外文翻译--基于线性预测方法的水声信道最小均方误差盲均衡（节选）

中文 3775字 毕业设计 (论文 )外文资料翻译 系 别 电子信息系 专 业 通信工程 班 级 姓 名 学 号 外文出处 J. Marine Sci. Appl. (2011) 10: 113-120 附 件 1. 原文； 2. 译文 2013年 03月 1 Blind Adaptive MMSE Equalization of Underwater Acoustic Channels Based on the Linear Prediction Method R Bragos, R Blanco-Enrich Abstract: The problem of blind adaptive equalization of underwater single-input multiple-output (SIMO)acoustic channels was analyzed by using the linear prediction method. Minimum mean square error (MMSE) blind equalizers with arbitrary delay were described on a basis of channel identification. Two methods forcalculating linear MMSE equalizers were proposed. One was based on full channel identification and realizedusing RLS adaptive algorithms, and the other was based on the zero-delay MMSE equalizer and realized usingLMS and RLS adaptive algorithms, respectively. Performance of the three proposed algorithms and comparison with two existing zero-forcing (ZF) equalization algorithms were investigated by simulations utilizing two underwater acoustic channels. The results show that the proposed algorithms are robust enough to channel order mismatch. They have almost the same performance as the corresponding ZF algorithms under a high signal-to-noise (SNR) ratio and better performance under a low SNR. 1 Introduction Time-varying characteristic and multi-path fading of underwater acoustic channels can induce severe inter symbol interference (ISI) in high data rate communication systems. Channel equalization applying adaptive filters is one of the techniques to mitigate the effects of ISI. Conventionally, the initialization of an adaptive filter is achieved by a known training sequence from a transmitter before data transmission, so that valuable channel capacity is reduced. Recently, blind equalization technique (Stojanovic, 1996) has attracted more and more attention. Compared with adaptive equalization technique, the major advantage of such technique is that no training sequence is needed to start up or restart the system whenever the communication breaks down unpredictably. Traditionally, symbol rate sampled channel output sequence is stationary and higher order statistics are used to estimate the channel and to calculate the equalizer. More recently, it has been shown that the channel output sequence is cyclostationary if the sampling rate exceeds the symbol rate, and then second-order statistics (SOS) 2 contain sufficient information to estimate most communication channels using cyclostationarity (Tong et al., 1994; Tong et al., 1995; Papadias and Slock, 1999). Based on the seminar work of Tong et al. (1994), many effective blind methods have been proposed for estimating the channel from the output of only second-order statistics. However, it turns out that these methods have much computational complexity or they are very sensitive to channel order mismatch (Moulines et al., 1995; Meraim et al., 1997; Liu et al., 1994; Alberge et al., 2002), which are major obstacles for their real-time implementations. The prediction error method offers an alternative to the class of techniques above. It was introduced by Slock (1994), Meraim et al. (1997), Ding (1997), Gesber and Duhamei (1997), Tugnait (1999) and offered great advantages over other SOS based techniques because of its robustness to channel order mismatch and low computational complexity. Based on multichannel linear prediction of the observations, zero-forcing (ZF) and minimum mean square error (MMSE) equalizers with arbitrary delay were investigated in Papadias and Slock (1999). Nevertheless, when calculating ZF equalizers, not only the (n+1)-step-ahead linear prediction of the noise-free channel output should be estimated, but also the backward linear prediction of some sufficient order M of the prediction error of the previous (n+1)-step-ahead linear predictor need to be carried out. When calculating MMSE equalizers, ZF equalizers should be worked out first and noise variance must be estimated correctly. These operations make the two kinds of equalizers very complicated and hard to realize. A computationally effective blind ZF equalization method has been discussed in Li and