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Second-Order Volterra System Identification With Noisy Input–Output Measurements

System identification with noisy input-output measurements has been dominantly addressed through the optimization of the mean-squared-error criterion (MSE), especially in adaptive filtering. MSE is known to provide models that approximate the conditional expectation of the target output given the input; however, when the input signal is also contaminated by noise - a frequent occurrence - MSE yields biased estimates of the model parameters with the severity of the bias dependent on the noise power. This drawback has been addressed in various ways, including errors-in-variables techniques. Recently, error whitening criterion (EWC) and associated adaptation algorithms were proposed to address this issue in linear system identification. We extend the applicability of the main concept behind EWC to the unbiased identification of order-2 Volterra series models of nonlinear dynamical systems. The extension does not apply to higher order Volterra models. The main contribution of this letter is a statistical criterion that can be utilized to identify analytically the true parameters of an order-2 Volterra model from noisy input-output data. We also support the theoretical results with simulations; however online learning algorithms that can be derived for the proposed criterion will not be addressed.