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 statistics**of** 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 **for**m 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, $(.) is the derivative **of** @(yn),that is,4(Yn) = @'(yn), (6)and it is sometimes called prediction error function.If an FIR filter is used as the equalizer, then (5) can beexpressed as(5)e(n+l) .=?(") - pXn4kn), ( 7)where is the coeficient vector **of** a blind equalizer after71-th iteration defined asand x, is the channel output vector at time n defined asATxn = [xn+N,'.., xn, ''' , Zn-N] .Since all BIB0 channels can be approximated as amoving-average model with appropriate impulse response{LM,. .., ho,. . ., h ~ } the , channel output vector can be expressedasxn = UTan, (10)where U is a (2N+ 2M + 1) x (2N + 1) channel matrix definedas(9)With the above definitions, the channel output can be expressedin a compact **for**m:where Zdn) is t8he equalized system vector at time n defined asIt is obvious that an FIR channel can not be perfectly equalizedby an FIR equalizer, that is, there is no equalizationvector c such thatwhereUc = eM+N, (15)However, when the length **of** the equalizes is large enough,there exists a e such that 11 - eM+N 11 is very small.111. PROPERTIES OF PREDICTION ERROR FUIVCTIONBe**for**e analyzing the convergence behavior **of** blind equalizers,we first introduce some properties **of** the prediction errorfunction here. The following lemma considers two importantproperties to be used in subsequent discussions.Lemma: The prediction error function 4(.) has the followingtwo properties:1) When th,e parameters **of** a finite-length equalizer makeits cost function attain one **of** its minima, the output **of**the equalized system, &, satisfies(2) E{d@n)xn} 0, and(ai) UTF% is positive-definite,where the (2M + 2N + 1) x (2M + 2N + 11) matrix F isdefined as1F = -E{a,C$(&)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 minimum**of** 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 there**for**e, C = c,. For Godard algorithm[6],$(y) = y(y2 - r) with r = M, there**for**e,andwhereE{qqa,)a;} = 3u4 - m4, (31)E{$'(%)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 minimum**of** 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 validation**of** 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 **for**mula (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, uni**for**mlydistributed 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 coefficients**of** 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 **of**learning curves are almost within one standard deviation **of**the 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 **of**011equalizers, 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 **of**oman equalizer, we have to consider the trade-**of**f 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-size**of** 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

~ ”.,”typical telephone channel [16]. The center-tap initializationstrategy[5] is used **for** blind equalization algorithm.When the Sat0 algorithm is used, the theoretical relationshipbetween the total distortion and the length **of** equalizer**for** different step sizes are illustrated in Figure 5 (a), which 1indicates that the optimum length **of** Sat0 equalizer **for** this a ~ ~ l ochannel is between 15 and 25 dependent upon the step-size.Figure 5 (b) demonstrates the comparison between the theoreticalresults **of** Df + D, and simulated results **for** step sizep = 0.002.The calculation and simulation results are given in Figure6 **for** Godard algorithm. Because g(l)/f(l) **for** Godard algorithm(0.169) is less than that **for** Sat0 algorithm (0.250) **for**4-level PAM input, Godard algorithm should have less distortionthan Sat0 should according to (48), which is confirmedby comparing Figure 5 and 6.111 (hlFig. 4. (a) The impulse response, and (b) the frequency response **of**channel 11.Fig. 5. Total distortion **of** equalized system (a) theoretical results **for**different step size p, (b) simulation results **for** p = 0.002, using Satoalgorithm.We have studied the static and dynamic convergence behavior**of** adaptive blind equalizers in PAM digital communicationsystems based on the first-order approximation to thecost function **of** blind algorithms under the independence assumption.Most **of** the analysis results presented here can beextended to QAM digital communication systems. Our analysisresult indicates that **for** a given channel and step-size, thereis an optimal length **for** an equalizer to minimize the intersynibolinterference. The results imply that a longer-lengthblind equalizer does not necessarily outper**for**m a shorter one,as contrary to what is conventionally conjectured. The analysisresults presented in this paper can be directly employedin the design **of** blind equalizer in practical communicationsystems.I’”’---em(ai)__---Fig. 6. Total distortion **of** equalized system (a) theoretical results **for**different step size p, (b) simulation results **for** p = 0.002, usingGodard algorithm.REFERENCES[l] A. Benveniste, M. Goursat, and G. Ruget. “Robust identification **of**a nonminimum phase system: blind adjustment **of** a linear equalizerin data communications,” IEEE Trans. on Automatic Control, AC-[2][3][4][5][6][7][8][9]25:385-399, June 1980.A. Benveniste and M. Goursat, “**Blind** equalizers,” IEEE Transactionson Communications, COM-32: 871-882, August, 1982.R. Cusani and A. Laurenti, “Convergence analysis **of** the CMAblind equalizer,” IEEE Transactions on Communications, COM-43:1304-1307, Feb./March/April 1995.Z. Ding, R. A. Kennedy, B. D. 0. Anderson and C. R. 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