n!¸)]T
hI (n)"[hI 0(n), hI 1(n),2, hI L(n)]T. In view of Eq. (3.1), let b(n)"hI T(n)U(n)#w(n),
(3.2)
we have the conditional mean E[b(n)DU(n)]"E[hI T(n)]U(n)#E[w(n)]"0,
(3.3)
and the conditional variance E[b*(n)b(n)DU(n)] "UH(n)E[hI *(n)hI T(n)]U(n)#E[Dw(n)D2] L " + E[hI *k(n)hI k(n)]Du(n!k)D2#p2w k/0 L " + R I k(0)Du(n!k)D2#p2 , w h k/0
(3.4)
where the superscript * denotes the complex conjugate, and the superscript H denotes the transpose and complex conjugate. Remark 2. (a) The first equality in Eq. (3.4) is due to the assumption AS1. (b) The second equality in Eq. (3.4) is due to the uncorrelated scattering property of hI k(n) (see Eq. (2.3)). Combining Eqs. (3.1) and (3.2), we have y(n)"hTU(n)#b(n).
(3.5)
Given a set of training data X(N!1)"My(0), y(1),2, y(N!1),u(0), u(1),2, u(N!1)N,
(3.6)
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
we can obtain the least squares estimate hK of h from Eq. (3.5) by minimizing +N~1 n/0 Db(n)D2 with respect to h. Thus hK is given by [13,21]
G
H
N~1 ~1 N~1 * hK " + U*(n)UT(n) + U (n)y(n). n/0 n/0
(3.7)
Proposition 1. Let y(n), u(n) be related by Eqs. (2.1) and (3.1). Given a set of training data X(N!1) in Eq. (3.6), the estimate hK in Eq. (3.7) converges to h as N tends to infinity in the mean square sense (mss) and with probability one (wp1) under the assumption AS1.
47
Remark 3. The estimates hK and hK can be iteratively obtained by RLS or LMS algorithm. In this study, the RLS algorithm is employed to estimate h and h for simulation examples in Section 5. Let us denote p2hk"E[Dhk(n)D2]. Having obtained these estimates hK and hK , we can obtain the estimate pL 2hk of p2hk by pL 2hk"DhK kD2#RK hI k(0), for k"0,1,2,¸. These estimated statistical characteristics will be used to design an optimal and realizable equalization filter in the next section.
Proof. See Appendix A. The second step in the estimation procedure is to estimate RhI k(0) for k"0, 1,2, ¸. Having obtained the estimate hK , we can form the residual sequence bK (n)"y(n)!hK TU(n),
(3.8)
where n"0, 1,2, N!1. Observing Eq. (3.4), let g(n)"DbK (n)D2!p2w!hTu(n),
(3.9)
where h"[RhI 0(0), RhI 1(0),2, RhI L(0)]T and u(n)"[ Du(n)D2, Du(n!1)D2,2, Du(n!¸)D2]T. Then the estimate hK of h can be obtained by minimizing +N~1Dg(n)D2 with respect to h. Thus hK is given n/0 by [13,21]
G
H
N~1 ~1N~1 + u(n)[DbK (n)D2!p2w]. hK " + u(n)uT(n) n/0 n/0 (3.10) Proposition 2. Let y(n), u(n) be related by Eq. (2.1) or Eq. (3.1). Given a set of training data X(N!1) in Eq. (3.6), the estimate hK in Eq. (3.10) converges to h as N tends to infinity in the mss and wp1 under the assumptions AS1—AS3. Proof. See Appendix B.
4. Optimal equalization filter design under multipath Rician fading channels This section proposes a novel approach to design an optimal and realizable equalization filter for multipath Rician fading channels with stochastic tap coefficients. Based on statistical characteristics of input signal, channel noise, and fading channel, the optimal equalization filter is designed to minimize the MSE from the viewpoint of frequency domain. For notational simplicity, we derive the design procedure under the assumption of known statistical characteristics. But for simulation examples in next section, we design the optimal equalization filter based on the estimated statistical characteristics of fading channels under investigation. To treat this equalization problem (see Fig. 2), Eq. (2.1) can be rewritten in the following form: y(n)"H(q~1;n)u(n)#w(n),
(4.1)
where q~1 represents the backward shift operator, i.e., q~1u(n)"u(n!1), and H(q~1;n)" +Lk/0hk(n)q~k. Given the received signal y(n), the problem is to find a stable and causal filter F(q~1) to reconstruct the input signal uL (n)"F(q~1)y(n),
(4.2)
such that the following MSE J"Eh,u,w[De(n!l)D2] "Eh,u,w[Du(n!l)!uL (n)D2]
(4.3)
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
Fig. 2. The proposed receiver.
is minimized, where e(n) denotes the reconstruction error, and the MSE J is regarded as the reconstruction performance index. Remark 4. Depending on the value of l, the equalization filter can be a filter (l"0), a smoother (l'0), or a predictor (l(0). In order to improve the reconstruction performance, the smoothing case is considered.
L L L " + (RhI i(0)#DhiD2)# + + hih*j z~i`j. i/0 i/0 j/0,jEi (4.7) The performance index J in Eq. (4.5) can be rewritten as
Q
From Eqs. (4.1), (4.2) and (4.3), we get e(n!l)"(q~l!F(q~1)H(q~1;n))u(n) (4.4)
!F(q~1)w(n).
L L L " + E[Dhi(n)D2]# + + E[hi(n)]E[h*j (n)]z~i`j i/0 i/0 j/0,jEi
1 J" M[1!z~lF*(z~1)H*(z~1) 2pj @z@/1 !zlF(z~1)H(z~1) #H*(z~1)H(z~1)F*(z~1)F(z~1)]p2 u dz #F*(z~1)F(z~1)p2wN . (4.8) z
Because both w(n) and hk(n) are uncorrelated with u(n) (AS1), according to Parseval’s theorem, the reconstruction performance index J can be rewritten as
The problem now lies in how to find an optimal equalization filter Fo(z~1) such that Eq. (4.8) is minimized. In the sequel, calculus of variation technique in complex domain, Cauchy theory and spectral factorization method are employed to treat this optimization problem. Let
Q
1 J" ME [(z~l!F(z~1)H(z~1;n))*(z~l 2pj @z@/1 h !F(z~1)H(z~1;n))]p2u dz #F*(z~1)F(z~1)p2wN , z
(4.5)
where {@z@/1 denotes the counter-clockwise integration around the unit circle. Since only the coefficients of H(z~1;n) are stochastic processes, let us denote H(z~1)"Eh[H(z~1;n)] L L " + Eh[hk(n)z~k]" + hkz~k k/0 k/0 and H*(z~1)H(z~1) "Eh[H*(z~1;n)H(z~1;n)] "E
C
h
D
L L + + h (n)h*(n)z~izj i j i/0 j/0
(4.6)
F(z~1)"Fo(z~1)#eg(z~1),
(4.9)
where Fo(z~1) denotes the optimal equalization filter to be derived, g(z~1) is any realizable function with all poles in DzD(1, and e is an arbitrarily small real number. Then the MSE in Eq. (4.8) can be rewritten as
Q
1 J" M[1!z~l(F*o(z~1)#eg*(z~1))H*(z~1) 2pj @z@/1 !zl(F (z~1)#eg(z~1))H(z~1) o * #H (z~1)H(z~1)(F*(z~1) o #eg*(z~1))(Fo(z~1)#eg(z~1))]p2u #[F*o(z~1)#eg*(z~1)][Fo(z~1) dz (4.10) #eg(z~1)]p2wN . z
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
Based on the calculus of variation technique, the MMSE must satisfy the following condition:
K
LJ Le
(4.11)
"0, e/0
i.e.
