# Learning Characteristics for General Class of Adaptive Blind Equalizer

Learning Characteristics for General Class of Adaptive Blind Equalizer

where {s,} is the impulse response of the equalized systemrelated to h, and c, byand a, is the input symbol vector at time n defined askIn blind equalization, the original sequence is unknown tothe receiver except for its probabilistic or statistical properties.A blind equalization algorithm is usually devisedby minimizing a cost function consisting of the statisticsof the output of the equalizer y, which is a function of{. . ., s-1, SO, SI,...} or {..., cP1, co, cl,...}, The costfunction is usually of the form E{@(y,)}, where @(y,), afunction of y, is selected such that the cost function has theglobal minimum points at{sn} = f{s[n - nd]} for all nd = 0, fl, 3~2,. .. (4)A stochastic gradient algorithm is used to miniimize the costfunction to obtain an on-line equalization algorithm, whichadjusts the k-th parameter of the equalizer at time n by4-(n+l) = 4,) -'k ck &(Yn)zn-k,where p is a small step size, \$(.) is the derivative of @(yn),that is,4(Yn) = @'(yn), (6)and it is sometimes called prediction error function.If an FIR filter is used as the equalizer, then (5) can beexpressed as(5)e(n+l) .=?(") - pXn4kn), ( 7)where is the coeficient vector of a blind equalizer after71-th iteration defined asand x, is the channel output vector at time n defined asATxn = [xn+N,'.., xn, ''' , Zn-N] .Since all BIB0 channels can be approximated as amoving-average model with appropriate impulse response{LM,. .., ho,. . ., h ~ } the , channel output vector can be expressedasxn = UTan, (10)where U is a (2N+ 2M + 1) x (2N + 1) channel matrix definedas(9)With the above definitions, the channel output can be expressedin a compact form:where Zdn) is t8he equalized system vector at time n defined asIt is obvious that an FIR channel can not be perfectly equalizedby an FIR equalizer, that is, there is no equalizationvector c such thatwhereUc = eM+N, (15)However, when the length of the equalizes is large enough,there exists a e such that 11 - eM+N 11 is very small.111. PROPERTIES OF PREDICTION ERROR FUIVCTIONBefore analyzing the convergence behavior of blind equalizers,we first introduce some properties of the prediction errorfunction here. The following lemma considers two importantproperties to be used in subsequent discussions.Lemma: The prediction error function 4(.) has the followingtwo properties:1) When th,e parameters of a finite-length equalizer makeits cost function attain one of its minima, the output ofthe equalized system, &, satisfies(2) E{d@n)xn} 0, and(ai) UTF% is positive-definite,where the (2M + 2N + 1) x (2M + 2N + 11) matrix F isdefined as1F = -E{a,C\$(&)a;},U2(17)with #(.) being the derivative of 4(.), &, = >lk EkXn-kand & being the equalizer coeficients meking the costfunction attain a minimum.2) For all integers n and ICx&undE{d'(an)a2,} > 0. (19)With the a'bove lemma, we are now able to analyze thestatic and dynamic convergence of adaptive blind equalizers.1021

IV. STATIC CONVERGENCE ANALYSISIf the equalizer is double-infinite, then at the global minimumof the cost function, the parameters of the equalizer{Ci} = {+L,>,(20)for some integer nd. However, only FIR blind equalizer isused in practical systems. In this case, smart initializationstrategies [4],[5], [9] will make the equalizer coefficients convergeto a minimum {E, : n = -N, .. . + ,0, . ., N } of the costfunction near the channel inverse such that in - a, is verysmall. Using first-order approximation to 4(.) at a,, we canprove the following theorem.Theorem 1: If an FIR equalizer is used to equalize an FIRchannel, then at the minimum near the channel inverse, theequalizer coefficient vector [E-N, . .. ,Eo, ..., ENIT can beexpressed asZ: = f(0)Ri'h. (21)whereandwithand-M+NRf & ZTFZM+N(23)From the above theorem, the equalizer coefficient vector atthe minimum of the cost function near the channel inverse isdetermined by (21).For the channel with impulse response vector h, the optimumequalizer (Wiener-Hopf filter) coefficient vector to minimizeE{(y, - a,)2} is given by [7]wherec, = R-lh, (26)R = ZTZ.Comparing (21) and (26), we have that the sufficient andnecessary condition for C = c, for any FIR channel isSince 3c is of full column rank for all no-zero h, Equation (28)impliesf (0) = fU), (29)which meansE{\$'(a,)a:} = ~{4(anW{a3. (30)For Sat0 algorithm[l4], decision-directed equalizers[lO], [12],\$'(a,) = 1, and therefore, C = c,. For Godard algorithm[6],\$(y) = y(y2 - r) with r = M, therefore,andwhereE{qqa,)a;} = 3u4 - m4, (31)E{\$'(%)U;} = 2m4, (32)m4 = ~(a4,). (33)Hence, if the channel input is binary, (30) is true and C = e,.Otherwise, E # c,.The distortion due to the finite-length of equalizer isA 2Df = 11 g-eM+N 112= II Zc-eM+N II .(34)With the increase of the length of the blind equalizer, theglobal minimum of the cost function adopted by the equalizationalgorithm will be closer to the channel inverse. Hence,the distortion Df will decrease.V. DYNAMICONVERGENCE ANALYSISWhen the blind equalization algorithms are implementedusing stochastic gradient method, as are most blind equalizers,the blind equalizers will have an extra distortionAE, = E(,) - C due to the gradient noise. Here, we study thestochastic dynamic convergence behavior of blind equalizerswhen the parameters of blind equalizers near the global minimumof the cost function. In our analysis, we will use theindependence assumption which assumes that a, and 6, arestatistically independent. Similar assumptions have also beenused in the convergence analysis of LMS algorithm, decisiondirectedequalizer, and Sato algorithm. The references [7],[12], [20],[21] have given some good justification on the validationof this assumption.By means of independence assumption, together with thefirst-order approximation, we are able to prove the followingdynamic convergence theorem.Theorem 2: LetRf A ZTF'H, (35)with the largest eigenvalue A ,withandR, ZTG'H, (36)- 1G = -E{a,\$2(&)aT}.02(37)1) For any FIR blind equalization algorithm mean convergencebehavior near the global minimum of the cost functionsatisfiesE{€,} = (I - p2R,)"E{6o}. (38)If the step-size p in iteration formula (5) or (7) satisfies(39)1022

