自适应均衡

5．Adaptive Equalization5. 1 Linear Optimum FilteringConsider the block diagram of Fig 5.1 for a linear discrete-time filter. The input consists of a time series u(0),u(1),u(2)……and the filter is itself characterized by the impulses response w0, w1, w2….. At some discrete time n, the filter produces a output y(n) which provides an estimate of a desired response denoted by d (n)Fig. 5.1The filter design problem is to select a set of filter coefficients w i such that, given a number of input samples u(0),u(1),u(2),……, the mean-square value of the estimation error e(n), defined as the difference between the desired response d(n) and the actual response y(n), is minimized.5. 2 Principle of OrthogonalityThe filter output y(n) at the time n is defined by the linear convolution sum*0()(),0,1,2k k y n w u n k n ∞==−=∑" (5.1)where the asterisk denotes complex conjugation. The estimation of d(n) is accompanied by an error()()()e n d n y n =−. (5.2)We choose to minimize the mean-square value of the estimation error e(n). We may thus define the cost function as the mean–squared error*2[()()][()]J E e n e n E e n ==(5.3)Let the k th filter coefficient be a complex number(0,1,2)k k kw a jb k =+="(5.4)Correspondingly, we define a gradient operator ∇ which may generate a gradient vector whose k th element is written as(0,1,2)k k kj k a b ∂∂∇=+=∂∂"(5.5)Thus for the situation at hand, applying this operator to the cost function J , the k th element of the gradient vector J ∇ is(),0,1,2k k kJ J J j k a b ∂∂∇=+=∂∂""(5.6)For the cost function J to attain its minimum value, all the elements of the gradient vector J ∇ must be simultaneously equal to zero, as shown by()00,1,2k J k ∇=="" ()5.7Under this set of conditions, the filter is said to be optimum in mean-squared-error sense. From (5.3) and (5.6), we have****()()()()()()(()())k k k k k e n e n e n e n J E e n e n j e n e n a a b b ⎡⎤∂∂∂∂∇=+++⎢⎥∂∂∂∂⎣⎦(5.8)Using Eqs. (5.2) and (5.4) we get the following partial derivatives.**()()ke n ju n k b ∂=−−∂ (5.9)Thus, from (5.8) and (5.9) we get the result*2()()k J E u n k e n ⎡⎤∇=−−⎣⎦(5.10)Let e 0 denote the special value of the estimation error that results when the filter operates in its optimum condition. Then, we have*0()()0 (0,1,2,.......).E u n k e n k ⎡⎤−==⎣⎦ (5.11)Note that for two random variables x and y, we have*[]0E xy orthogonal =⇔ .(5.11) states the following: The necessary and sufficient condition for the cost function J to attain its minimum value is that the corresponding value of the estimation error e 0(n) is orthogonal to each input sample that enters into the estimation of the desired response at the time n. This statement constitutes the**()()()()()()kkke n u n k a e n ju n k b e n u n k a ∂=−−∂∂=−∂∂=−−∂principle of orthogonality, it represents one of the most elegant theorems in the subject of linear optimum filtering.5. 3 Wiener-Hopf EquationsWe may reformulate the necessary and sufficient condition for optimality by substituting Eqs. (5.1) and (5.2) in (5.11).**0()()()0 (0,1,2,......)oi i E u n k d n w u n i k ∞=⎡⎤⎛⎞−−−==⎢⎥⎜⎟⎝⎠⎣⎦∑ (5.12)where w oi is the i th coefficient in the impulse response of the optimum filter. Expanding this equation and rearranging terms, we get**()()()()(0,1,2)oii owE u n k u n i E u n k d n k ∞=⎡⎤⎡⎤−−=−=⎣⎦⎣⎦∑" (5.13)The expectation *()()E u n k u n i ⎡⎤−−⎣⎦ is equal to the autocorrelation function of thefitter input for a lag of i - k . We thus express this expectation as*()()()r i k E u n k u n i ⎡⎤−=−−⎣⎦. (5.14)The expectation *()()E u n k d n ⎡⎤−⎣⎦ is equal to the cross-correlation function betweenthe filter input )(k n u − and the desired response d(n) for a lag of -k . We thus express this expectation as*()()() p k E u n k d n ⎡⎤−=−⎣⎦ (5.15)Accordingly, we get()()(0,1,2,)oii Wr i k p k k ∞=−=−=∑" (5.16)The equation (5.16) is called Wiener-Hopf equation.Practically, the filter weights are limited by a finite set . Then (5.16) reduces to a system of M simultaneous equations, as shown by10()()(0,1,2,,1)M oii wr i k p k k M −=−=−=−∑" (5.17)Let R denote the M-by-M correlation matrix of the inputs u(n), u(n-1), ……u(n-M+1) of the filter,[()()]E n n ′=R u u (5.18)where u (n) is the M-by-1 input vector:[]()(),(1),(1)Tn u n u n u n M =−−+u " (5.19)***(0)(1)(1)(1)(0)(2)(1)(2)(0)r r r M r r r M r M r M r −⎡⎤⎢⎥−⎢⎥=⎢⎥⎢⎥−−⎣⎦R "### (5.20) Let P denote the M-by-1 cross-correlation vector as*()()E n d n ⎡⎤=⎣⎦P u . (5.21)In expanded form, we have[](0),(1),,(1)Tp p p M =−−P " (5.22)We may thus rewrite the Wiener-Hopf equations in the compact matrix form:0=RW P (5.23)where W 0 denotes the Mx1 optimum weight vector of the filter, that is,000010,1,,,,TM w w w −⎡⎤=⎣⎦W " (5.24)By pre-multiplying both sides of (5.23) by 1−R , the inverse of the correlation matrix, we obtain the optimum weight vector10−=W R P . (5.25)5. 