Q
1 MM!z~lg*(z~1)H*(z~1)!zlg(z~1)H(z~1) 2pj @z@/1 #H*(z~1)H(z~1)[g*(z~1)Fo(z~1) #F*o(z~1)g(z~1)]Np2u #[F*o(z~1)g(z~1) dz #g*(z~1)Fo(z~1)]p2wN "0. z
(4.12)
By symmetry, the above equation can be rewritten as
Q
#H*(z~1)H(z~1)g*(z~1)Fo(z~1)]p2u dz #g*(z~1)Fo(z~1)p2wN "0. z
(4.13)
Discard the constant 2/(2pj) and rearrange the terms in the integrand, then Eq. (4.13) can be rewritten as
Q
MFo(z~1)[H (z~1)H(z~1)p2u #p2w] @z@/1 dz "0. z
(4.14)
Let us perform the spectral factorization [1,2] d*(z~1)d(z~1)"H*(z~1)H(z~1)p2#p2 , u w
(4.15)
where d(z~1) is free of poles and zeros in DzD'1. Substituting Eq. (4.15) into Eq. (4.14) yields
Q
MFo(z~1)d(z~1)d*(z~1) @z@/1 !z~lH*(z~1)p2u Ng*(z~1)
zP*(z~1)"F (z~1)d(z~1)d*(z~1)!z~lH*(z~1)p2. 0 u (4.17) By Cauchy’s residue theorem, Eq. (4.16) holds only if P*(z~1) is analytic inside the unit circle on the z-plane [15]. Multiplying both sides of Eq. (4.17) by d~*(z~1) yields zP*(z~1)d~*(z~1) "F0(z~1)d(z~1)!z~lH*(z~1)d~*(z~1)p2u , (4.18) where d~*(z~1) is the inverse and conjugate of d(z~1). The second term of right-hand side in Eq. (4.18) can be decomposed as z~lH*(z~1)d~*(z~1)p2u "Mz~lH*(z~1)d~*(z~1)p2u N` where M ) N`, denoting the causal and stable part of M ) N, is analytic outside the unit circle on the z-plane, and M ) N~, denoting the anti-causal and stable part of M ) N, is analytic inside the unit circle on the z-plane. Substituting Eq. (4.19) into Eq. (4.18) and rearranging the items, we get zP*(z~1)d~*(z~1)#Mz~lH*(z~1)d~*(z~1)p2u N~
Note that the left-hand side of Eq. (4.20) is analytic inside the unit circle, while the right-hand side is analytic outside the unit circle. Hence, the only solution is that both of them are zero at the same time. According to this analysis, we can obtain the optimal equalization filter F0(z~1) as F (z~1)"Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1). (4.21) 0 u ` Remark 5. Observing Eq. (4.15), if the channel noise power tends to zero, i.e., p2 P0, we have w d*(z~1)d(z~1)KH*(z~1)H(z~1)p2u .
dz "0. z
(4.19)
"F0(z~1)d(z~1)!Mz~lH*(z~1)d~*(z~1)p2u N`. (4.20)
*
!z~lH*(z~1)p2u Ng*(z~1)
Let
#Mz~lH*(z~1)d~*(z~1)p2u N~,
2 M[!z~lg*(z~1)H*(z~1) 2pj @z@/1
49
(4.16)
(4.22)
If the transmission channel is further assumed to be a time-invariant minimum phase system, then we
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
can obtain d(z~1)KH(z~1)p . (4.23) u In this situation, the optimal equalization filter Fo(z~1) in Eq. (4.21) can be derived by F0(z~1)Kz~lH~1(z~1). (4.24) Hence, under these two conditions mentioned above, the optimal equalization filter Fo(z~1) in Eq. (4.24) is just the delayed inverse filter of the transmission channel.
Table 1 Design algorithm of the proposed optimal equalization filter Step 1
Obtain H(z~1) from Eq. (4.6) and H*(z~1)H(z~1) from Eq. (4.7).
Step 2
Perform the spectral factorization in Eq. (4.26) to obtain d(z~1) and d*(z~1).
Step 3
Decompose z~lH*(z~1)d~*(z~1)p2 by u z~lH*(z~1)d~*(z~1)p2"Mz~lH*(z~1)d~*(z~1)p2N # u u ` Mz~lH*(z~1)d~*(z~1)p2N . u ~ Obtain the optimal equalization filter F (z~1)"Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1). 0 u `
Step 4
Remark 6. From Eq. (4.10), we get
K
L2J Le2
Q
1 " Mg*(z~1)g(z~1)H*(z~1)H(z~1)p2u 2pj @z@/1 e/0 dz #g*(z~1)g(z~1)p2wN '0 z (4.25)
is always positive. The necessary and sufficient condition for the optimal equalization filter Fo(z~1) in Eq. (4.21) is guaranteed. Remark 7. From Eq. (4.7), the spectral factorization in Eq. (4.15) can be rewritten as follows:
C
L d*(z~1)d(z~1)" + (RhI i(0)#DhiD2) i/0 L L # + + h h*z~i`j p2#p2 . u w i j i/0 j/0,jEi (4.26)
D
A numerical computation algorithm of the above spectral factorization has been found in [10]. Remark 8. With the optimal equalization filter in Eq. (4.21), the MMSE given by Appendix C is 1 J.*/" 2pj
Q
MMz~lH*(z~1)d~*(z~1)p2u N*~ @z@/1
]Mz~lH*(z~1)d~*(z~1)p2N u ~ * * #[d (z~1)d(z~1)!H (z~1)H(z~1)p2u ] dz ][d*(z~1)d(z~1)]~1p2u N . z
(4.27)
Remark 9. The value of MMSE J in Eq. (4.27) .*/ can be directly calculated by the residue integration method. From the above analysis, a complete design algorithm of the optimal equalization filter for multipath Rician fading channels with stochastic tap coefficients is summarized in Table 1.
5. Simulation examples In this section, two simulation examples are given to illustrate the performance of the proposed design method. For comparison, a conventional adaptive decision feedback equalizer (DFE) using RLS adaptation algorithm is also designed. Before demonstrating the simulation examples, the operations of our proposed design method and the conventional adaptive DFE will be described, respectively. The operations of the proposed design algorithm are explained as follows. Initial training mode. During the initial training period, the training data are used to estimate the channel’s statistical characteristics. Then based on these estimated statistical characteristics, an optimal equalization filter is derived. The detailed design steps are described below. Step 1. Collect the training data X(N!1) in Eq. (3.6), where N is the data length during the initial training period and N"1000 for the simulation examples in this study.
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
Step 2. From Eq. (3.5), obtain the estimate hK using RLS adaptation algorithm. Step 3. From Eq. (3.9), obtain the estimate hK using RLS adaptation algorithm. Step 4. Based on hK and hK obtained at Steps 2 and 3, derive the optimal equalization filter according to the design procedure in Table 1. Operating mode. Equalize the received data sequence, y(n), using the optimal equalization filter derived at the initial training mode and decode the equalized data sequence uL (n) using the nearestneighbor criterion. Also, the operations of the adaptive DFE are explained as follows. Initial training mode. During the initial training period, the training data are used to adapt the coefficients of the DFE by recursively minimizing the sum of squares of the differences between the desired values (input samples) and the equalized values of the received samples. The detailed design steps are described below. Step 1. Collect the training data X(N!1) in Eq. (3.6). Step 2. Run RLS algorithm to estimate a constant vector u by minimizing N~1 + DuL (n)!u(n)D2, n/0 where u"[g(!K ),2, g(!1),g(0), g(1),2, g(K )]T 1 2 and 0 K2 uL (n)" + g(k)y(n!k)# + g(k)u(n!k). k/~K1 k/1 Operating mode. Based on the estimated vector u at initial training mode, decode the equalized data sequence uL (n) using the nearest-neighbor criterion, where uL (n)"+0 g(k)y(n!k)# k/~K1 +K2 g(k)uN (n!k) and uN (n) being a valid 16-QAM k/1 symbol sequence for our simulation examples is the decoded data sequence.
