The Overlapping Problem Of Nonlinear Components In Physiological System Addressed By Maximum Length Sequences

Living systems usually exhibit complex and nonlinear behaviors, which can be characterized by a mathematical model carefully tuned to represent the relationship between the input and output data. A linear system is capable of determining the input and output relationships through an impulse response function; however, for a nonlinear system, a higher order transfer function has to be used for this purpose. A nonlinear system can be typically modeled by Volterra or an equivalent Wiener series expansion, in which the Volterra or Wiener kernels to be estimated can fully define the system characteristics.The kernel estimation for such nonlinear system usually requires the input signal to be a long Gaussian white noise to completely activate the underlying system. Under such conditions, Lee and Schetzen proposed a convenient cross-correlation method widely used to estimate the kernel functions. In several circumstances, particularly for a variety of living biological systems, input signals are constrained as a series of impulse trains instead of continuous signals, such as the Gaussian white noise. For instance, the auditory system is usually studied by stimulating the ear with a series of click sounds to activate the corresponding neurons in the cochlea and neural pathway to evaluate the hearing integrity. A well-studied impulse train for the input is a pseudo-random binary sequence called maximum length sequence (short for MLS), which has an important role in nonlinear system identification. The correlation property of an MLS is analogous to a Gaussian white noise such that to model the system by borrowing the idea of the cross-correlation method for Gaussian white noise input is possible. Hence, the binary kernels are defined using cross-correlation method for MLS inputs.However, the binary kernel slices — derived by making use of the shift-and-product property of the MLS — are all laid in the first-order cross-correlation function between the MLS input and the system response, that is, the observed output. The specific location of any kernel slice in the cross-correlation function is determined through a complex shift function that cannot be explicitly determined. If the kernel slices are improperly arranged such that overlaps among neighboring slices occur, then the kernel estimation is inevitably distorted..In the present study, we introduce a well-established model based on kernel functions for input of the MLS can be used to estimate nonlinear binary kernel slices using cross-correlation method and focusing on the characteristics of kernel slices overlapping problem. The major works of this thesis include three projects as follows:(1) Investigate the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input MLS and the system output. Techniques that may help minimize overlap among kernel slices include multiply the length of the input, sparsely the impulse train that reduced number of available kernel slices, separately estimated kernel function and properly arrange the adjacent slices lie along the time axis of the first-order cross-correlation function.With appropriate constraints, we derived the relation of binary kernels to Volterra kernel, and explained the physical significance of binary kernel slices. Unlike Volterra and Wiener kernel, the even-order diagonal values of binary kernel simply do not exist. This is because higher-order product sequences of MLS input, if there are no relative shifts, reduce to the all-one sequence. These missing values reflect the incomplete characterization of a nonlinear system using binary kernel.(2) We addressed the overlap problem through a new strategy using an inverse-repeat maximum length sequence (short for IR-MLS). We provided and proved several relevant properties of the IR-MLS allowing the estimation of the binary kernel slices. By examining the special shift-and-product properties of the IR-MLS derived for odd and even shifts, the odd- and even-order kernel slices can be separately represented in the cross-correlation functions, such that the chance of overlapping significantly decreases.(3) Consideration the shift function of primitive polynomial and its reciprocal primitive polynomial is symmetrical in a way, the computational burden to calculate the specific value of a shift function can reduce 50%. The conventional method to avoid overlap is to increasing the order of MLS, we pointed out that increasing the order cannot ensure to avoid overlap, but decreasing the probability. We proposed a systematic method available for properly selecting a sequence to avoid overlap based on the structure of MLS is determined through a primitive polynomial. Lastly, a second-order nonlinear Wiener model system was simulated to demonstrate the process of the proposed method.