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编号： 毕业设计(论文)外文翻译 （原文） 院 （系）： 应用科技学院 专 业： 电子信息工程 学生姓名： 农 惜 童 学 号： 0701130204 指导教师单位： 应用科技学院 姓 名： 赵 响 职 称： 讲 师 2011年 6 月 12 日 桂林电子科技大学毕业设计（论文）原文用纸 第 24 页 共 24 页 Adaptive Multiuser Channel Estimation using Reduced Abstract This paper presents an adaptive multiuser channel estimator using thereduced-Kalman least-mean-square (RK-LMS) algorithm. The frequency-selective fading channel is modeled as a tapped-delay-line filter with smoothly time-varying Rayleigh distributed tap coefficients. The multiuser channel estimator based on minimum-mean-square-error (MMSE) criterion is used to predict the filter coefficie- nts. We also present its convergence characteristics and tracking performance using the RK-LMS algorithm.Unlike the previously available Kalman filtering algorithm based approach (Chen, Chen IEEE Trans Signal Process 49(7): 1523–1532, 2001) the incorporation of RK-LMS algorithm reduces the computational complexity of multi- user channel estimator used in the code division multiple access wireless systems. The computer simulation results are presented to demonstrate the substantial improvement in its tracking performance under the smoothly time-varying environment. Keywords： Kalman filter LMS algorithm MMSE Estimation Multipath fading 1 Introduction Code-Division-Multiple-Access (CDMA) system is a spread spectrum technique, which is being widely applied in cellular mobile and other high data rate wireless communication systems for the efficient usage of bandwidth. During the last two decades, the commercial use of mobile communication has increased due to the demand for large capacity in terms of the number of users. However, the major limiting factors for capacity in the CDMA systems are intersymbol-interference (ISI) due to the multipath propagation through linear dispersive media ,multiple access interference (MAI) due to the usage of non-orthogonal codes, and additive white Gaussian noise (AWGN). On the reverse link from mobile to base station, the received composite signal also includes ISI due to the past symbols of other active users. The problem of MAI can be resolved using the orthogonal codes in synchronous system.However it becomes unavoidable factor under the multipath reception conditions, because the signature (assigned code to each user) sequences for different users are non-orthogonal at different timing offsets at the receiver. Thus, the unpredictable nature of ISI and MAI limits the capacity and performance of the system under time-varying environment. The conventional strategy for handling the time-varying fading channel is to design an adaptive equalizer using the Kalman filtering algorithm [1]. In Ref. [2], S. McLaughlin has employed a channel estimator for the design of Kalman algorithm based equalizer. The per-formance of this equalizer depends on the accuracy of channel estimation. L. Chen et al.have proposed a receiver design technique for the multiuser detection in Ref. [3], in which the channel model can be directly derived from the Doppler spread of fading channel. For robust adaptive design, this Kalman algorithm based approach considers channel estimation errors and model uncertainties to improve the performance. But, it requires the knowledge of maximum Doppler spread of the multipath fading channel accurately at the receiver. In another approach [4], B. Chen et al. have proposed a linear-trend tracking algorithm and a multiuser detection algorithm, which uses the self-tuning scheme to track the time-varying fading.Moreover, the nonlinear limitation functions are embedded into the multiuser channel estimator and detector to mitigate the noise effects. The robustness is achieved at the cost of increased computational complexity. It is seen in above techniques [3,4] that the channel estimator plays an important role in the overall performance of receiver. The Kalman filtering algorithm is considered to be optimum for the channel estimation in time-varying environments [5,6], but its computational complexity and requirement of the knowledge of system model may often preclude its use. To avoid the online Riccati updating in the ordinary Kalman adaptation laws, L. Lindbom et al. have proposed the Wiener-LMS algorithm for improved tracking performance [7,8]. The major limitation of this technique is that it requires prior information about the model of dynamics of the time-varying parameters (hypermodel). Moreover, the complexity of this algorithm is directly coupled to the choice of hypermodel. Therefore, this technique has found limited applications. In another approach [9], S. Gazor has simplified the Kalman algorithm to obtain a two-step LMS-type (G-LMS) adaptive algorithm for the channel estimation problem, in which