51
Periodic retraining#operating mode. It is well known that the ‘run-away’ phenomenon may occur to the conventional adaptive DFE. Hence, a periodic retraining # operating mode is necessary. For simulation examples in this section, a periodic retraining period of 14 samples is used, alternating with an operating period of 128 samples. Remark 10. Unlike the conventional adaptive DFE, the designed optimal equalization filter is a fixed one which needs not to adapt the parameters of filter after the initial training period. Remark 11. It is worth noting that the conventional adaptive DFE needs a periodic retraining period to avoid a possible ‘run-away’ phenomenon. On the contrary, our proposed design method only needs training data during the initial training period. Hence, our proposed design method needs less training data than the conventional adaptive DFE. Remark 12. The respective number of taps in the forward and feedback filters of the conventional adaptive DFE is 5 and 2 for the simulation examples. This DFE is able to converge within 14 symbol periods [5,17]. Remark 13. The proposed design method does not work for any M-ary phase modulated signal in that AS3 (E[Du(n)D4]O(E[Du(n)D2])2) is violated. However, if the statistical characteristics of fading channels are known, we still can design the optimal equalization filter based on known statistical characteristics. In the sequel, we shall illustrate and compare the performances of these two different design algorithms mentioned above by giving two simulation examples. In these two simulation examples, the smoothing lag l"3, the input signal is a 16-QAM data sequence and the average signal-to-noise ratio 2 +. (SNR) is defined as SNR"10 log E*@y(n)~w(n)@ 10 E*@w(n)@2+ Example 1. Consider a two path (i.e., ¸"1 in Eq. (2.1)) fading channel, the parameters in Eqs. (2.2) and (2.4) are given by h "0.8#0.6j, 0
52
B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
Fig. 3. (a) Estimation of Real(h ); (b) estimation of Imag(h ); (c) estimation of Real(h ); (d) estimation of Imag(h ); (e) estimation of R M 0(0); 0 0 1 1 h (f) estimation of R M (0). h1
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
53
h "0.4#0.3j, o "o "0.95, p2 "4]10~4, p2 v0 v1 1 0 1 "2]10~4. Example 2. Consider a four path (i.e. ¸"3 in Eq. (2.1)) fading channel, the parameters in Eqs. (2.2) and (2.4) are given by h "1, h " 0 1 !0.4!0.5j, h "0.16#0.2j, h "!0.08j, 2 3 o "o "o "o "0.95, p2 "p2 "p2 "p2 " v0 v1 v2 v3 0 1 2 3 4]10~4. For these two examples, our purpose is twofold. The first purpose is to show the performance of our proposed estimation method. In Fig. 3(a)—(f) we show the recursive estimate of statistical characteristics of fading channel in Example 1. The true values are also indicated by the solid line. All the simulations are implemented with SNR"20 dB. These results show that after the 400 samples, the estimated statistical characteristics converge to the true values. The second purpose is to illustrate the performance of the proposed design method by plotting the respective symbol error probability (SEP) as a function of the SNR. Each different SNR is achieved by tuning the channel noise power p2 . w Fig. 4 (Fig. 5) shows the SEP of Example 1 (Example 2) for the proposed design method (solid line) and the conventional adaptive DFE (dashed line). The
Fig. 5. Symbol error probability versus SNR for Example 2. Solid line (——) for the proposed design method, dashed line (----) for the conventional adaptive DFE.
SEP is obtained by averaging 100 independent Monte Carlo (MC) simulation runs for each SNR and the data length of each MC realization is 2048 points (i.e. the data length during 16 operating periods for the conventional adaptive DFE). From the simulation results shown in Fig. 4 (Fig. 5), the performance of the proposed design method surpasses that of adaptive DFE at about 10 (6) dB for higher SNR.
6. Conclusion
Fig. 4. Symbol error probability versus SNR for Example 1. Solid line (——) for the proposed design method, dashed line (----) for the conventional adaptive DFE.
This study has proposed a novel design method to treat the estimation and equalization problems of multipath Rician fading channels with stochastic tap coefficients. In the proposed design method, a two-step estimation procedure using RLS adaptation algorithm was employed to estimate the mean and variance of stochastic tap coefficients. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter minimizing the MSE was designed from the viewpoint of frequency domain to reconstruct the transmitted sequence. Furthermore, calculus of variation technique, spectral factorization method and Cauchy’s residue theorem were employed in the derivation procedure of the optimal equalization filter. Also, an expression for the MMSE was
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
derived. This is the first time to design a fixed filter to achieve the optimal signal reconstruction for multipath fading channels under investigation. Unlike conventional adaptive equalization schemes, which adapt the parameters of filter during the initial training period and retraining periods, the proposed optimal equalization filter needs not to adapt the parameters of filter after the initial training period. Therefore, it is more easy to implement and
more suitable for real time signal reconstruction under multipath Rician fading channels. To illustrate the performance of the proposed design method, a conventional adaptive DFE using RLS adaptation algorithm was also designed for comparison. The simulation results have confirmed the superior performance of the proposed design method.
Appendix A. Proof of Proposition 1 Substituting Eq. (3.1) into Eq. (3.7), we get
G
H G H G
H
~1 1 N~1 * 1 N~1 * + U (n)UT(n) + U (n)[UT(n)h#UT(n)hI (n)#w(n)] (A.1) N N n/0 n/0 ~1 1 N~1 * ~1 1 N~1 * 1 N~1 * 1 N~1 * + U (n)UT(n) + U (n)UT(n)hI (n) # + U (n)UT(n) + U (n)w(n) . (A.2) "h# N N N N n/0 n/0 n/0 n/0 It can be shown that the entries of U*(n)UT(n), U*(n)UT(n)hI (n) and U*(n)w(n) have finite memory and variance. According to Lemma B.1 in [21], the braced terms in Eq. (A.2) converge to their expected values as N tends to infinity in the mss and wp1, i.e., hK "
G
H G
H G
H
hK Ph#R~1 as NPR in the mss and wp1. UU RUUhI #R~1 UU RUw
(A.3)
Under the assumption AS1, RUU"p2I, RUUhI "RUUE[hI (n)]"0 and RU "E[U*(n)]E[w(n)]"0. Hence, we u w can conclude from Eq. (A.3) that hK converges to h as N tends to infinity in the mss and wp1. Also, we can conclude that hK is a strongly consistent estimate [13] of h.
Appendix B. Proof of Proposition 2 Substituting Eq. (3.1) into Eq. (3.8), we get bK (n)"UT(n)(h!hK )#UT(n)hI (n)#w(n).
(B.1)
Hence, DbK (n)D2"[UT(n)(h!hK )#UT(n)hI (n)#w(n)]*[(h!hK )TU(n)#hI T(n)U(n)#w(n)] "UH(n)(h!hK )*(h!hK )TU(n)#UH(n)(h!hK )*hI T(n)U(n)#UH(n)(h!hK )*w(n)#UH(n)hI *(n)(h!hK )TU(n) #UH(n)hI *(n)hI T(n)U(n)#UH(n)hI *(n)w(n)#w*(n)(h!hK )TU(n)#w*(n)hI T(n)U(n)#w*(n)w(n). Substituting Eq. (B.2) into Eq. (3.10), we get
G
H G
H
~1 1 N~1 N~1 hK " N + u(n)uT(n) + u(n)UH(n)(h!hK )*(h!hK )TU(n) N n/0 n/0
(B.2)
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
G G G G G G G G
H H H H H H H H
G G G G G G G G
55
H
~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)(h!hK )*hI T(n)U(n) N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)(h!hK )*w(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)(h!hK )TU(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)hI T(n)U(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)w(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)(h!hK )TU(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)hI T(n)U(n) # N N n/0 n/0 ~1 1 N~1 ~1 1 N~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)w(n) ! + u(n)uT(n) + u(n)p2 . # (B.3) w N N N N n/0 n/0 n/0 n/0 According to Proposition 1 and following the similar analyses in Proposition 1, the braced terms in Eq. (B.3) converge to their expected values as N tends to infinity in the mss and wp1. Hence, under the assumptions AS1—AS3, the first—fourth and sixth—eigth terms tend to zero vector, the ninth and tenth terms tend to R~1E[u(n)]p2 , and the fifth term tends to h as NPR in the mss and wp1, where R , a positive definite rr w rr matrix, is defined by #
C
H
H
H
H
H
H H G
E[Du(n)D4]
R "E [u (n) uT (n)]" rr
p4 u F
p4 2 u } } }
}
p4 u F
H G
D
p4 u E[Du(n)D4]
.
H
(B.4)
2 p4 p4 u u From the above analyses, we conclude that hK converges to h as N tends to infinity in the mss and wp1. Hence, hK is a strongly consistent estimate [13] of h. Remark 14. In view of Eq. (B.4), if assumption AS3 (E[Du(n)D4]O(E[Du(n)D2])2) does not hold, All the elements in R will be equal to p4. In such case, R will be a singular matrix. Subsequently, hK will not be guaranteed u rr rr to converge to h as NPR. Therefore, assumption AS3 is required to guarantee that hK will always converge to h in the mss and wp1 as NPR.
Appendix C. Derivation of Eq. (4.27) Before obtaining the MMSE, we rewrite Mz~lH*(z~1)d~*(z~1)p2N "z~lH*(z~1)d~*(z~1)p2!Mz~lH*(z~1)d~*(z~1)p2N . u ~ u u `
(C.1)
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
Thus, Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N "H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4 u ~ u ~ u !z~lH*(z~1)p2Mz~lH*(z~1)d~*(z~1)p2N* d~*(z~1)!zlH(z~1)p2Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1) u u ` u u ` #Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N . u ` u ` Using the relationship in Eq. (4.21), Eq. (C.2) can be reduced as
(C.2)
Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N "H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4 u ~ u ~ u !z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)d*(z~1)d(z~1). (C.3) u u 0 0 0 0 Now, replacing F(z~1) with F (z~1), the MMSE in Eq. (4.8) becomes 0 dz 1 Mp2!z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)[H*(z~1)H(z~1)p2#p2 ]N J " u w z u u 0 u 0 0 0 .*/ 2pj @z@/1
Q
1 " 2pj
Q
1 J " .*/ 2pj
Q
1 " 2pj
Q
dz Mp2!z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)d*(z~1)d(z~1)N . u 0 u u 0 0 0 z @z@/1 Associating the result in Eq. (C.3), the Eq. (C.4) can be rewritten as
(C.4)
MMz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N #p2 u u ~ u ~ @z@/1 dz !H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4N u z MMz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N #[d*(z~1)d(z~1) u ~ u ~ @z@/1 dz !H*(z~1)H(z~1)p2][d*(z~1)d(z~1)]~1p2N . u u z
References [1] Y.L. Chen, B.S. Chen, Minimax robust deconvolution filters under stochastic parametric and noise uncertainties, IEEE Trans. Signal Process. SP-42 (1) (January 1994) 32—45. [2] B.-S. Chen, S.-C. Peng, Deconvolution filter design with consideration of channel sensitivity, Signal Processing 39 (1—2) (1994) 149—166. [3] G. D’Aria, R. Piermarini, V. Zingarelli, Fast adaptive equalizers for narrow-band TDMA mobile radio, IEEE Trans. Veh. Techno1. 40 (2) (May 1991) 392—404. [4] H. Hashemi, The indoor radio propagation channel, Proc. IEEE 81 (7) (July 1993) 943—967. [5] S. Haykin, Adaptive Filter Theory, Prentice Hall, Englewood Cliffs, NJ, 1985. [6] R.A. Iltis, A.W. Fuxjaeger, A digital DS spread-spectrum receiver with joint channel and doppler shift estimation, IEEE Trans. Commun. 39 (8) (August 1991) 1255—1267.
(C.5)
[7] W.C. Jakes, Microwave Mobile Communications, Wiley, New York, 1974. [8] M.C. Jeruchim, P. Balaban, K.S. Shanmugan, Simulation of Communication Systems, Plenum, New York, 1992. [9] I. Korn, M-ary CPFSK-DPD with L-diversity maximum ratio combining in Rician fast-fading channels, IEEE Trans. Veh. Techno1. 45 (4) (November 1996) 613—621. [10] V. Kucera, Discrete Linear Control: The Polynomial Equation Approach, Wiley, New York, 1979. [11] H. Lan, C.J. Burkley, Fading rate and multipath delay effects on signal acquisition using sliding correlation in a multi-user environment, Wireless Personal Communications 2 (1995) 235—244. [12] F. Ling, J.G. Proakis, Adaptive lattice decision feedback equalizer — Their performance and application to timevariant multipath channels, IEEE Trans. Commun. COM-33 (4) (April 1985) 348—356.
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57 [13] L. Ljung, T. So¨derstro¨m, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983. [14] D.W. Matolak, S.G. Wilson, Detection for a statistically known, time-varying dispersive channel, IEEE Trans. Commun. 44 (12) (December 1996) 1673—1683. [15] I.L. Nagrath, M. Gopal, Control Systems Engineering, Wiley, New York, 1982. [16] J.G. Proakis, Digital Communications, 2nd edn., McGrawHill, New York, 1989. [17] S.U.H. Qureshi, Adaptive equalization, Proc. IEEE 73 (September 1985) 1349—1387. [18] B. Rosenberg, A survey of stochastic parameter regression, Ann. Econ. and Soc. Meas. 2 (1973) 381—398. [19] R. Sharma, W.D. Grover, W.A. Krzymien, Forward-error-control (FEC) — Assisted adaptive equalization for digital cellular mobile radio, IEEE Trans. Veh. Techno1. 42 (1) (February 1993) 94—102.
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[20] W.H. Sheen, G.L. Stu¨ber, MLSE equalization and decoding for multipath-fading channels, IEEE Trans. Commun. 39 (10) (October 1991) 1455—1464. [21] T. So¨derstro¨m, P. Stoica, System Identification, Series in Systems and Control Engineering, Prentice-Hall, Englewood Cliffs, NJ, London, 1989. [22] H. Suzuki, A statistical model for urban radio propagation, IEEE Trans. Commun. COM-25 (July 1977) 673—680. [23] G.M. Vitetta, D.P. Taylor, U. Mengali, Double-filtering receivers for PSK signals transmitted over Rayleigh frequency—flat fading channels, IEEE Trans. Commun. 44 (6) (June 1996) 686—695. [24] G.W. Wornell, Spread-response precoding for communication over fading channels, IEEE Trans. Inform. Theory 42 (2) (March 1996) 488—501.
Estimation and equalization of multipath Rician fading channels with stochastic tap coefficients Bor-Sen Chen*, Yih-Jinn Huang, Sin-Chung Chen Department of Electrical Engineering, National Tsing-Hua University, Hsin-Chu 300, Taiwan, ROC Received 21 August 1996; received in revised form 10 March 1998
Abstract This study investigates the estimation and equalization problems of multipath Rician fading channels with stochastic tap coefficients. The time-varying tap coefficients of fading channels under investigation are typically modeled as complex-valued Gaussian processes. First, least squares estimation methods are used to estimate the mean and variance of the stochastic tap coefficients. Then, based on these estimated statistical characteristics, an optimal equalization filter is designed to recover the transmitted sequence. In the proposed design method, a realizable equalization filter is constructed to minimize the mean square error from the perspective of frequency domain. Also, both calculus of variation technique and spectral factorization method are employed in the design procedure. To exhibit the superior performance of the proposed design method, some illustrative simulations are presented. ( 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Die vorliegende Studie untersucht das Scha¨tzproblem und das Entzerrungsproblem fu¨r Rice’sche Schwundkana¨le mit Mehrwegeausbreitung und stochastischen lmpulsantwortkoeffizienten. Die zeitvarianten lmpulsantwortkoeffizienten der untersuchten Schwundkana¨le werden typischer Weise als komplexwertige Gau{prozesse modelliert. Zuerst werden Kleinste-Quadrate-Methoden benutzt, um Mittelwert und Varianz der stochastischen lmpulsantwortkoeffizienten zu scha¨tzen. Danach wird auf der Grundlage dieser gescha¨tzten statistischen Charakteristiken ein optimales Entzerrerfilter zur Ru¨ckgewinnung der u¨bertragenen Folge entworfen. ln dem vorgestellten Entwurfsverfahren wird ein realisierbares Entzerrerfilter konstruiert, das den mittleren quadratischen Fehler im Frequenzbereich minimiert. Auch werden in dem Entwurfsverfahren sowohl die Variationsrechung als auch die Methode der spektralen Faktorisierung eingesetzt. Um die ho¨here Leistung der vorgeschlagenen Entwurfsmethode zu zeigen, werden einige illustrative Simulationen vorgestellt. ( 1998 Elsevier Science B.V. All rights reserved. Re´sume´ Cette e´tude porte sur les proble`mes d’estimation et d’e´galisation de canaux a` e´vanouissement multi-chemins de type Rice avec des coefficients de filtre ale´atoires. Les coefficients de filtre variant dans le temps des canaux a` e´vanouissement analyse´s sont typiquement mode´lise´s par des processus gaussiens a` valeurs complexes. Tout d’abord, les me´thodes
—————— * Corresponding author. E-mail: [email protected]. 0165-1684/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 7 ) 0 0 0 5 6 - 5
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
d’estimation aux moindres carre´s sont utilise´es pour estimer la moyenne et la variance des coefficients de filtre ale´atoires. Ensuite, sur la base des estime´es de ces caracte´ristiques statistiques, un filtre d’e´galisation optimal est conc7 u pour recouvrer la se´quence transmise. Dans la me´thode de conception propose´e, un filtre d’e´galisation re´alisable est construit pour minimiser l’erreur quadratique moyenne dans le domaine fre´quentiel. De plus, la technique du calcul des variations et la me´thode de factorisation spectrale sont employe´es dans la proce´dure de conception. Quelques exemples de simulation illustratifs sont pre´sente´s pour mettre en lumie`re les performances supe´rieures de la me´thode de conception propose´e. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Multipath Rician fading channel; Wide sense stationary and uncorrelated scattering
1. Introduction Signal reconstruction of the transmitted sequence in the presence of intersymbol interference (ISI) and channel noise is an important research topic for digital communication systems. The ISI is caused by nonideal channel characteristic and the channel noise comes from quantization, external corruption, etc. Both ISI and channel noise are two major impeding factors for high speed data transmission. The role of the equalization filter is to reconstruct the transmitted sequence by removing the effect of ISI and eliminating the corruption of channel noise. In general, an optimal equalization filter can be designed when the transfer function of the transmission channel is precisely known and the statistical properties of the input signal and channel noise are also exactly given. However, in practical digital communication systems, the transmission channels can not be precisely estimated beforehand. Furthermore, the transmission channels may have randomly time-varying impulse responses corrupted by channel noises. This characterization serves as a model for signal transmission over many communication channels [4,6,7,16]. The time-varying impulse responses of these channels are a consequence of the constantly changing physical characteristics of the media or the time variations in the structure of the medium. As a result of such variations, the nature of the transmission channel varies with time. Moreover, the time variations appear to be unpredictable to the user of the channel. Therefore it is reasonable to characterize the time-varying transmission channel statistically.
Multipath fading channels with Rician (or Rayleigh) distribution have been used to characterize the statistical properties of time-varying channels for a long time [9,11,12,22—24], each tap coefficient of multipath fading channels is usually modeled as a complex-valued Gaussian process. To treat the equalization problem for such a timevarying channel, conventional adaptive equalization schemes using recursive least squares (RLS) adaptation algorithm are usually employed to track the channel’s variations and equalize the received signals [3,19]. In [14], Matolak and Wilson develop a maximum likelihood estimator (MLE) to detect the transmitted sequence for a statistically known Rayleigh fading channel. This study proposes a novel design method to treat the estimation and equalization problems for multipath Rician fading channels with stochastic tap coefficients. In the proposed design method, least squares estimation methods are employed to estimate the mean and variance of stochastic tap coefficients. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter which minimizes the mean square error (MSE) is designed from the viewpoint of frequency domain to reconstruct the transmitted sequence. This is the first time to develop a closed form filter to achieve the optimal signal reconstruction under multipath fading channels. Unlike conventional adaptive equalization schemes, the proposed optimal equalization filter is of fixed form which needs not to adapt parameters of filter after the initial training period. Therefore, it is easy to implement and suitable for real time signal reconstruction. Furthermore, calculus of variation technique, spectral factorization method and Cauchy’s residue theorem are employed
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
in the derivation procedure of the optimal equalization filter. Also, an expression for the minimum mean square error (MMSE) is derived. Finally, some illustrative simulations are given to exhibit the superior performance of the proposed design method. The organization of this paper is as follows. In Section 2, the problem description is presented and some modeling aspects of multipath Rician fading channels are discussed. In Section 3, least squares estimation methods are employed to estimate the statistical characteristics of fading channels under investigation. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter minimizing the MSE is derived in Section 4. Two simulation examples are given in Section 5 to exhibit the performance of the proposed design method. Finally, a brief conclusion is given in Section 6.
2. System description and problem formulation Digital signal transmission through a multipath fading channel is usually described by Fig. 1(a). In this figure, the i.i.d., equiprobable input sequence u(n) with zero mean and variance p2u is transmitted every ¹s seconds through a communication system consisting of the transmitter’s pulse shaping filter, the multipath fading channel, the receiver’s matched
45
filter and the sampler. It is well known that the received discrete-time signal, y(n), is given by [8,16,20,24] L y(n)" + hk(n)u(n!k)#w(n), (2.1) k/0 where w(n) denotes the additive white circular complex-valued Gaussian noise with zero mean and variance p2w and hk(n) is the time-varying impulse response of the overall system including transmitter/receiver filters and fading channel (see Fig. 1(b)). In this study, the time-varying impulse response hk(n) under investigation can be expressed in the following form: hk(n)"hk#hI k(n), (2.2) where hk is a non-zero constant mean and hI k(n) is wide-sense stationary and uncorrelated scattering (WSSUS) [6,9,16,24], i.e., E[hI *(n)hI (n!m)]"R I k(m)d(k!l), (2.3) k l h where d[n] is the unit sample sequence defined by
G
d(n)"
1, n"0, 0, otherwise,
and for each k, hI k(n) is a circular complex-valued Gaussian sequence generated by [6,12,20] hI k(n)"okhI k(n!1)#vk(n),
Fig. 1. (a) Signal transmission system through multipath fading channel. (b) An equivalent discrete-time system of (a).
(2.4)
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
where v (n) is an i.i.d., circular complex-valued k Gaussian sequence with zero mean and variance p2k and the parameter o is arbitrarily chosen close v k to unity so that hI (n) represents a generic low-pass k process. From Eq. (2.4), R I k(m) in Eq. (2.3) can be h calculated by
k"0,1,2,¸. Associating Eqs. (2.1) and (2.2), we have
o@m@ k RhI k(m)" p2 . 1!DokD2 vk
h"[h0, h1,2, hL]T,
Remark 1. If the coefficients hk(n) in Eq. (2.2) have non-zero constant means, then the time-varying channel is called the Rician fading channel [4,9]. Otherwise, it is called the Rayleigh fading channel [4,16,24].
and
The following assumptions are imposed on Eqs. (2.1), (2.2), (2.3) and (2.4): AS1: Both w(n) and h (n) are uncorrelated with the k input sequence u(n). AS2: w(n) is uncorrelated with vk(n) for k"0,1,2,¸. Subsequently, w(n) is uncorrelated with hI k(n) and hk(n) for k"0,1,2,¸. AS3: The input sequence u(n) has finite E[Du(n)D4] and E[Du(n)D4]O(E[Du(n)D2])2. For such a time-varying fading channel, the goal of this study is to estimate the channel’s statistical characteristics including each tap coefficient’s mean and variance. Then based on the estimated statistical characteristics, an optimal and realizable equalization filter which minimizes the MSE is designed from the viewpoint of frequency domain to reconstruct the transmitted sequence.
3. Estimation of the channel’s statistical characteristics In this section, a two-step estimation procedure is introduced to determine the channel’s statistical characteristics including each tap coefficient’s mean and variance. This estimation procedure is a generalization of a two-step procedure proposed by Rosenberg [18] to estimate the parameters of a stochastic coefficient regression model. The first step is to estimate each tap coefficient’s mean hk, for
y(n)"hTU(n)#hI T(n)U(n)#w(n),
(3.1)
where the superscript T denotes the transpose,
U(n)"[u(n), u(n!1),2, u(n!¸)]T
hI (n)"[hI 0(n), hI 1(n),2, hI L(n)]T. In view of Eq. (3.1), let b(n)"hI T(n)U(n)#w(n),
(3.2)
we have the conditional mean E[b(n)DU(n)]"E[hI T(n)]U(n)#E[w(n)]"0,
(3.3)
and the conditional variance E[b*(n)b(n)DU(n)] "UH(n)E[hI *(n)hI T(n)]U(n)#E[Dw(n)D2] L " + E[hI *k(n)hI k(n)]Du(n!k)D2#p2w k/0 L " + R I k(0)Du(n!k)D2#p2 , w h k/0
(3.4)
where the superscript * denotes the complex conjugate, and the superscript H denotes the transpose and complex conjugate. Remark 2. (a) The first equality in Eq. (3.4) is due to the assumption AS1. (b) The second equality in Eq. (3.4) is due to the uncorrelated scattering property of hI k(n) (see Eq. (2.3)). Combining Eqs. (3.1) and (3.2), we have y(n)"hTU(n)#b(n).
(3.5)
Given a set of training data X(N!1)"My(0), y(1),2, y(N!1),u(0), u(1),2, u(N!1)N,
(3.6)
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
we can obtain the least squares estimate hK of h from Eq. (3.5) by minimizing +N~1 n/0 Db(n)D2 with respect to h. Thus hK is given by [13,21]
G
H
N~1 ~1 N~1 * hK " + U*(n)UT(n) + U (n)y(n). n/0 n/0
(3.7)
Proposition 1. Let y(n), u(n) be related by Eqs. (2.1) and (3.1). Given a set of training data X(N!1) in Eq. (3.6), the estimate hK in Eq. (3.7) converges to h as N tends to infinity in the mean square sense (mss) and with probability one (wp1) under the assumption AS1.
