Time-Frequency Analysis for Non-Stationary Signals

In this dissertation, the traditional Gabor represantation with sinusoidal basis functions, which is widely used in the time-frequency analysis of non-stationary signals, is extended to the Fractional Gabor expansion with fractionally modulated basis functions. The completeness and biorthoganility conditions of the analysis and synthesis basis sets of the expansion are derived. Then, a discrete fractional Gabor expansion, that can be used to analyze discrete-time signals, is obtained by sampling the continuous-time represantation. By deriving the completeness and biorthoganility conditions, the discrete fractional Gabor expansion can be implemented on a computer to analyze discrete-time signals. Furthermore, to increase the time-frequency resolution of the signal represantation, time-scaled version of a mother window are used to obtain Multi-window fractional Gabor expansion. Finally an evolutionary spectral analysis approach is given for the time-varying spectral analysis of non-stationary signals. The connection between spectral representation and the fractional Gabor coefficient is established and the evolotionary kernel is estimated. Evolutionary spectrum is calculated as the magnitude square of this time-varying kernel. Thus, a time-frequency signal represantation and a time-varying spectrum is obtained simultaneously. Using the proposed multi-window fractional evolutionary spectral method, positive time-frequency very spectra with high resolution can be calculated.