This paper presents a new analysis of power complementary filters using the state-space representation. Our analysis is based on the bounded-real Riccati equations that were developed in the field of control theory. Through this new state-space analysis of power complementary filters, we prove that the sum of the controllability/observability Gramians of a pair of power complementary filters is represented by a constant matrix, which is given as a solution to the bounded-real Riccati equations. This result shows that power complementary filters possess complementary properties with respect to the Gramians, as well as the magnitude responses of systems. Furthermore, we derive new theorems on a specific family of power complementary filters that are generated by a pair of invertible solutions to the bounded-real Riccati equations. These theorems show some interesting relationships of this family with respect to the Gramians, zeros, and coefficients of systems. Finally, we give a numerical example to demonstrate our results.

This paper describes a design technique of perfect reconstruction (PR) two-channel IIR filter bank. M.J.T. Smith et al., gave two types of PR IIR filter bank systems. One is the system such that the analysis and synthesis filters with nonlinear phase are implemented with all-pass polyphase filters and satisfy the power complementary condition approximately. The other is the system such that all the analysis and synthesis filters have liner phase responses and do not satisfy the power complementary condition. To improve coding performance, we propose a filter bank system such that all the analysis and synthesis filters have linear phase and satisfy the power complementary condition approximately.