Quantum theta functions and Gabor frames for modulation spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change

Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition

In this paper, the adaptive chirplet decomposition combined with the Wigner-Ville transform and the empirical mode decomposition combined with the Hilbert transform are employed to process various non-stationary signals (strong ground motions and structural responses). The efficacy of these two adaptive techniques for capturing the temporal evolution of the frequency content of specific seismic signals is assessed. In this respect, two near-field and two far-field seismic accelerograms are analyzed. Further, a similar analysis is performed for records pertaining to the response of a 20-story steel frame benchmark building excited by one of the four accelerograms scaled by appropriate factors to simulate undamaged and severely damaged conditions for the structure. It is shown that the derived joint time–frequency representations of the response time histories capture quite effectively the influence of non-linearity on the variation of the effective natural frequencies of a structural system during the evolution of a seismic event; in this context, tracing the mean instantaneous frequency of records of critical structural responses is adopted. The study suggests, overall, that the aforementioned techniques are quite viable tools for detecting and monitoring damage to constructed facilities exposed to seismic excitations

A Comparative Study of Time-Frequency Representations for Fault Detection in Wind Turbine

To reduce the cost of wind energy, minimization and prediction of maintenance operations in wind turbine is of key importance. In variable speed turbine generator, advanced signal processing tools are required to detect and diagnose the generator faults from the stator current. To detect a fault in non-stationary conditions, previous studies have investigated the use of time-frequency techniques such as the Spectrogram, the Wavelet transform, the Wigner-Ville representation and the Hilbert-Huang transform. In this paper, these techniques are presented and compared for broken-rotor bar detection in squirrel-cage generators. The comparison is based on several criteria such as the computational complexity, the readability of the representation and the easiness of interpretatio

Time-frequency methods for coherent spectroscopy

Time-frequency decomposition techniques, borrowed from the signal-processing field, have been adapted and applied to the analysis of 2D oscillating signals. While the Fourier-analysis techniques available so far are able to interpret the information content of the oscillating signal only in terms of its frequency components, the time-frequency transforms (TFT) proposed in this work can instead provide simultaneously frequency and time resolution, unveiling the dynamics of the relevant beating components, and supplying a valuable help in their interpretation. In order to fully exploit the potentiality of this method, several TFTs have been tested in the analysis of sample 2D data. Possible artifacts and sources of misinterpretation have been identified and discussed

The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde