最大最小位相分解に基づくpre－inverse適応システムと収束条件

摘要

未知系の伝達関数の逆数を推定することを目的として，未知系の前段に適応フィルタを配置した構成の適応システムは一般にpre壷verse適応システムと呼ばれる．未知系の伝達関数そのものは適応トランスパーサルフィルタにより容易に推定できるが，その逆数の実現性の保証がないために種々の適応システムが検討されてきた．その代表例が魚hered-Ⅹアルゴリズムを用いた適応システムである．このシステムは推定誤差を有する補助の適応フィルタの出力を用いて前段の適応フィルタを制御するために，正しい適応動作は期待できない．これに対し最大最小位相分解手法を用いれば，各々の伝達関数の逆数が遅延を含む安定な伝達関数となるために，補助の適応フィルタは不要となりこれにより正しい適応動作が期待できる．本論文ではこの手法に基づく適応システムを提案し，その収束条件を定式化する．適応システムの出力が常に入力信号の遅延に等しいという性質を用いてヘッセ行列を求め，これが正定値となることを示す．この結果，収束条件が定式化される．これらの理論検討結果は計算機シミュレーションにより厳密に検証される．%In general, the adaptive system using the structure in which an adaptive filter is placed before the unknown system to estimate the inverse transfer function of the unknown system is called pre-inverse adaptive system. The transfer function of the unknown system can be easily estimated by an adaptive transversal filter. Since there is no guarantee that the inverse transfer function of the estimated transfer function can be realized, various adaptive system has been studied. The representative example is an adaptive system that used the filtered-x adaptive algorithm. The correct adaptive performance cannot be expected since this system controls the main adaptive filter by using the output signal of the auxiliary adaptive filter that has the estimation error. In the maximum and minimum phase decomposition approach, the inverse of each transfer function becomes a stable transfer function that contains the delay so that the auxiliary adaptive filter becomes unnecessary and the correct adaptive performance can be expected, hi this paper we propose an adaptive system and formulate the convergence condition of this adaptive system. Hessian is derived using the property that the output signal of the adaptive system always equals to the delay version of the input signal, and it is shown that Hessian becomes a positive definite. As a result, the convergence condition is formulated. Finally, these results of theoretical consideration are verified strictly by computer simulation.