4 Channel EqualizationIn this section we consider a temporal signal-processing problem, namely, that of channel equalization is an application of wiener filtering theory.In a wireless communication system, inter symbol interference (ISI) arises due to multipath transmission channels. This depressive’s nature of the channel results in an overlap of adjacent signal pulses. If ISI is left unchecked, it can produce errors in the reconstructed data stream at the receiver output. An effective method for combatingthe system degradation due to ISI is to connect an equalizer in cascade with the channel as shown in Fig 5.2.Fig. 5.2(a) Block diagram of equalized channel.Fig. 5.2(b) Symmetric tapped-delay-line filter implementation of the equalizer.For equalizer symmetry, the total number of taps in the equalizer is chosen to be 2N+1, with the tap-weights denoted by 101,,,,N N w w w w w −−"". The impulse response of the equalizer is therefore()()Nkk Nh t w t kT δ=−=−∑ (5.25)where T is the delay time of the equalize delay element.Suppose that the equalizer is connected with a channel whose impulse response is c(t). Let p(t) denote the impulse response of the combined system. p(t) is equal to the convolution of c(t) and h(t).()()*()()*()()*()()Nkk NNk k NNkk Np t c t h t c t w t kT w c t t kT w c t kT δδ=−=−=−==−=−=−∑∑∑ (5.26)Evaluating (5.26) at the sampling times t=nT, we get the discrete convolution sum()()()Nkk Np nT w c n k T =−=−∑. (5.27)Thus, for no inter-symbol interference , we require that⎩⎨⎧≠==0001)(n n nT p (5.28)Consequently, we have⎩⎨⎧±±±===Nn n nT p ,,2,1001)("We further let the n th sample of the impulse response c (t) be written as()n c c nT = (5.29)Then with (5.27) and (5.28) we obtain a set of 2N+1 simultaneous equations:101,2,Nk n kk Nn w cn N−=−=⎧=⎨=±±±⎩∑" (5.30)Equivalently, in matrix form we may write01121012111010121011211000100N N N N N N N N N N N NN NN N c c c c c w c c c c c w c c c c c w c c c c c w c c c c c w −+−−−−−−−−−−−−−+−++−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦""#####""""""""##""(5.31)A equalizer described by Eq. (5.30) or (5.31) is referred to as a zero-forcing equalizer.5. 5 Adaptive Equalizer5. 5. 1 The System DescriptionThe zero-forcing equalization described above works well in laboratories, where we have access to the system to be equalized, in which case we know the system coefficients 101,....,,,,......,N N c c c c c −− that are needed for the solution of (5.31). In a telecommunication environment, however, the channel is usually time varying. Therefore there is need for adaptive equalization.The process of equalization is said to be adaptive when the equalizer adjusts itself continuously and automatically by operating on the input signal. This equalization can be achieved by training the equalizer with the guidance of a satiable training sequence transmitted through the channel so as to adjust the equalizer parameters to optimum values.There one two modes of operation of an adaptive equalizer, namely, the training mode and a decision directed mode, as shown in Fig.5.3.During the training mode, a known sequence is transmitted and the synchronized version of this signal is generated in the receiver, where it is applied to the adaptiveequalizer as the desired response; the tap-weights of the equalizer are adjusted with LMS algorithm.In order to proceed with our analysis, we study following mathematic basics.5. 5. 2 Mathematic Basics5. 5. 2. 1 Constrained Optimization TheoremLet *(,)f W W be a real-valued function of vector–valued complex variable W in which the dependence on the variable and its conjugate is explicit. By treating W andW * as independent variables, the quantity pointing in the direction of the maximum rate of change of *(,)f W W is *().f ∇W WTo find the stationary points of a scalar-valued function of a complex-valued vector, we must solve.