Undermodeling-Error Quantification for Quadratically Nonlinear System Identification in the Short-Time Fourier Transform Domain

In this paper, we introduce an estimation error analysis for quadratically nonlinear system identification in the short-time Fourier transform (STFT) domain. The identification scheme consists of a parallel connection of a linear component, represented by crossband filters between subbands, and a quadratic component, which is modeled by multiplicative cross-terms. We mainly concentrate on two types of undermodeling errors. The first is caused by employing a purely linear model in the estimation process (i.e., nonlinear undermodeling), and the second is a consequence of restricting the number of estimated crossband filters in the linear component. We derive analytical relations between the noise level, nonlinearity strength, and the obtainable mean-square error (mse) in subbands. We show that for low signal-to-noise ratio (SNR) conditions, a lower mse is achieved by allowing for nonlinear undermodeling and utilizing a purely linear model. However, as the SNR increases, the performance can be generally improved by incorporating a nonlinear component into the model. We further show that as the SNR increases, a larger number of crossband filters should be estimated to attain a lower mse, whether a linear or nonlinear model is employed. Experimental results support the theoretical derivations.