# Stochastic Gaussian and non-Gaussian signal modeling

#### Abstract

This thesis introduces a methodology for modeling stochastic signals that have either Gaussian or approximately bell-shaped non-Gaussian distribution. The synthesized model can be used to generate stochastic signals that approximate both the power spectral density (PSD) and the probability density function (pdf) of the original stochastic signal. The new methodology is based on non-linear transformations, filter banks and autoregressive-moving average (ARMA) models. Because the stochastic signals modeled can have Gaussian distribution or approximately bell shaped non-Gaussian distribution, normality tests such as sample skewness, sample kurtosis, Kolmogorov-Smirnov test, and Shapiro-Wilk test are also used. Many methods have been proposed in the literature for model order estimation and parameter estimation in ARMA modeling. For Gaussian signals, the methods are usually based on second order statistics and, for non-Gaussian, the methods are usually based on higher order statistics. However, in the literature review conducted, a simple method was not found for modeling either Gaussian or non-Gaussian signals that can: (1) consistently estimate true model orders; (2) approximate peaky power spectral densities; (3) approximate the pdf of the original input signal; and (4) have fast computation using parallel architectures. The problem of modeling stochastic non-Gaussian signals is solved here by first using an appropriate non-linear function to transform the input signal to Gaussian. If the input stochastic signal is already Gaussian, then this first step is skipped. The Gaussian signal is then decomposed using an adaptive filter bank which simplifies the model order estimation, and which enables modeling of each subband component using a fixed order ARMA model with only two poles and two zeros. The use of decimated filter bank decomposition also simplifies the modeling of signals that have peaky PSD because it generates subband components that have a smoother PSD than the Gaussian input signal and that can be approximated with low order ARMA. The adaptive filter bank gives perfect reconstruction and implements a dyadic tree decomposition similar to that used in adaptive wavelet packets. After the modeling process is completed, the system can be used to generate stochastic signals that approximate both the PSD and the pdf of the original input signal. Independent random sequences with Gaussian distribution are applied to the input of the ARMA models and then filtered with the corresponding filters of the synthesis filter bank. The new components are then added to synthesize a stochastic signal that approximates the PSD of the Gaussian input signal. If the original input signal was non-Gaussian, then the corresponding inverse non-linear transformation is finally applied. This new stochastic signal approximates both the PSD and the pdf of the original stochastic signal. The new methodology was extensively tested using test Gaussian sequences generated by known ARMA models and stochastic non-Gaussian sequences provided by the Army Research Laboratory (ARL). The non-Gaussian test sequences used had skewness in the range of -0.3 to +0.3 and kurtosis in the range 1.5 to 5.5. All the sequences tested in this thesis indicate that the new methodology can be used to model both the PSD and the pdf of stochastic Gaussian or bell-shaped non-Gaussian sequences.

#### Subject Area

Electrical engineering

#### Recommended Citation

Yerramothu, Madhu Kishore, "Stochastic Gaussian and non-Gaussian signal modeling" (2008). *ETD Collection for University of Texas, El Paso*. AAI1461171.

https://scholarworks.utep.edu/dissertations/AAI1461171