47
Remark 3. The estimates hK and hK can be iteratively obtained by RLS or LMS algorithm. In this study, the RLS algorithm is employed to estimate h and h for simulation examples in Section 5. Let us denote p2hk"E[Dhk(n)D2]. Having obtained these estimates hK and hK , we can obtain the estimate pL 2hk of p2hk by pL 2hk"DhK kD2#RK hI k(0), for k"0,1,2,¸. These estimated statistical characteristics will be used to design an optimal and realizable equalization filter in the next section.
Proof. See Appendix A. The second step in the estimation procedure is to estimate RhI k(0) for k"0, 1,2, ¸. Having obtained the estimate hK , we can form the residual sequence bK (n)"y(n)!hK TU(n),
(3.8)
where n"0, 1,2, N!1. Observing Eq. (3.4), let g(n)"DbK (n)D2!p2w!hTu(n),
(3.9)
where h"[RhI 0(0), RhI 1(0),2, RhI L(0)]T and u(n)"[ Du(n)D2, Du(n!1)D2,2, Du(n!¸)D2]T. Then the estimate hK of h can be obtained by minimizing +N~1Dg(n)D2 with respect to h. Thus hK is given n/0 by [13,21]
G
H
N~1 ~1N~1 + u(n)[DbK (n)D2!p2w]. hK " + u(n)uT(n) n/0 n/0 (3.10) Proposition 2. Let y(n), u(n) be related by Eq. (2.1) or Eq. (3.1). Given a set of training data X(N!1) in Eq. (3.6), the estimate hK in Eq. (3.10) converges to h as N tends to infinity in the mss and wp1 under the assumptions AS1—AS3. Proof. See Appendix B.
4. Optimal equalization filter design under multipath Rician fading channels This section proposes a novel approach to design an optimal and realizable equalization filter for multipath Rician fading channels with stochastic tap coefficients. Based on statistical characteristics of input signal, channel noise, and fading channel, the optimal equalization filter is designed to minimize the MSE from the viewpoint of frequency domain. For notational simplicity, we derive the design procedure under the assumption of known statistical characteristics. But for simulation examples in next section, we design the optimal equalization filter based on the estimated statistical characteristics of fading channels under investigation. To treat this equalization problem (see Fig. 2), Eq. (2.1) can be rewritten in the following form: y(n)"H(q~1;n)u(n)#w(n),
(4.1)
where q~1 represents the backward shift operator, i.e., q~1u(n)"u(n!1), and H(q~1;n)" +Lk/0hk(n)q~k. Given the received signal y(n), the problem is to find a stable and causal filter F(q~1) to reconstruct the input signal uL (n)"F(q~1)y(n),
(4.2)
such that the following MSE J"Eh,u,w[De(n!l)D2] "Eh,u,w[Du(n!l)!uL (n)D2]
(4.3)
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B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
Fig. 2. The proposed receiver.
is minimized, where e(n) denotes the reconstruction error, and the MSE J is regarded as the reconstruction performance index. Remark 4. Depending on the value of l, the equalization filter can be a filter (l"0), a smoother (l'0), or a predictor (l(0). In order to improve the reconstruction performance, the smoothing case is considered.
L L L " + (RhI i(0)#DhiD2)# + + hih*j z~i`j. i/0 i/0 j/0,jEi (4.7) The performance index J in Eq. (4.5) can be rewritten as
Q
From Eqs. (4.1), (4.2) and (4.3), we get e(n!l)"(q~l!F(q~1)H(q~1;n))u(n) (4.4)
!F(q~1)w(n).
L L L " + E[Dhi(n)D2]# + + E[hi(n)]E[h*j (n)]z~i`j i/0 i/0 j/0,jEi
1 J" M[1!z~lF*(z~1)H*(z~1) 2pj @z@/1 !zlF(z~1)H(z~1) #H*(z~1)H(z~1)F*(z~1)F(z~1)]p2 u dz #F*(z~1)F(z~1)p2wN . (4.8) z
Because both w(n) and hk(n) are uncorrelated with u(n) (AS1), according to Parseval’s theorem, the reconstruction performance index J can be rewritten as
The problem now lies in how to find an optimal equalization filter Fo(z~1) such that Eq. (4.8) is minimized. In the sequel, calculus of variation technique in complex domain, Cauchy theory and spectral factorization method are employed to treat this optimization problem. Let
Q
1 J" ME [(z~l!F(z~1)H(z~1;n))*(z~l 2pj @z@/1 h !F(z~1)H(z~1;n))]p2u dz #F*(z~1)F(z~1)p2wN , z
(4.5)
where {@z@/1 denotes the counter-clockwise integration around the unit circle. Since only the coefficients of H(z~1;n) are stochastic processes, let us denote H(z~1)"Eh[H(z~1;n)] L L " + Eh[hk(n)z~k]" + hkz~k k/0 k/0 and H*(z~1)H(z~1) "Eh[H*(z~1;n)H(z~1;n)] "E
C
h
D
L L + + h (n)h*(n)z~izj i j i/0 j/0
(4.6)
F(z~1)"Fo(z~1)#eg(z~1),
(4.9)
where Fo(z~1) denotes the optimal equalization filter to be derived, g(z~1) is any realizable function with all poles in DzD(1, and e is an arbitrarily small real number. Then the MSE in Eq. (4.8) can be rewritten as
Q
1 J" M[1!z~l(F*o(z~1)#eg*(z~1))H*(z~1) 2pj @z@/1 !zl(F (z~1)#eg(z~1))H(z~1) o * #H (z~1)H(z~1)(F*(z~1) o #eg*(z~1))(Fo(z~1)#eg(z~1))]p2u #[F*o(z~1)#eg*(z~1)][Fo(z~1) dz (4.10) #eg(z~1)]p2wN . z
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
Based on the calculus of variation technique, the MMSE must satisfy the following condition:
K
LJ Le
(4.11)
"0, e/0
i.e.
Q
1 MM!z~lg*(z~1)H*(z~1)!zlg(z~1)H(z~1) 2pj @z@/1 #H*(z~1)H(z~1)[g*(z~1)Fo(z~1) #F*o(z~1)g(z~1)]Np2u #[F*o(z~1)g(z~1) dz #g*(z~1)Fo(z~1)]p2wN "0. z
(4.12)
By symmetry, the above equation can be rewritten as
Q
#H*(z~1)H(z~1)g*(z~1)Fo(z~1)]p2u dz #g*(z~1)Fo(z~1)p2wN "0. z
(4.13)
Discard the constant 2/(2pj) and rearrange the terms in the integrand, then Eq. (4.13) can be rewritten as
Q
MFo(z~1)[H (z~1)H(z~1)p2u #p2w] @z@/1 dz "0. z
(4.14)
Let us perform the spectral factorization [1,2] d*(z~1)d(z~1)"H*(z~1)H(z~1)p2#p2 , u w
(4.15)
where d(z~1) is free of poles and zeros in DzD'1. Substituting Eq. (4.15) into Eq. (4.14) yields
Q
MFo(z~1)d(z~1)d*(z~1) @z@/1 !z~lH*(z~1)p2u Ng*(z~1)
zP*(z~1)"F (z~1)d(z~1)d*(z~1)!z~lH*(z~1)p2. 0 u (4.17) By Cauchy’s residue theorem, Eq. (4.16) holds only if P*(z~1) is analytic inside the unit circle on the z-plane [15]. Multiplying both sides of Eq. (4.17) by d~*(z~1) yields zP*(z~1)d~*(z~1) "F0(z~1)d(z~1)!z~lH*(z~1)d~*(z~1)p2u , (4.18) where d~*(z~1) is the inverse and conjugate of d(z~1). The second term of right-hand side in Eq. (4.18) can be decomposed as z~lH*(z~1)d~*(z~1)p2u "Mz~lH*(z~1)d~*(z~1)p2u N` where M ) N`, denoting the causal and stable part of M ) N, is analytic outside the unit circle on the z-plane, and M ) N~, denoting the anti-causal and stable part of M ) N, is analytic inside the unit circle on the z-plane. Substituting Eq. (4.19) into Eq. (4.18) and rearranging the items, we get zP*(z~1)d~*(z~1)#Mz~lH*(z~1)d~*(z~1)p2u N~
Note that the left-hand side of Eq. (4.20) is analytic inside the unit circle, while the right-hand side is analytic outside the unit circle. Hence, the only solution is that both of them are zero at the same time. According to this analysis, we can obtain the optimal equalization filter F0(z~1) as F (z~1)"Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1). (4.21) 0 u ` Remark 5. Observing Eq. (4.15), if the channel noise power tends to zero, i.e., p2 P0, we have w d*(z~1)d(z~1)KH*(z~1)H(z~1)p2u .
dz "0. z
(4.19)
"F0(z~1)d(z~1)!Mz~lH*(z~1)d~*(z~1)p2u N`. (4.20)
*
!z~lH*(z~1)p2u Ng*(z~1)
Let
#Mz~lH*(z~1)d~*(z~1)p2u N~,
2 M[!z~lg*(z~1)H*(z~1) 2pj @z@/1
49
(4.16)
(4.22)
If the transmission channel is further assumed to be a time-invariant minimum phase system, then we
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can obtain d(z~1)KH(z~1)p . (4.23) u In this situation, the optimal equalization filter Fo(z~1) in Eq. (4.21) can be derived by F0(z~1)Kz~lH~1(z~1). (4.24) Hence, under these two conditions mentioned above, the optimal equalization filter Fo(z~1) in Eq. (4.24) is just the delayed inverse filter of the transmission channel.