*()0f ∇=W W (5.32)A typical optimization problem can be stated as followsmin *(,)f W W subjectto *(,)=g W W 0 (5.33)The classic approach to solving this constrained optimization problem is the method of Lagrange multipliers . The Lagrangian of this problem is defined to be()()()***,,T L f ∗′=++W,W λg W W λg W W (5.34)Here L must be a real number. To solve the problem of (5.33), we require*L ∇=W 0(5.35)The gradient vector *L ∇W is defined as()*25.36L L ∗∂∇=∂w WFor example, if L ′=W RW then*L ∇=W *22L∂=∂RW W . (5.37)5. 5. 2. 2 Eigen AnalysisLet R denote a MxM matrix of complex elements. If following relationship holds,H =R R (5.38)we say that R is a Hermitian matrix . For a Hermitian matrix R , we wish to find an Mx1 vector q that satisfies the conditionλ=Rq q . (5.39)For MxM matrix R there will be M such vectors along with M different values for λ. We call λ aneigenvalue of the matrix R, and q an eigenvector of the matrix R . Let i λ denote the i th eigenvalue of the matrix R . Also, let i q be a nonzero vector suchthati i i λ=Rq q (5.40)The vector i q is the eigenvector associated with the eigenvalue i λ. The eigenvector 12,M q q q "" are orthogonal to each others, i.e.()0H i j q q i j =≠ (5.41)Let 12,M q q q "" be the eigenvector corresponding to the distinct eigenvaluesM λλλ""21, of the MxM matrix R , respectively. Define MxM matrix[]12,M =Q q q q "". (5.42)where1H i j i ji j=⎧=⎨≠⎩q q . (5.43) Define a MxM diagonal matrix()12,M diag λλλΛ="". (5.44)Then the matrix R may be diagonalized as followsH =ΛQ RQ . (5.45)(5.43) indicates that the eigenvectors 12,M q q q "" form an orthonormal set. By definition, the eigenvectors satisfy the equations()()1,2, 5.46i i ii M λ==Rq q ""From (5.42) and (5.44), we may rewrite the set of M equations (5.46) as a single matrix equation.ΛRQ =Q (5.47)Owing to the orthonormal nature of the eigenvectors , as defined in Eq. (5.43) , we find thatH Q Q =I . (5.48)Equivalently, we may write()1 5.49H−=Q QA matrix that has this property called a unitary matrix. Thus, from (5.47) and (5.49), we obtainH =ΛQ RQ (5.50)Then multiplying both sides of Eq. (5.47) by 1−Q , we have1HMH i i ii λ==Λ=∑R Q Q q q(5.51)Furthermore ,1111HM Hi i i iλ−−==Λ=∑R Q Q q q (5.52)5. 5. 2. 3 Mean-Squared ErrorAssume u is an input signal vector of an adaptive equalizer,()(),(1),(),(1),()Tn u n N u n u n u n u n N =++−−⎡⎤⎣⎦u """" (5.53)W is the equalizer weight vector[]()101,,, 5.54TN N w w w w w −−=W """"The mean-squared error is expressed as2()()H J E d n u n ⎡⎤=−⎢⎥⎣⎦W . (5.55) Expand Eq. (5.55)()()()()()()()*2***2()()()()()()H H T H H T H H H d J E d n n d n n E d n d n n d n n n n σ⎡⎤=−−⎢⎥⎣⎦⎡⎤=−−+⎣⎦=+−+W u W u W u W u W u W u P W W P W RW(5.56)where22()d E d n σ⎡⎤=⎣⎦(5.75) *()()E n d n ⎡⎤=⎣⎦P u (5.58) ()()H E n n ⎡⎤=⎣⎦R u u (5.59)Note that *()T n W u is a 1x1 scalar. Then()**()()()TT T Hn n n ==W u W u u W(5.60)The mean squared error at the time n is2()()()()()H H H d J n n n n n σ=+−+P W W P W RW (5.61)5. 5. 3 The Steepest-Descent AlgorithmThe updated weight vector is computed by using the simple reclusive relation*()1(1)()()2n n n J n μ+=−∇W W W (5.62)where []101()()(),(),()()TN N n w n w n w n w n w n −−=W """".The factor21used merely for the propose of canceling a factor 2 that appears in the formula**()()()2()22()n J n J n n n ∂∇=∂=−+W W P RW (5.63) Then[]1n n n μ+W()=W()+P -RW() (5.64)5. 5. 4 The Least-Mean-Square (LMS) Adaptive AlgorithmIn reality, the exact measurements of the gradient Vector are almost impossible. Consequently, the gradient vector must be estimated from the available data. The simplest choice of estimators for R and P is to use instantaneous estimates that are based on sample values of the input signal vector and desired response, i.e.()()()ˆH n n n =Ru u ()()*ˆ()n n d n =Pu (5.65) Correspondingly, the instantaneous estimate of the gradient vector is**()ˆ()2()()2()()()H n J n n d n n n n ∇=−+W u u u W (5.66) Substituting the estimate of (5.66) for *()()n J n ∇W in the steepest-descent algorithm in (5.62), we obtain a new recursive relation for the weight vector:**(1)()()()()()()()()Hn n n d n n n n n e n μμ⎡⎤+=+−⎣⎦=+W W u u W W u (5.67)where*()()()()T e n d n n n =−u W (5.68)(5.67) is referred to as LMS adaptive algorithm. To ensure the convergence, the iteration constant μ should be selected as20maxμλ<<(5.59) where max λ is the largest eigenvalue of the matrix R .(Simon Haykin, Adaptive Filter Theory, 3rd Edition, Prentice-Hall, 1996.)。