The Complex Subband Decomposition and its Application to the Decimation of Large Adaptive Filtering Problems

In this paper, we show that a near perfect reconstruction (NPR) $M$-channel filterbank with a diagonal system inserted between the analysis and synthesis filterbanks may be used to decompose a finite impulse response (FIR) system of order $L$ into $M$ complex subband components, each of order $L/K$, where $K$ is the downsampling rate. This decomposition is at the expense of using complex arithmetic for the subband processing. The theory surrounding the proposed filterbank structure leads to a new understanding of subbanded adaptive filtering implementations. It also leads naturally to a delayless subbanded adaptive filter scheme. Using conditions on the analysis and synthesis filters, the formulas for the subband components and their respective properties are developed. Simulation results for an acoustic echo cancellation (AEC) example are given to support the developed theory.