Table 1 Design algorithm of the proposed optimal equalization filter Step 1
Obtain H(z~1) from Eq. (4.6) and H*(z~1)H(z~1) from Eq. (4.7).
Step 2
Perform the spectral factorization in Eq. (4.26) to obtain d(z~1) and d*(z~1).
Step 3
Decompose z~lH*(z~1)d~*(z~1)p2 by u z~lH*(z~1)d~*(z~1)p2"Mz~lH*(z~1)d~*(z~1)p2N # u u ` Mz~lH*(z~1)d~*(z~1)p2N . u ~ Obtain the optimal equalization filter F (z~1)"Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1). 0 u `
Step 4
Remark 6. From Eq. (4.10), we get
K
L2J Le2
Q
1 " Mg*(z~1)g(z~1)H*(z~1)H(z~1)p2u 2pj @z@/1 e/0 dz #g*(z~1)g(z~1)p2wN '0 z (4.25)
is always positive. The necessary and sufficient condition for the optimal equalization filter Fo(z~1) in Eq. (4.21) is guaranteed. Remark 7. From Eq. (4.7), the spectral factorization in Eq. (4.15) can be rewritten as follows:
C
L d*(z~1)d(z~1)" + (RhI i(0)#DhiD2) i/0 L L # + + h h*z~i`j p2#p2 . u w i j i/0 j/0,jEi (4.26)
D
A numerical computation algorithm of the above spectral factorization has been found in [10]. Remark 8. With the optimal equalization filter in Eq. (4.21), the MMSE given by Appendix C is 1 J.*/" 2pj
Q
MMz~lH*(z~1)d~*(z~1)p2u N*~ @z@/1
]Mz~lH*(z~1)d~*(z~1)p2N u ~ * * #[d (z~1)d(z~1)!H (z~1)H(z~1)p2u ] dz ][d*(z~1)d(z~1)]~1p2u N . z
(4.27)
Remark 9. The value of MMSE J in Eq. (4.27) .*/ can be directly calculated by the residue integration method. From the above analysis, a complete design algorithm of the optimal equalization filter for multipath Rician fading channels with stochastic tap coefficients is summarized in Table 1.
5. Simulation examples In this section, two simulation examples are given to illustrate the performance of the proposed design method. For comparison, a conventional adaptive decision feedback equalizer (DFE) using RLS adaptation algorithm is also designed. Before demonstrating the simulation examples, the operations of our proposed design method and the conventional adaptive DFE will be described, respectively. The operations of the proposed design algorithm are explained as follows. Initial training mode. During the initial training period, the training data are used to estimate the channel’s statistical characteristics. Then based on these estimated statistical characteristics, an optimal equalization filter is derived. The detailed design steps are described below. Step 1. Collect the training data X(N!1) in Eq. (3.6), where N is the data length during the initial training period and N"1000 for the simulation examples in this study.
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
Step 2. From Eq. (3.5), obtain the estimate hK using RLS adaptation algorithm. Step 3. From Eq. (3.9), obtain the estimate hK using RLS adaptation algorithm. Step 4. Based on hK and hK obtained at Steps 2 and 3, derive the optimal equalization filter according to the design procedure in Table 1. Operating mode. Equalize the received data sequence, y(n), using the optimal equalization filter derived at the initial training mode and decode the equalized data sequence uL (n) using the nearestneighbor criterion. Also, the operations of the adaptive DFE are explained as follows. Initial training mode. During the initial training period, the training data are used to adapt the coefficients of the DFE by recursively minimizing the sum of squares of the differences between the desired values (input samples) and the equalized values of the received samples. The detailed design steps are described below. Step 1. Collect the training data X(N!1) in Eq. (3.6). Step 2. Run RLS algorithm to estimate a constant vector u by minimizing N~1 + DuL (n)!u(n)D2, n/0 where u"[g(!K ),2, g(!1),g(0), g(1),2, g(K )]T 1 2 and 0 K2 uL (n)" + g(k)y(n!k)# + g(k)u(n!k). k/~K1 k/1 Operating mode. Based on the estimated vector u at initial training mode, decode the equalized data sequence uL (n) using the nearest-neighbor criterion, where uL (n)"+0 g(k)y(n!k)# k/~K1 +K2 g(k)uN (n!k) and uN (n) being a valid 16-QAM k/1 symbol sequence for our simulation examples is the decoded data sequence.
51
Periodic retraining#operating mode. It is well known that the ‘run-away’ phenomenon may occur to the conventional adaptive DFE. Hence, a periodic retraining # operating mode is necessary. For simulation examples in this section, a periodic retraining period of 14 samples is used, alternating with an operating period of 128 samples. Remark 10. Unlike the conventional adaptive DFE, the designed optimal equalization filter is a fixed one which needs not to adapt the parameters of filter after the initial training period. Remark 11. It is worth noting that the conventional adaptive DFE needs a periodic retraining period to avoid a possible ‘run-away’ phenomenon. On the contrary, our proposed design method only needs training data during the initial training period. Hence, our proposed design method needs less training data than the conventional adaptive DFE. Remark 12. The respective number of taps in the forward and feedback filters of the conventional adaptive DFE is 5 and 2 for the simulation examples. This DFE is able to converge within 14 symbol periods [5,17]. Remark 13. The proposed design method does not work for any M-ary phase modulated signal in that AS3 (E[Du(n)D4]O(E[Du(n)D2])2) is violated. However, if the statistical characteristics of fading channels are known, we still can design the optimal equalization filter based on known statistical characteristics. In the sequel, we shall illustrate and compare the performances of these two different design algorithms mentioned above by giving two simulation examples. In these two simulation examples, the smoothing lag l"3, the input signal is a 16-QAM data sequence and the average signal-to-noise ratio 2 +. (SNR) is defined as SNR"10 log E*@y(n)~w(n)@ 10 E*@w(n)@2+ Example 1. Consider a two path (i.e., ¸"1 in Eq. (2.1)) fading channel, the parameters in Eqs. (2.2) and (2.4) are given by h "0.8#0.6j, 0
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Fig. 3. (a) Estimation of Real(h ); (b) estimation of Imag(h ); (c) estimation of Real(h ); (d) estimation of Imag(h ); (e) estimation of R M 0(0); 0 0 1 1 h (f) estimation of R M (0). h1
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
53
h "0.4#0.3j, o "o "0.95, p2 "4]10~4, p2 v0 v1 1 0 1 "2]10~4. Example 2. Consider a four path (i.e. ¸"3 in Eq. (2.1)) fading channel, the parameters in Eqs. (2.2) and (2.4) are given by h "1, h " 0 1 !0.4!0.5j, h "0.16#0.2j, h "!0.08j, 2 3 o "o "o "o "0.95, p2 "p2 "p2 "p2 " v0 v1 v2 v3 0 1 2 3 4]10~4. For these two examples, our purpose is twofold. The first purpose is to show the performance of our proposed estimation method. In Fig. 3(a)—(f) we show the recursive estimate of statistical characteristics of fading channel in Example 1. The true values are also indicated by the solid line. All the simulations are implemented with SNR"20 dB. These results show that after the 400 samples, the estimated statistical characteristics converge to the true values. The second purpose is to illustrate the performance of the proposed design method by plotting the respective symbol error probability (SEP) as a function of the SNR. Each different SNR is achieved by tuning the channel noise power p2 . w Fig. 4 (Fig. 5) shows the SEP of Example 1 (Example 2) for the proposed design method (solid line) and the conventional adaptive DFE (dashed line). The
Fig. 5. Symbol error probability versus SNR for Example 2. Solid line (——) for the proposed design method, dashed line (----) for the conventional adaptive DFE.
