【正文】 . The considered adaptive ?ltering problem consists in trying to adjust a set of weighting coef?cients so that the system output, kTkk XWy ? , tracks a reference signal, assumed as kkTkk nXWd ?? * ,where kn is a zero mean Gaussian noise with the variance 2n? ,and *kW is the optimal weight vector (Wiener vector). Two cases will be considered: WWk ?* is a constant (stationary case) and *kW is timevarying (nonstationary case). In nonstationary case the unknown system parameters( . the optimal vector *kW )are time variant. It is often assumed that variation of *kW may be modeled as Kkk ZWW ??? ** 1 is the zeromean random perturbation, independent on kX and kn with the autocorrelation matrix ? ? IZZEG ZTkk 2??? .Note that analysis for the stationary case directly follows for 02?Z? .The weighting coef?cient vector converges to the Wiener one, if the condition from [1, 2] is satis?ed. De?ne the weighting coef?cientsmisalignment, [1–3], *kkk WWV ?? . It is due to both the effects of gradient noise (weighting coef?cients variations around the average value) and the weighting vector lag (difference between the average and the optimal value), [3]. It can be expressed as: ? ?? ? ? ?? ?*kkkkk WWEWEWV ???? , (2) According to (2), the ith element of kV is: (3) where ? ?? ?kWbias i is the weighting coef?cient bias and ??ki? is a zeromean random variable with the variance 2? .The variance depends on the type of LMSbased algorithm, as well as on the external noise variance 2n? .Thus, if the noise variance is constant or slowlyvarying, 2? is time invariant for a particular ? ? ? ?? ? ? ?? ? ? ? ? ?? ?? ?? ?? ? ? ?kkWbi as kWEkWkWkWEkViiiiiii ??? ???? * LMSbased algorithm. In that sense, in the analysis that follows we will assume that 2? depends only on the algorithm type, . on its parameters. An important performance measure for an adaptive ?lter is its mean square deviation (MSD) of weighting coef?cients. For the adaptive ?lters, it is given by, [3]: ? ?kTkk VVEM S D ??? lim. 3. Combined adaptive ?lter The basic idea of the bined adaptive ?lter lies in parallel implementation of two or more adaptive LMSbased algorithms, with the choice of the best among them in each iteration [9]. Choice of the most appropriate algorithm, in each iteration, reduces to the choice of the best value for the weighting coef?cients. The best weighting coef?cient is the one that is, at a given instant, the closest to the corresponding value of the Wiener vector. Let ? ?qkWi , be the i ?th weighting coef?cient for LMSbased algorithm with the chosen parameter q at an instant k. Note that one may now treat all the algorithms in a uni?ed way (LMS: q ≡ 181。). LMSbased algorithm behavior is crucially dependent on q. In each iteration there is an optimal value qopt , producing the best performance of the adaptive al gorithm. Analyze now a bined adaptive ?lter, with several LMSbased algorithms of the same type, but with different parameter q. The weighting coef?cients are random variables distributed around the ??kWi* ,with ? ?? ?qkWbias i , and the variance 2q? , related by [4, 9]: ? ? ? ? ? ?? ? qiii qkWbi askWqkW ????? , * , (4) where (4) holds with the probability P(κ), dependent on κ. For example, for κ = 2 and a Gaussian distribution,P(κ) = (two sigma rule). De?ne the con?dence intervals for ? ? ]9,4[,qkWi : ? ? ? ? ? ?? ?qiqii qkWkqkWkD ??? 2,2, ??? (5) Then, from (4) and (5) we conclude that, as long as ? ?? ? qi qkWb ia s ???, , ? ? ? ?kDkW ii ?* , independently on q. This means that, for small bias, the con?dence intervals, for different sq? of the same LMSbased algorithm, of the same LMSbased algorithm, intersect. When, on the other hand, the bias bees large, then the central positions of the intervals for different sq? are far apart, and they do not intersect. Since we do not have apriori information about the ? ?? ?qkWbias i , ,we will use a speci?c statistical approach to get the criterion for the choice of adaptive algorithm, . for the values of q. The criterion follows from the tradeoff condition that bias and variance are of the same order of magnitude, . ? ?? ? ? ?4, qi qkWb ia s ???. The proposed bined algorithm (CA) can now be summarized in the following steps: Step 1. Calculate ? ?qkWi , for the algorithms with different sq? from the prede?ned set ? ??, 2qqQ