A Functional Analysis Approach to Subband System Approximation and Identification

The subband system identification method consists of identifying a linear system in the time-frequency domain. This technique can be also used to approximate a linear system in the same domain. In both cases, it has the advantage of having a very high numerical efficiency; however, analyzing such a technique is not trivial, and the best setup for subband system approximation and identification is not clear. In this paper, we propose a functional analysis setting to the analysis of the subband technique, which leads us to some results on both subband system approximation and identification. Concerning system approximation, we provide an analytical expression to calculate the optimal subband approximation of a given full-band system, when the quality of the approximation is measured by the power of the output error signal, assuming a white input signal. We also provide a tight approximation error bound, for a given subband configuration, which applies in the case where the unknown system to be approximated is known to be the finite-impulse response (FIR) of a given order. Concerning system identification, we provide a novel identification strategy that consists of identifying a “low quality” subband model and use it to build the required model for either subband or delayless reconstruction. This identification strategy reduces the computational complexity of the identification process and yields significantly smaller asymptotic residual errors, when compared with the existing methods.