SEP is obtained by averaging 100 independent Monte Carlo (MC) simulation runs for each SNR and the data length of each MC realization is 2048 points (i.e. the data length during 16 operating periods for the conventional adaptive DFE). From the simulation results shown in Fig. 4 (Fig. 5), the performance of the proposed design method surpasses that of adaptive DFE at about 10 (6) dB for higher SNR.
6. Conclusion
Fig. 4. Symbol error probability versus SNR for Example 1. Solid line (——) for the proposed design method, dashed line (----) for the conventional adaptive DFE.
This study has proposed a novel design method to treat the estimation and equalization problems of multipath Rician fading channels with stochastic tap coefficients. In the proposed design method, a two-step estimation procedure using RLS adaptation algorithm was employed to estimate the mean and variance of stochastic tap coefficients. Then based on these estimated statistical characteristics, an optimal and realizable equalization filter minimizing the MSE was designed from the viewpoint of frequency domain to reconstruct the transmitted sequence. Furthermore, calculus of variation technique, spectral factorization method and Cauchy’s residue theorem were employed in the derivation procedure of the optimal equalization filter. Also, an expression for the MMSE was
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derived. This is the first time to design a fixed filter to achieve the optimal signal reconstruction for multipath fading channels under investigation. Unlike conventional adaptive equalization schemes, which adapt the parameters of filter during the initial training period and retraining periods, the proposed optimal equalization filter needs not to adapt the parameters of filter after the initial training period. Therefore, it is more easy to implement and
more suitable for real time signal reconstruction under multipath Rician fading channels. To illustrate the performance of the proposed design method, a conventional adaptive DFE using RLS adaptation algorithm was also designed for comparison. The simulation results have confirmed the superior performance of the proposed design method.
Appendix A. Proof of Proposition 1 Substituting Eq. (3.1) into Eq. (3.7), we get
G
H G H G
H
~1 1 N~1 * 1 N~1 * + U (n)UT(n) + U (n)[UT(n)h#UT(n)hI (n)#w(n)] (A.1) N N n/0 n/0 ~1 1 N~1 * ~1 1 N~1 * 1 N~1 * 1 N~1 * + U (n)UT(n) + U (n)UT(n)hI (n) # + U (n)UT(n) + U (n)w(n) . (A.2) "h# N N N N n/0 n/0 n/0 n/0 It can be shown that the entries of U*(n)UT(n), U*(n)UT(n)hI (n) and U*(n)w(n) have finite memory and variance. According to Lemma B.1 in [21], the braced terms in Eq. (A.2) converge to their expected values as N tends to infinity in the mss and wp1, i.e., hK "
G
H G
H G
H
hK Ph#R~1 as NPR in the mss and wp1. UU RUUhI #R~1 UU RUw
(A.3)
Under the assumption AS1, RUU"p2I, RUUhI "RUUE[hI (n)]"0 and RU "E[U*(n)]E[w(n)]"0. Hence, we u w can conclude from Eq. (A.3) that hK converges to h as N tends to infinity in the mss and wp1. Also, we can conclude that hK is a strongly consistent estimate [13] of h.
Appendix B. Proof of Proposition 2 Substituting Eq. (3.1) into Eq. (3.8), we get bK (n)"UT(n)(h!hK )#UT(n)hI (n)#w(n).
(B.1)
Hence, DbK (n)D2"[UT(n)(h!hK )#UT(n)hI (n)#w(n)]*[(h!hK )TU(n)#hI T(n)U(n)#w(n)] "UH(n)(h!hK )*(h!hK )TU(n)#UH(n)(h!hK )*hI T(n)U(n)#UH(n)(h!hK )*w(n)#UH(n)hI *(n)(h!hK )TU(n) #UH(n)hI *(n)hI T(n)U(n)#UH(n)hI *(n)w(n)#w*(n)(h!hK )TU(n)#w*(n)hI T(n)U(n)#w*(n)w(n). Substituting Eq. (B.2) into Eq. (3.10), we get
G
H G
H
~1 1 N~1 N~1 hK " N + u(n)uT(n) + u(n)UH(n)(h!hK )*(h!hK )TU(n) N n/0 n/0
(B.2)
B.-S. Chen et al. / Signal Processing 68 (1998) 43—57
G G G G G G G G
H H H H H H H H
G G G G G G G G
55
H
~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)(h!hK )*hI T(n)U(n) N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)(h!hK )*w(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)(h!hK )TU(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)hI T(n)U(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)UH(n)hI *(n)w(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)(h!hK )TU(n) # N N n/0 n/0 ~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)hI T(n)U(n) # N N n/0 n/0 ~1 1 N~1 ~1 1 N~1 1 N~1 1 N~1 + u(n)uT(n) + u(n)w*(n)w(n) ! + u(n)uT(n) + u(n)p2 . # (B.3) w N N N N n/0 n/0 n/0 n/0 According to Proposition 1 and following the similar analyses in Proposition 1, the braced terms in Eq. (B.3) converge to their expected values as N tends to infinity in the mss and wp1. Hence, under the assumptions AS1—AS3, the first—fourth and sixth—eigth terms tend to zero vector, the ninth and tenth terms tend to R~1E[u(n)]p2 , and the fifth term tends to h as NPR in the mss and wp1, where R , a positive definite rr w rr matrix, is defined by #
C
H
H
H
H
H
H H G
E[Du(n)D4]
R "E [u (n) uT (n)]" rr
p4 u F
p4 2 u } } }
}
p4 u F
H G
D
p4 u E[Du(n)D4]
.
H
(B.4)
2 p4 p4 u u From the above analyses, we conclude that hK converges to h as N tends to infinity in the mss and wp1. Hence, hK is a strongly consistent estimate [13] of h. Remark 14. In view of Eq. (B.4), if assumption AS3 (E[Du(n)D4]O(E[Du(n)D2])2) does not hold, All the elements in R will be equal to p4. In such case, R will be a singular matrix. Subsequently, hK will not be guaranteed u rr rr to converge to h as NPR. Therefore, assumption AS3 is required to guarantee that hK will always converge to h in the mss and wp1 as NPR.
Appendix C. Derivation of Eq. (4.27) Before obtaining the MMSE, we rewrite Mz~lH*(z~1)d~*(z~1)p2N "z~lH*(z~1)d~*(z~1)p2!Mz~lH*(z~1)d~*(z~1)p2N . u ~ u u `
(C.1)
56
B.-S. Chen et al. / Signal Processing 68 (1998) 43–57
Thus, Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N "H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4 u ~ u ~ u !z~lH*(z~1)p2Mz~lH*(z~1)d~*(z~1)p2N* d~*(z~1)!zlH(z~1)p2Mz~lH*(z~1)d~*(z~1)p2N d~1(z~1) u u ` u u ` #Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N . u ` u ` Using the relationship in Eq. (4.21), Eq. (C.2) can be reduced as
(C.2)
Mz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N "H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4 u ~ u ~ u !z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)d*(z~1)d(z~1). (C.3) u u 0 0 0 0 Now, replacing F(z~1) with F (z~1), the MMSE in Eq. (4.8) becomes 0 dz 1 Mp2!z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)[H*(z~1)H(z~1)p2#p2 ]N J " u w z u u 0 u 0 0 0 .*/ 2pj @z@/1
Q
1 " 2pj
Q
1 J " .*/ 2pj
Q
1 " 2pj
Q
dz Mp2!z~lF* (z~1)H*(z~1)p2!zlF (z~1)H(z~1)p2#F* (z~1)F (z~1)d*(z~1)d(z~1)N . u 0 u u 0 0 0 z @z@/1 Associating the result in Eq. (C.3), the Eq. (C.4) can be rewritten as
(C.4)
MMz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N #p2 u u ~ u ~ @z@/1 dz !H*(z~1)d~*(z~1)H(z~1)d~1(z~1)p4N u z MMz~lH*(z~1)d~*(z~1)p2N* Mz~lH*(z~1)d~*(z~1)p2N #[d*(z~1)d(z~1) u ~ u ~ @z@/1 dz !H*(z~1)H(z~1)p2][d*(z~1)d(z~1)]~1p2N . u u z
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(C.5)
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