Time-frequency analysis and design of signals: The Wigner distribution and its applications

Time frequency representations provide us with an efficient way to process non-stationary signals and systems with time-dependent spectrum. In this thesis, a tutorial review of several time-frequency representations is presented. The historical development and the properties are discussed with the emphasis being placed on the Wigner distribution (WD).An efficient way to calculate the Cramer-Rao lower bound for the time-varying frequency estimation is obtained. The theoretical performance of the WD when applied to the estimation of this time-varying frequency is derived and the simulation examples are provided.Using the properties of the instantaneous spectrum, new algorithms are developed which can accurately estimate the frequencies of the non-stationary signals with fast varying amplitude. A particular example of the applications is the frequency estimation in Search and Rescue Satellite (SARSAT). For tests on real SARSAT signals, our new algorithms exhibit superior performance to other spectrum estimation methods.The Cramer-Rao bound (CRB) for the joint time delay and Doppler shift estimation is derived and it is proved that the joint estimates of these two parameters using time-frequency representations are optimum under high SNR in the sense that it is unbiased and the variance reaches the CRB. Several efficient methods are developed to obtain the optimum signal for the joint time delay and Doppler shift estimation under different practical constraints.Three important applications of time-frequency representations are studied. They are the tracking of instantaneous frequency, high resolution estimation of frequencies in the non-stationary environment, and the design of signals for optimum simultaneous estimation of time delay and Doppler shift.