Parameter Recovery for Transient Signals

Transient signals naturally arise in numerous disciplines for which the decay rates and amplitudes carry some informational significance. Even when the decay rates are known, solving for the amplitudes results in an ill-conditioned formulation. Transient signals in the presence of noise are further complicated as the signal-to-noise ratio asymptotically decreases in time. In this thesis the Discrete-Time Transient Transform and the Discrete Transient Transform are defined in order to represent a general signal using a linear combination of decaying exponential signals. A common approach to computing a change of basis is to make use of the dual basis. Two algorithms are proposed for generating a dual basis: the first algorithm is specific to a general exponential basis, e.g., real exponential or harmonically related complex exponential bases are special cases of the general exponential basis, while the second algorithm is usable for any general basis. Several properties of a transient domain representation are discussed. Algorithms for computing numerically stable approximate transient spectra are additionally proposed. The inherent infinite bandwidth of a continuous-time transient signal motivates in part the development of a framework for recovering the decay rates and amplitudes of a discrete-time lowpass filtered transient signal. This framework takes advantage of existing parameter modeling, identification, and recovery techniques to determine the decay rates while an alternating projection method utilizing the Discrete Transient Transform determines the amplitudes. Thesis Supervisor: Alan V. Oppenheim Title: Ford Professor of Engineering