the channel coefficients are assumed to be time-varying according to a first-order Markov process. This algorithm supersedes the conventional LMS algorithm because of its ability to combat the lag noise [10]. It does not require prior information about the time-variations of the true system. However, it requires relatively longer training period, during which, the oscillatory behaviour of the algorithm accounts for the high value of residual MMSE. In this work, an adaptive multiuser channel estimator is developed using the two-step RK-LMS algorithm, which combats the nonstationarity introduced by the channel varia-tions. Moreover, it achieves better performance over the smoothly time-varying channels by reducing the lag during tracking process, where the conventional LMS based approach fails to perform well. The proposed multiuser channel estimator exploits the Kalman filtering algorithm to reduce the computational complexity in comparison with the previous methods presented in Ref. [3,4]. The paper is organized as follows. In Sect. 2, we first describe the CDMA system model,and also give details about the mathematical formulation of frequency-selective channel model for the multipath fading environment. The smoothly time-varying Rayleigh fading channel coefficients are considered to be the autoregressive (AR) process. Section 3 includes the proposed RK-LMS adaptive algorithm based multiuser channel estimator and analytical results for its adaptation characteristics (see [11]). Simulation results are presented in Sect. 4.Finally, conclusions and future scope are given in Sect. 5. 2. System model In the following, we consider a spread spectrum binary communication system, employing normalized modulation waveforms s1(t) , s2 (t) , ..., sk (t), such that skt=j=0N-1cjkψ(t-jTc) （1） where, Cjkis the j th chip 1/N in the spreading code sequence of kth user, Tc is the chip period, N is the length of spreading code sequence in terms of the chip periods, 1/N is the energy normalization factor, and ψ (t) is the real transmitted chip waveform shape,which has unit energy in the time interval 0 ≤ t ≤ Tc i.e.,ψ(t) = 0 for t / ∈ [0, Tc].The transmitted bandpass signal for kth user may be written as xkt=ReAkibkisk(t-iTb)ejwct=Rexk(t)ejωct (2) where, bk(i) is a real valued transmitted data symbol 1, sk(t) is the spreading signature sequence of user k, Ak is the amplitude level（ Ak= 2pk） , Tb is the symbol period (with Tb = NTc), and ωc is the carrier frequency. Each user’s transmitted signal (with signal power level Pk ) is assumed to pass through an independent frequency-selective Rayleigh fading channel, which transforms the bandpass signal for kth user as rkt=Re2Pkibkil=0Lk-1ϒlktskt-iTb-τlkejωct (3) =Rerk(t)ejωct where, rk(t) is the equivalent lowpass signal, Lk is the number ofmultipaths for kth user, and the complex quantity ϒlk(t) = |ϒlk(t)|e-jωcτlk represents the complex attenuation factor of lth path. The system suffers from severe ISI, as the value of (Lk− 1) approaches N [12]. Therefore, the channel order (Lk− 1) is kept less than the processing gain N (i.e., the max-imum delay spread of channel is smaller relative to the symbol period). We further assume that the fading channel response changes at the symbol rate. We define the total delay as τlk=Ωk+ tlk.For kth user,Ωk is the delay with respect to the desired user, and tlk is the propagation delay. Using (1) and (3), the demodulated equivalent lowpass signal at the receiver for kth user can be written as rkt=2Pkij-0N-1bkicjkgkt;t-iN+jTc (4) where gk(t ; τ ) =l=0Lk-1ϒlktψ（τ-τlk）, which is the convolution of the equivalen lowpass impulse response l=0Lk-1ϒlktδ（τ-τlk）, of multipath fading channel and the lowpass chip waveform ψ (τ ). Therefore, this fading channel model is analogous to the tapped-delay-line filter model for the time-varying frequency-selective channel by virtue of analogy between the multipath transmission [4]andISI[3], respectively. The Lk coefficients of the tapped-delay-line filter are assumed to be time-varying according to an AR process. Assume that gk(t ; τ ) has finite support of length LTc for k = 1, 2, ..., K i.e., gk (t ; τ ) = 0 for τ ≮[0, LTc]. As chip level processing is considered in this work, therefore the data symbol at chip level is defined as bk (i ) = bk(iN) = bk (iN + 1) = = bk (iN + j ) for 0 ≤ j ≤ N − 1. For mathematical simplicity, we replace bk(i) with bk(iN + j ) in Eq. 4,such that rkt=2Pkij-0N-1bkiN+jcjkgkt;t-iN+jTc (5) If K active users are present in the system, then the equivalent lowpass composite received signal after demodulation is represented as rkt=K=1Krkt+z(t) (6) where, z (t) is the zero-mean lowpass AWGN with two-sided power spectral density N0(due to thermal noise at receiver), which does not include