Constrained solutions to IFPPS by finite-dimensional approximation

Abstract

The inverse Frobenius-Perron problem (IFPP) refers to the probabilistic modelling problem that requires the design of a discrete-time, one-dimensional dynamical system (i) that is ergodic and (ii) generates a zero-input time response that possesses prescribed statistical metrics as specifications. Systems designed using solutions to the IFPP hold potential as highly efficient and versatile random signal generators as well as solutions to practical radar signal processing problems. Distinct formulations of the IFPP have appeared in the literature: IFPP-I: The system is designed such its zero-input response possesses a pre- scribed probability density function (PDF). IFPP-II: The system is designed such that its zero-input response simultaneously possesses a prescribed PDF and a prescribed power spectral density (PSD). IFPP-III: An unknown system is reconstructed from sequences of PDF estimates of its zero-input response, which are derived from measurements of the unknown system. The work presented in this thesis contributes towards IFPP-II and IFPP-III by developing novel methods that improve upon exiting solutions. These novel methods construct candidate systems that admit an exact finite-dimensional representation for the FPO, which governs the evolution of density functions of the system’s response. The finite dimensionality of the FPO yields practical analytical methods for designing dynamical systems with prescribed statistical metrics as specifications, and for the consistent reconstruction of unknown dynamical systems. However, the structure of these systems imposes restrictions on the statistical metrics that are realisable, and only permit the approximation of arbitrary metrics that may be prescribed. Examples are presented that demonstrate the realisation of power spectra with prescribable poles and multiple peaks that are adjustable, and the reconstruction of iii unknown systems with consistent branch monotonicity and power spectrum mode characteristics, using the novel methods. Although the application-specific design and performance evaluation of random signal generators constructed using the proposed methods are not considered in this work, it is anticipated that these methods will be of interest in applications such as low-complexity random signal generation for Digital Radio Frequency Memory (DRFM) systems.