ディジタルフィルタの平衡形実現と最適実現の等価関係およびその時間領域設計への応用

抄録

Synthesis of state-space digital filters (SSDF) with small quantization errors is a very important problem in digital filter design, when filters are to be implemented with short register lengths and stable performances. Although synthesis methods of optimal realizations have been proposed, the structural properties of optimal realizations are still unknown in detail. Thus, optimal realizations can not be synthesized efficiently by existing methods.
The controllability and the observability gramians as well as the second order modes of digital filters play an important role in the synthesis of optimal realizations. Therefore, it is reasonable to investigate these quantities first in order to gain some insight into the structural properties of optimal realizations. This paper shows that optimal realizations are scaled and rotated balanced realizations, which have been studied extensively in the linear system context. This relation between optimal realizations and balanced realizations suggests that optimal realizations could have most of the properties which balanced realizations have. One of the most important properties is that optimal realizations are limit-cycle free, and this is proved simply by using the above relation. Applying these results to digital filter design, we propose a direct design method in the time domain. This method simplifies the traditional two-step (approximation and synthesis) design into a one-step design with less computational complexity. The resulting filters can approximate given impulse responses closely, and they are suboptimal and limit-cycle free. The efficiency of the direct design method is shown by a numerical example.