interference due to other users.The demodulated lowpass signal is filtered using the chip waveform matched filter [13], andsubsequently sampled within the spreading limit (implicitly) at the chip rate to give riN+n=(iN+n)Tc(iN+n+1)Tcrtψ(t-(iN+n)Tc)dt =K=1KrkiN+n+z(iN+n) (7) Since the short codes used in the above described CDMA system are cyclic (pe -riodi) in nature, therefore the received signal sample r(iN + n) is a cyclostationary process. More-over, the transmission through themultipath fading channel is also peri -odically time-varying.For kth user, the discrete-time received signal in i th data symbol interval’s nth chip is repre-sented as riN+n=2pkl=0L-1hlkiN+n-lbk(iN+nl)c((n-l)N)k (8) For n=0,1,2,…,N-1 z(iN+n)=(iN+n)Tc(iN+n+1)TcztψtiN+nTcdt (9) where, the expression (x)N denotes “x mod N” and hkiN+n,l0Tcgkt+iN+nTc;tlTcψtdt (10) We can rearrange the above equation to give hlkiN+n-l=hk(iN+n-l,l) (11) =lTcl+1Tcgkt+iN+n-lTc;tψ(t-lTc)dt In the above equation, the channel coefficient hlk is assumed to be constant for the data symbol duration Tb, i.e., hlk(i) = hlk(iN) = hlk(iN + 1) = = hlk(iN + n) for 0 ≤ n ≤ N − 1. If z(n) has autocorrelation 1/2E [z (n) z∗ (m)] =N0δ (n− m),then resulting average signal-to-noise ratio (SNR) of kth user is given by SNRkavg=Pkl=0Lk-lEhlk(i)2N0 (12) where, ()∗ denotes the complex conjugate operator.Without the loss of generality,we assume that all the active users are transmitting at the same signal power level, such that A1= A 2 == Ak = 1 i.e.,2pk = 1. The received signal vector rk(i ) consists of N consecutive stacked samples, where i is the data symbol index. The N 1 dimensional vector rk(i ) can be written as [rk(i ) =rk(iN) rk(iN+1)rk(iN+j) rk(i N+N-1)]T.The N 2L dimensional signature-sequence-matrix Ck(i),2L2L dimensional data-symbol-matrix Bk(i),and2L1dimensional multipath channel coefficient vector hk(i) for the kth user’s i th data symbol can be defined as where,IL is the LL dimensional identitymatrix, and[ h k(i) =[h0k(i)h1k (i) * h(L-1)k(i)]T,Using the Eq. 8, the received signal vector rk(i) can be rewritten in the matrix form as rki=CkiBkihki=Dkihki (13) where, Dk(i) is the N 2L dimensional chip-data-matrix. The composite signal vector where, Ck(i)=CiCi=C1C2…CkC1C2…Ck is an N 2KL dimensional matrix, ˆBi=diagb1iIL,b2iIL,…bKiIL[b1i-1IL,b2i-1IL,…,bKi-1IL] is a 2KL2KL dimensionalmatrix, hi(i) =[h1Tih2Ti…hkT(i)][h1Ti-1h2Ti-1…hkT(i-1)]T is a 2KL 1 dimensional matrix, Di(i) =[C1b1iIL …CkbkiIL|| {C1b1i-1IL…Ckbki-1IL}] is an N 2KL dimensional matrix,and Zi(i) =[z (iN) z (iN + 1) ... z (iN + j ) ... z (iN + N - 1)]Tdenotes the noise sample vector. In the next section, theEq. 14 is used to estimate themultipath fading channel response. 3 Adaptive Multiuser Channel Estimation In the literature, the unknown channel coefficients are often assumed to be a first-order Markov process for the tracking performance analysis of the LMS-based adaptive algorithms [3,9,10]. Therefore, the multipath channel coefficient vector of the kth user can be modeled by using the AR(1) process as where, the L L dimensional channel-correlation-coefficient matrix is ρ = diag [a0k,a1k,…,aL-1k]. The above model is valid for the fading channel only if the channel coherence time is large enough to estimate the channel response. The subscri- pt ()o denotes the optimum value. The scaling factor alk denotes the state transition coefficient of the kth userin l th path. According to Jakes’model [14], this factor may be defined as the correlation coefficient. In Eq. 15, ωk(i)=[ωk0iωk1i…ωkL-1(i)]T is a zero-mean white noise process vector with the covariance matrix σωkl2IL, which results in the uncorrelated tap coefficients of the multipath fading channel (wide-sense stationary uncorrelated scattering channel). If we consider the first-order weight increment vector hk,0(i-1)=hk,0(i)- hk,0(i-1) correlated with the vector hk,0(i), then this redundant information may be used to predict the vector hk,0(i) because hk,0(i) = d｛-ρhk,0i-1+ωk(i)｝;where the LL dimensional scaling matrix is d=diag[ (1-a0k)-1,(1-a1k)-1,…,(1-aL-1k )-1]. Similarly, the second-order weight increment vector hk,0(i-1) = hk,0(i)- hk,0(i-1) may be used to approximate the state equation for the time-varying fading environment. In the following, we assume that alk = a ,i.e., the correlation coefficients of all the fading channels are equal.In the multiuser scenario, the state equation may be defined as Such that h0i=Ah0i-1+W(i) with A = aI ,where W(i) is a zero-mean white noise process vector, and I is a 2KL 2KL dimensional identity matrix. 