Fan (2000). It is based on two linear prediction models, one is used to calculate the zero-delay ZF equalizer and the other is used to calculate ZF equalizers with arbitrary delay on the basis of the first one. However, only ZF equalizers are presented. In Giannaki andHalford (1997), an approach for directly estimating nonzerodelay MMSE equalizers was given. Nevertheless, the first coefficient of the channel response must be known a priori and noise variance should be estimated correctly. In order to improve the performance of blind equalizers without the aforementioned limitations, two methods for finding linear MMSE equalizers with arbitrary delay are presented in this paper. One is based on full channel identification and realized using RLS adaptive algorithm, the other is based on the zero-delay MMSE equalizer and realized using LMS and RLS adaptive algorithms, respectively. 3 Simulation results show that the proposed methods are robust to channel order mismatch and they have better performance than the corresponding ZF algorithms under low SNR. For the whole paper, vectors and matrices are boldface small and capital letters, respectively. The notations (.) ， (.)T ， .H , (.) stand for the trace,transpose, conjugate transpose, and the Moore-Penrose pseudoinverse, respectively. LIis the LL identity matrix and0MNis the MN zero matrix. .E denotes the statistical expectation. 2 Problem formulation Consider a linear time-invariant communication channel.The received baseband signal y(t) can be expressed as ( ) ( ) ( ) ( ) ( )lsly t x t v t s h t l T v t (1) where lsdenotes the symbol emitted by the digital source at time slTwith sTbeing the symbol duration; ()ht the overall complex baseband equivalent impulse response of the transmitter filter, unknown channel and the receiver filter; ()xt the channel output without noise; and ()vt the channel noise that is assumed to be stationary as well as uncorrelated with ls. The following assumptions are held throughout this paper: 1) The symbol sequence lsis stationary sub-Gaussian signal with zero-mean and unit-variance. 2) The noise ()vt is Gaussian with variance 2v. 3) ()ht is causal and has finite support 0hsLT. 4) The subchannels have no common zeros. The oversampling factor is assumed to be L and the initial sampling time instant is. The oversampled received signal can now be represented as 0 00( ) ( ) ( )st t n T L l s s sly t s h t n T L l T v t n T L (2) Let 0( ) ( )sx n x t n T L ，0( ) ( )sy n y t n T L 0( ) ( )sv n v t n T L ，0( ) ( )sh n h t n T L (3) then Eq.(2) becomes 4 ( ) ( ) v ( ) ( ) ( )lly n s h n l L n x n v n (4) Define ( ) ( )ix n x n L i， ( ) ( )iy n y n L i ( ) ( )ih n h n L i， ( ) ( )iv n v n L i (5) where i = 0, L 1 . Then the single-input single-output (SISO) system of Eq.(4) has an equivalent SIMO description as follows, ( ) ( ) v ( ) ( ) ( )i l i i ily n s h n l n x n v n (6) Define the following symbol rate vector, 01y ( ) ( ) , , ( ) TLn y n y n K, 01( ) ( ) , , ( ) TLx n x n x n K 01( ) ( ) , , ( ) TLh n h n h n K , 01( ) ( ) , , ( ) TLv n v n v n K (7) The Eq.(4) can be represented in a vector form ( ) ( ) ( ) ( ) ( )lly n s h n l v n x n v n (8) Furthermore, it can be represented as the following matrix form, ( ) ( ) ( ) ( ) ( )N S N N Ny n H n v n x n v n (9) where H is a ()hN L N Lblock Toeplitz matrix, ()sn is a ( ) 1hNLvector and ()Nxn, ()Nvn， ()Nynare 1NL vctors as follows ( 0 ) ( )( 0 ) ( )hhh h LHh h LKM O M (10) ( ) ( ) ( 1 ) TTTNx n x n x n N K ( ) ( ) ( 1 ) TTTNy n y n y n N K ( ) ( ) ( 1 ) TTTNv n v n v n N K 1() h TN n n N Lx n s s K(11) 5 3 The proposed methods 3.1 ZF equalizers and MMSE equalizers Consider the fractionally spaced FIR linear equalizer shownin Fig.1, where ()ign for 01iLK is the equalizer with order gL of the ith subchannel. In the absence of noise, one natural choice is to require ( ) ( )n w n dw for some integer delay d with 0,hgd L L. This type of equalizer is known as zero-forcing. More precisely, a ZF equalizer is described by 1 ()00()hLL diillh l g （ m-l ） = ( m - d ) (12) where superscript (d) refers to the delay d . Choose 1gNLin Eq.(10), then Eq.