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This paper proposes an impulse response modeling in presence of input andnoisy output of a linear time-invariant (LTI) system. The approach utilizesRelative Entropy (RE) to choose the optimum impulse response estimate, optimumtime delay and optimum impulse response length. The desired RE is theKulback-Lielber divergence of the estimated distribution from its unknown truedistribution. A unique probabilistic validation approach estimates the desiredrelative entropy and minimizes this criterion to provide the impulse responseestimate. Classical methods have approached this system modeling problem fromtwo separate angles for the time delay estimation and for the order selection.Time delay methods focus on time delay estimate minimizing various proposedcriteria, while the existing order selection approaches choose the optimumimpulse response length based on their proposed criteria. The strength of theproposed RE based method is in using the RE based criterion to estimate boththe time delay and impulse response length simultaneously. In addition,estimation of the noise variance, when the Signal to Noise Ratio (SNR) isunknown is also concurrent and is based on optimizing the same RE basedcriterion. The RE based approach is also extended for online impulse responseestimations. The online method reduces the model estimation computationalcomplexity upon the arrival of a new sample. The introduced efficient stoppingcriteria for this online approaches is extremely valuable in practicalapplications. Simulation results illustrate precision and efficiency of theproposed method compared to the conventional time delay or order selectionapproaches.

Relative Entropy (RE) Based LTI System Modeling Equipped with time delay Estimation and Online Modeling