3.1 RK-LMS Algorithm Based Channel Estimator The application of Kalman algorithm [15–17] for multiuser channel estimation is based on the following equations hii=hii-1+Kiri-re(i) (17) where, the Kalman gain or blending factor is Ki=P(i|i-1)DT(i) *[D(i)p(i|i-1)DTi+RZ]-1, and the estimated signal vector is denoted as rei=CiBihii-1=Dihii-1 (18) We note that D(i) is considered to be a deterministic process in the development of Kalman algorithm, but it is considered to be a random process in the analysis of algorithm [9]. The covariance matrix of the prediction error (in time update) is P (i |i − 1 ) = AP (i − 1 |i − 1 ) AT +Qω .The covariance matrix of the estimation error (in measurement update) is P (i |i ) =I − K(i ) D (i ) P (i |i − 1 ),where Rz =EzizH(i) and Qω = EWiWH(i) are the measurement noise and the process noise covar- iance matrices respectively. It is clear that the above described Kalman adaptive algorithm based approach is computationally complex. Therefore, the Kalman algorithm is reduced to give a computationally efficient two-step LMS-type algorithm, which precludes the matrix inversion operation in the Riccati update equation. The proposed RK-LMS algorithm based multiuser channel estimator is as follows hii=hii-1+μDTiri-rei (19) where, the scalar parameter is step size, which controls the convergence and stability of the adaptive algorithm. The Kalman gainK(i) is replaced by DT(i) ,and the apriori estimate h(i |i − 1) of h(i ) is defined as hii-1=hi-1i-1+βhi-1i-1 （20） In the above equation, the first-order weight increment vector is estimated as hii=hi-1i-1+αμDT(i)ri-re(i) with hii-1=hi-1i-1 (21) where, 0 ≤α< 1 is a smoothing parameter, which controls the lag in tracking the time-varying system [9]. The estimated first-order weight increment vector is scaled with a real valued control parameter 0 ≤β< 1, which controls the oscillatory behaviour of algorithm.The analytical and simulation results presented inRef. [11]manifest that the value of β should be tuned below the threshold value (depending on the value of α and ), which ensures stability in the convergence mode. Further it has been shown by computer simulations that for a single-user system, the proposed algorithm provides ∼ 14 dB performance advantage in channel estimation over the conventional LMS algorithm by reducing the mean square error in tracking mode. However, we have considered a multiuser system in the present work. Let E h(i|i) is the ensemble average of h(i|i)[10,11]. The estimated multiuser channel coefficient vector can be defined in terms of the optimum channel coefficient vector h0(i), tracking noise vector hTi=hii-Eh(i|i) , and lag noise vector hLi=Ehii-h0(i) as hii=h0i+hTi+hL(i) For the above procedure, the estimation of first-order weight increment vector (21) helps in reducing the lag noise i.e., hL(i) in tracking, but application of the conventional LMS algorithm in steps (19) and (21) contributes minor gradient noise in addition to the residual lag noise. The reduction in the lag noise results due to improved tracking in the time-varying nonstationary environment. Therefore, the Eq. 18–21 are used to estimate the multipath channel response for K users The reduced Kalman/LMS algorithm is computationally comparable to the conventiona LMS adaptive algorithm. Moreover, the RK-LMS algorithm does not require the knowledge of channel correlation coefficient a. In addition, the substitution of variable step size μ(i) results in the development of a new family of the two-step LMS-type algorithm based adaptive channel estimators. 3.2 Analysis of Multiuser Channel Estimator In this section, we study the mean convergence behaviour and tracking characteristics of the RK-LMS algorithm based adaptive multiuser channel estimator. Using the channel estimation Eq. 14, we redefine the received composite signal vector in terms of the optimum weight vector as ri=Dih0i+z(i). Let the state (channel coefficient) error vector be Φi=△hi△hi=hii-h0ihii-h0i (22) The above equation is solved using the RK-LMS algorithm (see Appendix A) to give Φi=TiΦi-1+Ni (23) Where,T(i)=I-μRi I-μRi-αμRi I-μRi with Ri=DTiDTi (24) Ni=μDTizi+1-βμRihi-1i-1-1-βhi-1i-1αμDTizi+α1-βμRihi-1i-1-h0i-1 (25) The above weight error vector may be considered as the output of a recursive linear system with the state transition matrix T (i ) and the input vector N (i ). Since z(i) is a zero-mean statistically independent random process vector i.e., Ez(i)=0, and R=EDTiD(i)is the average value of matrix R(i), therefore the ensemble average of the weight error vector (i) gives the mean error variation. We can write EΦi=TiEΦi-1+Ni (26) Where The square matrix R is spectrally factorized to give Q-1∧Q,where Q is the modal matrix consisting of eigenvectors as column vectors, and is the spectral matrix with only eigen-values at its main diagonal i.e., ∧= diagλ1,λ2,…,λq,…,λ2KL. The multiplication of the modal matrix Q on the

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