(12) can be written as ,1T d Z F dH g e (13) where ,gg ( 0 ) TTTd Z F d dgg K （ L ）ector of the equalizer taps corresponding to delay d and ( 1) 1gLL, ( ) ( )01( ) ( ) ( ) TdddLg l g l g l K, ( 1) 1hgLL vector with an 1 as the (d +1) th element and zeros elsewhere. The existence of ZF equalizers d ,ZF g has been proven (Giannaki and Halford, 1997; Slock and Papadias,1995) if the subchannels have no common zeros and 1ghLLIt can be written in the following expression, , ( : , 1 )T H Hd Z F x xg H d R (14) where xRis the (:, 1)Hd th column of the matrix H . 6 As ZF equalizers do not address noise suppression, another kind of equalizer called blind MMSE equalizers has been proposed to find ( ) : 0 , 1 1ig n i LKsuch that 2( ) ( )dJ E s n s n d is minimized, where 1 ()00( ) ( ) ( )gLP diiiln g l y n ls (15) Take the complex derivative with respect to the unknown equalizer taps and set them to zero, MMSE equalizer with arbitrary delay d is obtained, 2, ( : , 1 ) ( )T H Hd M M S E S y yg H d R (16) where ( ) ( )Hy y N NR E y n y n. RLS and cyclic LMS algorithms15 have been proposed to recursively calculate the equalizer taps. However, they can only be used to calculate zero-delay MMSE equalizer based on the assumption of the knowledge of h(0). If it is modified to get MMSE equalizers with arbitrary delay, noise variance of the received data must be estimated correctly, which makes it impractical for realization. Fortunately, according to Eq.(16), if the correct estimates of ()ihnare obtained, then (:, 1)Hd will be available so that RLS algorithm can be used to recursively calculate ,d MMSEg.Based on this idea, channel identification becomes a majorissue here. For the existing SOS based channel identification methods, most of them are sensitive to channel order mismatch or computationally complicated. The prediction error method offers an alternative to the channel identification. In the following sections, linear prediction based channel identification and equalization methods will be presented. 3.2 MMSE equalizers with arbitrary delay based on linear prediction Consider the following one-step-ahead linear prediction problem 1 1 1( ) ( ) ( 1 ) ( )N N L N Nn y n P y n I P y n (17) where ()n is a 1L prediction error vector and 1NP is a L L(N 1) prediction matrix. Minimizing the prediction error covariance leads to the following optimization problem, 7 111m i n ( ) ( ) m i nNNHHLNpPt r E n n t r I P(18) The solution of the optimization problem is the optimal predictor. Suppose that1NPis the optimal linear predictor in the noiseless case and hNL, then the following relationship can be derived (Li and Fan, 2000; Chow et al.,2002) 1 ( 0 ) 0LNI P H h K (19) and 1m i n ( ) ( ) ( 0 ) ( )N HHPE E n n h h n (20) In real application situations, the exact channel order is not known a priori. Rewrite the matrix PN1 as ( 1 ) ( 2 ) ( 1 )1 1 1 1NN N N NP P P P K (21) (1 )1(1) ( 0 )Nh P h ( 1 ) ( 2 )11( 2 ) (1 ) ( 0 )NNh P h P h K ( 1 ) ( 1 )11( ) ( 1 ) ( 1 )Nh e s t N h e s t N h e s th L P h L P h L N K Eq.(20) shows that the prediction error covariance is a rank-one matrix. Any column of this matrix can be used as the estimate of h(0). Then according to Eq.(22), the whole channel response can be calculated recursively so that the estimate of (:, 1)Hd is obtained. Notice that hNL hn should be satisfied to ensure good estimation result. Through the analysis above and combining Eq.(16), Eq.(17),Eq.(20) and Eq.(21), the following linear prediction based RLS algorithm, namely MMSE-RLS-1 for simplicity, is given for computing blind MMSE equalizer with arbitrary delay. Step 1: Initialization 1 ( 1 ) ( 1 )11( 0 )L N L NQI where 1is a small positive constant. 1 ( 1 )( 0 ) 0N L L NP , (0) 0LLE ,2 21( 0 )L N L NQI 1 ( 1 ) 10N L Ny ,1(0) 0Ly ,10N LNy where 2is a small positive constant. For each time instant n = 1, 2, ., perform Step 2 to Step 5. 8 Step 2: Get the optimal linear predictor PN1 , 11( 1 ) 1 : ( 2 ) , TTTNNy y n y L N 11( 1)NNy n y 1111 1 1 1( 1 ) ( 1 )()( 1 ) ( 1 ) ( 1 )NHNNQ n y nKny n Q n y n 11( ) ( ) ( 1 ) ( 1 )NNn y n P n y n 1 1 1( ) ( 1 ) ( ) ( )HNNP n P n n K n 1 1 1 1 111( ) ( 1 ) ( ) ( 1 ) ( 1 )HNQ n Q n K n y n Q n Step 3: Estimate (0)h1( ) ( 1 ) ( ) ( )HE n E n n n .The column of E(n) with the largest norm is taken as the estimate of h(0). Step 4: Calculate the estimates of (1 ) ( 2 ) ( 3 )h h h Kusing Eq.