No abstract

The Gabor expansion, which maps the time domain signal into the joint time and frequency domain, has long been recognized as a very useful tool in signal processing. Its applications, however, were limited due to the difficulties associated with selecting the Gabor coefficients. Because timeshifted and frequency-modulated elementary functions in general do not constitute an orthogonal basis, the selections of the Gabor coefficient are not unique. One solution to this problem, developed by Bastiaans, is to introduce an auxiliary biorthogonal function. Then, the Gabor coefficient is computed by the usual inner product rule. Unfortunately, it is not easy to determine the auxiliary biorthogonal function for an arbitrary given synthesis function and sampling pattern. While less success was found in the continuous case, we present a discrete solution in this paper, which is named the discrete Gabor transform (DGT). For a given synthesis window and sampling pattern, computing the auxiliary biorthogonal function of the DGT is nothing more than solving a linear system. The DGT presented applies for both finite as well as infinite sequences. Using the advantages of the nonuniqueness of the auxiliary biorthogonal function at oversampling, we further introduce the so-called orthogonal-like DGT. As the DFT (a discrete realization of the continuous-time Fourier transform), the DGT introduced provides a feasible vehicle to implement the useful Gabor expansion.

No abstract

SUMMARYTo verify the importance of the non-stationary frequency characteristic of seismic ground motion, a joint time-frequency analysis technique of time signals, called chirplet-based signal approximation, is developed to extract the non-stationary frequency information from the recorded data. The chirplet-based signal approximation is clear in concept, similar to Fourier Transform in mathematical expressions but with di erent base functions. Case studies show that the chirplet-based signal approximation can represent the joint time-frequency variation of seismic ground motion quite well. Both the random models of uniform modulating process and evolutionary process are employed to generate artiÿcial seismic waves. The joint time-frequency modulating function in the random model of evolutionary process is determined by chirplet-based signal approximation. Finally, non-linear response analysis of a SODF system and a frame structure is performed based on the generated artiÿcial seismic waves. The results show that the non-stationary frequency characteristic of seismic ground motion can signiÿcantly change the non-linear response characteristics of structures, particularly when a structure goes into collapse phase under seismic action. It is concluded that non-stationary frequency characteristic of seismic ground motion should be considered for the assessment of seismic capacity of structures.

No abstract