thenE{c(,)) -+ E and E{s(")} --+ S, (40)2) The equalizer coefficient vector c(") -+ Z. is not consistentand ut the equilibrium near the minimum of the cost function,the correlation matrix R, of E is uniquely determined by thefollowing Lyupunov equationif 01Rf& -+ R,Rf = pR,, (41)P < AmazD2.From the above theorem, the distortion of the equalizedsystem due to gradient noise isD, E{\/ s -S [I2}= E{ll 7-k 112}= t r [ 3tTR, X]= tr[RR,].When an FIR equalizer is so long that {E,a,} , thenRf z Rf, R, M R,,where we have used the definitionsandM in}, {& M(43)R, 2 3tTG3t, (44)VI. COMPUTER SIMULATIONS AND CONCLUSIONSince approximation has been used in our tlheoretical analysis,we shall check the validity of our theory by computer simulations.Two computer simulation examples are presentedin this section.Example I:The channel input sequence {a,} is independent, uniformlydistributed over {fa,f3a}(a = l/& to make E{ai} =1). The impulse response of the channel is h, = 0.3"u[n]with 4.1 being unit step function. An FIFL equalizer withcoefficients 1% and c1 is used to compensate for the channeldistortion. The initial value of the equalizer coefficient vector(49)The Sat0 algorithm[l4] is first used to adjust the coefficientsof the equalizer. When the step-size p = 0.002, 10trials of learning curves of c(,) are shown in Figuire 2. In thisfigure, the t,hick solid line is the theoretical average learningcurve, the thick dot-dash lines are the theoretical onestandard-deviationlines. According to this figure, 10 trials oflearning curves are almost within one standard deviation ofthe theoretical average learning curves for Saio algorithm.Similar simulations have also been done for Godard algorithm[6].The simulation results are shown in Figure 3, whichalso confirm our theoretical analysis.For the blind equalization algorithms with f(0) = f(l), Rf =f(1)R. Using (473, we haveFig. 2. 10 trials of learning curves of (a) CO, and (b) c1 for Sato algorithmusing 1.1 = 0.002.For those blind equalizers with f(0) # f (l),used to approximately estimate the average distortion introducedby gradient noise. According to (48), D, is proportionalto the step-size p and the length of equalizer N. But,on the other hand, step-size affects the convergence speed of011equalizers, i.e. the larger the p, the faster it converges if p is(.VI(48) can also be -~...... .in the allowable range. Hence, when we select the step-size ofoman equalizer, we have to consider the trade-off between these " *-CIIIIIu- 'm '_ " *""*-d,*lUI,.two factors.As we have seen, there are two Sources of distortion. One is Fig. 3. in trials of learning curves of (a) CO, and l(b) for GodardDf in (4.24) due to the finite length of an equalizer, another algorithm using p = 0.002.is D, in (48) due to the gradient noise. Once the step-sizeof a blind equalizer is set, there must be an optimum length Example 2:that can be found for an FIR equalizer to minimize the to- The channel input sequence in this example has the sametal distortion D = Df + D, since with the increase of the stat,istical property as in Example 1. The channel impulseequalizer length, Df decreases while D, increases.and frequency response are shown in Figure 4, which is a1023