【正文】 s performers of the troupe still tour the region39。 結(jié)論 組合 LMS 算法，在自適應(yīng)系統(tǒng) 中 將這些參數(shù)變化的跟蹤 與 算法的良好性能結(jié)果相結(jié)合，是 自 適應(yīng)過(guò)程 中 選擇 的 更好的算法，一直到穩(wěn)定狀態(tài)時(shí) 需要從最優(yōu)值與最小方差算法的加權(quán)系數(shù)的偏差。 所有對(duì) CA 參數(shù)的改進(jìn)和 是相同的， 對(duì) VS LMS 算法 [6]，有關(guān)的參數(shù)值是變化的 且具有符號(hào)的連續(xù)性， m0 = 11， m1 = 7。 （ a）優(yōu)化加權(quán)時(shí)變向量 ： 提出 的想法可能被應(yīng)用到 SA 算法 的 非平穩(wěn)情況。需要注意的是 第 200 和第 400 迭代，該 LMS 算法可以采取任何步長(zhǎng) 根據(jù) 不同的 認(rèn)識(shí) 。 簡(jiǎn)單的分析表明，在 CA 增加最多的 操作步驟 ，添加了 N（ L?1)和 N(L?1) IF 決定增補(bǔ)， 而且 需要 添加一些硬件以滿(mǎn)足組成算法 。 通過(guò)增加更多的觀察 ,這兩個(gè)極端之間 ,我們可以稍微改進(jìn)算法的瞬態(tài)行為。 從一個(gè)最大的差異值算法走向與差異較小的值。 ? ? ? ? ? ?? ? qiii qkWbi askWqkW ????? , * （ 4） （ 4） 中 的概率 P(κ)依賴(lài) κ的值 . 例如 κ= 2的 高斯分布， P(κ)= （ 兩個(gè)σ規(guī)則 ）。最好的加權(quán)系數(shù)是 1，即在給定的時(shí)刻，向相應(yīng)的維納矢量值最接近。 定義加權(quán) 錯(cuò)位系數(shù)， [1–3], *kkk WWV ?? 。}是預(yù)期值的估計(jì) 。 每一種基于 LMS 的 算法都至少 有 一個(gè)參數(shù) 在 適應(yīng)過(guò)程 （ LMS 算法 和 符號(hào)算法 ，加強(qiáng)和 GLMS 平滑系數(shù)，各種參數(shù)對(duì) 變步長(zhǎng) LMS 算法的影響 ） 中 被 預(yù)先 定義。 μ = . The optimal vectors is generated according to the presented model with ?Z? ,and with κ = 2. In the first 30 iterations the variance was estimated according to (7), and CA takes the coefficients of SA with μ (SA1). Figure 2(a) shows the AMSD characteristics for each algorithm. In steady state the CA does not ideally follow the SA3 with μ/8, because of the nonstationary problem nature and a relatively small difference between the coefficient variances of the SA2 and SA3. However,this does not affect the overall performance of the proposed algorithm. AMSD for each considered algorithm was: AMSD = (SA1,μ), AMSD = (SA2,μ/2), AMSD = (SA3, μ/8) and AMSD = (Comb). (b) Comparison with VS LMS algorithm [6]: In this simulation we take the improved CA (9) from , and pare its performance with the VS LMS algorithm [6], in the case of abrupt changes of optimal vector. Since the considered VS LMS algorithm[6] updates its step size for each weighting coefficient individually, the parison of these two algorithms is meaningful. All the parameters for the improved CA are the same as in . For the VS LMS algorithm [6], the relevant parameter values are the counter of sign change m0 = 11,and the counter of sign continuity m1 = 7. Figure 2(b)shows the AMSD for the pared algorithms, where one can observe the favorable properties of the CA, especially after the abrupt changes. Note that abrupt changes are generated by multiplying all the system coefficients by ?1 at the 2021th iteration (Fig. 2(b)). The AMSD for the VS LMS was AMSD = , while its value for the CA (CoLMS) was AMSD = . For a plete parison of these algorithms we consider now their calculation plexity, expressed by the respective increase in number of operations with respect to the LMS algorithm. The CA increases the number of requres operations for N additions and N IF the VS LMS algorithm, the respective increase is: 3N multiplications, N additions, and at least 2N IF decisions. These values show the advantage of the CA with respect to the calculation plexity. 6. Conclusion Combination of the LMS based algorithms, which results in an adaptive system that takes the favorable properties of these algorithms in tracking parameter variations, is the course of adaptation procedure it chooses better algorithms, all the way to the steady state when it takes the algorithm with the smallest variance of the weighting coefficient deviations from the optimal value. Acknowledgement. This work is supported by the Volkswagen Stiftung, Federal Republic of Germany. 基于 LMS 算法的自適應(yīng)組合濾波器 摘要 ： 提出了 一種 自適應(yīng) 組合 濾波器 。 Abstract: A bined adaptive ?lter is proposed. It consists of parallel LMSbased adaptive FIR ?lters and an algorithm for choosing the better among them. As a criterion for parison of the considered algorithms in the proposed ?lter, we take the ratio between bias and variance of the weighting coef?cients. Simulations results con?rm the advantages of the proposed adaptive ?lter. Keywords: Adaptive ?lter, LMS algorithm, Combined algorithm,Bias and variance tradeoff 1． Introduction Adaptive ?lters have been applied in signal processing and control, as well as in many practical problems, [1, 2]. Performance of an adaptive ?lter depends mainly on the algorithm used for updating the ?lter weighting coef?cients. The most monly used adaptive systems are those based on the Least Mean Square (LMS) adaptive algorithm and its modi?cations (LMSbased algorithms). The LMS is simple for implementation and robust in a number of applications [1–3]. However, since it does not always converge in an acceptable manner, there have been many attempts to improve its performance by the appropriate modi?cations: sign algorithm (SA) [8], geometric mean LMS (GLMS) [5], variable stepsize LMS(VS LMS) [6, 7]. Each of the LMSbased algorithms has at least one parameter that should be de?ned prior to the adaptation procedure (step for LMS and SA。(k) is changed [6, 7]. 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 (weighti