Time-Frequency Distributions: Approaches for Incomplete Non-Stationary Signals

There are many sources of waveforms or signals existing around us. They can be natural phenomena such as sound, light and invisible like elec- tromagnetic fields, voltage, etc. Getting an insight into these waveforms helps explain the mysteries surrounding our world and the signal spec- tral analysis (i.e. the Fourier transform) is one of the most significant approaches to analyze a signal. Nevertheless, Fourier analysis cannot provide a time-dependent spectrum description for spectrum-varying signals-non-stationary signal. In these cases, time-frequency distribu- tions are employed instead of the traditional Fourier transform. There have been a variety of methods proposed to obtain the time-frequency representations (TFRs) such as the spectrogram or the Wigner-Ville dis- tribution. The time-frequency distributions (TFDs), indeed, offer us a better signal interpretation in a two-dimensional time-frequency plane, which the Fourier transform fails to give. Nevertheless, in the case of incomplete data, the time-frequency displays are obscured by artifacts, and become highly noisy. Therefore, signal time-frequency features are hardly extracted, and cannot be used for further data processing. In this thesis, we propose two methods to deal with compressed observations. The first one applies compressive sensing with a novel chirp dictionary. This method assumes any windowed signal can be approximated by a sum of chirps, and then performs sparse reconstruction from windowed data in the time domain. A few improvements in computational com- plexity are also included. In the second method, fixed kernel as well as adaptive optimal kernels are used. This work is also based on the as- sumption that any windowed signal can be approximately represented by a sum of chirps. Since any chirp ’s auto-terms only occupy a certain area in the ambiguity domain, the kernel can be designed in a way to remove the other regions where auto-terms do not reside. In this manner, not only cross-terms but also missing samples’ artifact are mitigated signifi- cantly. The two proposed approaches bring about a better performance in the time-frequency signature estimations of the signals, which are sim- ulated with both synthetic and real signals. Notice that in this thesis, we only consider the non-stationary signals with frequency changing slowly with time. It is because the signals with rapidly varying frequency are not sparse in time-frequency domain and then the compressive sensing techniques or sparse reconstructions could not be applied. Also, the data with random missing samples are obtained by randomly choosing the samples’ positions and replacing these samples with zeros.