Relative Entropy (RE)-Based LTI System Modeling Equipped With Simultaneous Time Delay Estimation and Online Modeling

This paper proposes a novel and efficient method of impulse response modeling in presence of input and noisy output of a linear time-invariant (LTI) system. The approach utilizes Relative Entropy (RE) to provide the impulse response estimate of the system with an optimum length as well as an optimum time delay. Classical methods for this system modeling use two separate steps for the time delay estimation and for the impulse response length selection. Existing time delay methods focus on various proposed criteria, while the existing order selection approaches choose the optimum impulse response length based on their own criteria that are different from the time delay approaches. The strength of the proposed RE based method is in using only “one” criterion, the RE based criterion, to estimate both the time delay and the impulse response length simultaneously. The desired RE is the Kullback-Leilber divergence of the estimated distribution from its unknown true distribution. A unique probabilistic validation approach estimates this unavailable desired relative entropy and minimizes this criterion to provide the impulse response estimate. In addition, estimation of the noise variance, when the Signal to Noise Ratio (SNR) is unknown, is concurrent and is based on optimizing the same RE based criterion. This work elaborates the critical role of the data length and the SNR in data based LTI system modeling. The approach is also extended for online impulse response estimation. The proposed online method reduces computational complexity of the offline model estimation upon the arrival of a new sample. The introduced efficient stopping criterion for the online approach is extremely valuable in practical applications. Simulation results depict superiority of the RE based approach as a time delay estimator or as an order selection approach compared to the conventional methods. They also illustrate precision and efficiency of the proposed method compared to the state of the art methods in simultaneous time delay estimation and order selection. Not only RE based method outperforms the competing approaches, but also is shown to be more robust to the variations of the SNR.