(22). Step 5: Form vector :, 1Hd and calculate dgn( ) ( 1 : ( 1 ) , 1 ) TTTNNy y n y L N 2222( 1 )()( 1 )NHNNQ n yKny Q n y2 2 2 221( ) ( 1 ) ( ) ( 1 )HNQ n Q n K n y Q n 2( ) : , 1 THdg n H d Q n In fact, the algorithm can be modified as cyclic LMS. Though cyclic LMS algorithm has extremely low computational complexity, it has slower convergence rate than RLS algorithm and it is rather sensitive to amplitude error of the channel response estimation. Therefore, this algorithm will not be considered here but another modified MMSE equalizer will be realized using LMS and RLS adaptation. 3.3 MMSE equalizers with arbitrary delay based on zero-delay MMSE equalizer In order to enhance the performance of blind equalizers, MMSE equalizers are proposed here and LMS and RLS algorithms are developed to recursively calculate the equalizer taps. In the noise case, let the derivative of Eq.(18) with respect to PN1 9 equal to zero, the following equation can be obtained, 1 ( ) ( ) 1 0 0HL N y yI P E n n R K (23) 10 , 1( 0 ) ( ( ) ( ) )T H HM M S E L Ng h E n n I P (24) Combining Eq.(23) with Eq.(16), it shows that 110, 2( 0 ) ( 0 )( 0 ) ( 0 ) ( 0 )HHL N L NTZF Hh I P h I Pghh h (25) Compare it with Eq.(24), there is only a difference of computing the inversion of the first prediction error covariance ()En, which makes the performance of zero-delay MMSE equalizer better than that of zero-delay ZF equalizer. ZF equalizer with arbitrary delay can be deduced by a second linear prediction model ( ) ( ) ( )N N NJ n y n d P y n (26) where Npis the optimal linear predictor and NJis 1NLprediction error. Then there exists the following relationship: , 0 ,TTd Z F Z F Ng g P(27) While for the MMSE equalizer, there isnt this kind of compact expression. Li and Fan (2000) has shown thatdgcan be realized by the following minimization problem: , 0 , ,( ) ( ) ( )TTd Z F Z F N d Z F Nj n g y n d g y n (28) received data filtered by a d-delay ZF equalizer is equivalent to that the received data delayed by d filtered by a zero-delay ZF equalizer. Hence, the expression can be modified to get a d-delay MMSE equalizer based on the zero-delay MMSE equalizer using the similar minimization problem. , 0 , ,( ) ( ) ( )TTd M M S E M M S E N d M M S E Nj n g y n d g y n (29) Eq.(29) can be adaptively optimized using an LMS algorithm or RLS algorithm. Firstly, (0)h is estimated in section 3.2, and then the zero-delay MMSE equalizer ,d MMSEgis calculated by Eq.(24). After that,d MMSEgcan be updated recursively by an LMS algorithm to minimize ()iEnor simplify ,d MMSEg which results in the following tap adaptation equations: , 0 , ,( ) ( ) ( )TTd M M S E M M S E N d M M S E Nj n g y n d g y n *, , ,( ) ( 1 ) ( ) ( )d M M S E d M M S E d M M S Eg n g n j n y n (30) 10 , , ,( ) ( ) ( )TTd M M S E d M M S E N d M M S E Nj n g y n d g y n 3333( 1 ) ( )()( ) ( 1 ) ( )NHNNQ n y nKny n Q n y n3 3 3 331( ) ( 1 ) ( ) ( ) ( 1 )HNQ n Q n K n y n Q n *, , , 3( ) ( 1 ) ( ) ( )d M M S E d M M S E d M M S Eg n g n j n K n (31) 4 Simulation results In this section, simulation results are presented for the proposed algorithms MMSE-RLS-1, MMSE-LMS-2 and MMSE-RLS-3 described in the previous sections. The performance of the proposed algorithms is compared with the two existing algorithms (Li and Fan, 2000), which are both ZF algorithms and called Li-ZF-RLS and Li-ZF-LMS respectively for notational convenience. The two ZF algorithms have been proven to have faster convergence rate and lower ISI than many other existing algorithms. As a performance measurement, the residual ISI is estimated over 50 independent Monte Carlo runs and it is defined as 2210 2( ) m a x ( )1 0 l o gm a x ( )nnnc n c nI S Icn (32) 110 0 0( ) ( ) ( ) ( ) ( )gLLLi i i ii i ic n h n g n g j h n j (33) 4.1 Experiment 1: performance of the proposed algorithms in noise The performance of the proposed algorithms is considered in the presence of additive noise firstly. The channel used is a shallow sea channel (Zielinski et al., 1995) with carrier frequency of 10 kHz, bandwidth of 2 kHz and baud rate of 10