System identification and modelling of non-stationary signals

The modeling of non-stationary signals and the identification of linear, time-invariant (LTI) and linear, time-varying (LTV) systems are problems of great interest in diverse fields in signal processing. They have many applications in speech and image processing, geophysics, biostatistical signal processing, radar, medicine, and seismology. For each application, the identification or modeling results in a set of mathematical equations which can be used to understand the behavior of the system or signal. Most of the standard parameter estimation and LTI system identification algorithms available in the literature estimate only a spectrally equivalent minimum phase system. Although bispectrum has been recently applied in identification of non-minimum phase LTI systems, it requires the assumption of stationarity and restricts the process to have non-symmetric probability density. Therefore, when the input/output of the system are non-stationary, neither the power spectrum nor the bispectrum can handle this problem since they do not reflect the time variation of the process characteristics.;In the rest of the thesis, we have developed the following. In order to model non-stationary signals or identifying LTV systems, we propose four methods. The first method depends on the time-varying autocorrelation function of a non-stationary signal; the second one is by using the time-varying pseudo-autocorrelation; the third one is based on the time-varying cumulants; and the last one applies the time-varying sum-of-pseudo cumulants. Also we will show that if the output of the LTV system is corrupted by stationary/non-stationary noise with symmetric distribution, the time-varying coefficients of the system are identified using the time-varying cumulants and sum-of-pseudo cumulants algorithms, Finally, since speech is an interesting application of our theory, we have applied the time-varying autocorrelation and the time-varying sum-of-pseudo cumulants in modeling voiced/unvoiced speech segment.;In this thesis, some algorithms are proposed for LTI/LTV system identification and non-stationary signal modeling with or without additive noise, without restricting the signals to have non-symmetric probability densities. Our methodology for the identification and modeling is based on the theory of the evolutionary spectral and bispectral analysis and in particular the Wold-Cramer representation of non-stationary signals. Using the Wold-Cramer representation, a new time-dependent input/output spectral relationship for an LTI system with non-stationary input is obtained. The spectral relationship such obtained permits us to identify LTI systems with minor restriction on the input signal, and preserves not only the system magnitude but also its phase, thus non-minimum phase LTI systems can be identified. With small modification in the algorithm, we are able to identify LTI system with its output corrupted by stationary noise. A model for a non-stationary signal as the output of a cascade of an LTV with an LTI systems is developed, Also, an algorithm for decomposing a non-stationary signal into a stationary part and a non-stationary part is presented, using this time-dependent spectral relationship. The same relationship can be used to find the linear phase shift of a process. In the case when the noise is non-stationary with symmetric distribution, the effect of noise can be removed by introducing the evolutionary bispectrum.