Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design, lead to algorithms for reconstructing missing information via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of graph signals in the vertex and graph spectral domains, our approach generalizes the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform. One challenge we highlight is that the local structure in a small region of an inhomogeneous graph can drastically affect the uncertainty bounds, limiting the information provided by global uncertainty principles. Accordingly, we suggest new notions of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the atom's center vertex. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

This paper discusses the characteristics of time-frequency estimators to be used in theidentification of systems with localized non-linearities. The common idea underlying thisresearch is that, for certain classes of structural response signals, the availability ofa limited number of experimental data can be partially obviated by taking into account the“localisation” in time of the frequency components of the signals. Time-frequencytechniques for structural identification are reported that extend the definition ofinstantaneous time-frequency estimators and Gabor instantaneous estimators were extractedfrom non-stationary vibration signals. In order to foresee their validity on the basis ofmeasured data, methods were applied to seismic responses obtained from numerical testsconducted on steel frames. The results obtained made it possible to evaluate thecharacteristics of time-frequency identification techniques as well as their efficiencywhen applied to non-linear structures.

We examined the use and potential of quantifying instantaneous heart rate variability (HRV) using a joint time-frequency and time-domain methods. These new techniques are promising, because they provide tools to quantify nonstationary, beat-by-beat changes in HRV components, and are therefore flexible with respect to the design of experimental protocols. A smoothed pseudo-Wigner–Ville distribution (SPWVD) and a time-domain index using polynomial filtering produced fairly coherent estimates of band-specific HRV amplitudes, whereas SPWVD yielded additional information on the frequency characteristics of HRV. Instantaneous HRV appeared to have a complex and a frequency-specific relationship to cardiac activity and electrodermal activity. It is concluded that the time-frequency analysis of HRV is a very promising method for mapping transient changes in the frequency and amplitude characteristics of cardiac dynamics.