The Fractional Fourier Transform and Its Application to Fault Signal Analysis

To a large extent mathematical transforms are applied on a signal to uncover information that is concealed, and the capability of such transforms is valuable for signal processing. One such transforms widely used in this area, is the conventional Fourier Transform (FT), which decomposes a stationary signal into different frequency components. However, a major drawback of the conventional transform is that it does not easily render itself to the analysis of non-stationary signals such as a frequency modulated (FM) or amplitude modulated (AM) signal. The different frequency components of complex signals cannot be easily distinguished and separated from one another using the conventional FT. So in this thesis an innovative mathematical transform, Fractional Fourier Transform (FRFT), has been considered, which is more suitable to process non-stationary signals such as FM signals and has the capability not only of distinguishing different frequency components of a multi-component signal but also separating them in a proper domain, different than the traditional time or frequency domain.

The discrete-time FRFT (DFRFT) developed along with its derivatives, such as Multi-angle-DFRFT (MA-DFRFT), Slanted Spectrum and Spectrogram Based on Slanted Spectrum (SBSS) are tools belonging to the same FRFT family, and they could provide an effective approach to identify unknown signals and distinguish the different frequency components contained therein. Both artificial stationary and FM signals have been researched using the DFRFT and some derivative tools from the same family. Moreover, to accomplish a contrast with the traditional tools such as FFT and STFT, performance comparisons are shown to support the DFRFT as an effective tool in multi-component chirp signal analysis. The DFRFT taken at the optimum transform order on a single-component FM signal has provided higher degree of signal energy concentration compared to FFT results; and the Slanted Spectrum taken along the slant line obtained from the MA-DFRFT demonstration has shown much better discrimination between different frequency components of a multi-component FM signal.

As a practical application of these tools, the motor current signal has been analyzed using the DFRFT and other tools from FRFT family to detect the presence of a motor bearing fault and obtain the fault signature frequency. The conclusion drawn about the applicability of DFRFT and other derivative tools on AM signals with very slowly varying FM phenomena was not encouraging. Tools from the FRFT family appear more effective on FM signals, whereas AM signals are more effectively analyzed using traditional methods like spectrogram or its derivatives. Such methods are able to identify the signature frequency of faults while using less computational time and memory.