Compared to the classical theory of Fourier series and integral
transforms, the analysis of signals via the short-time Fourier (STFT) transform is a relatively young mathematical discipline. Known as time-frequency analysis, or Gabor analysis in it's discretized form, it starts from the so-called
sliding-window or spectrogram over phase space. Typically, this is the scalar product between the signal to be analyzed and a coherent state, i.e. a phase-space-shifted Gaussian or general Gabor window. This is one of the key aspects of the basis for the MP3 compression algorithm for digital music.
A good mathematical description of the questions arising in this context requires to use a Banach Gelfand Triple, using the space S 0 of functions which have integrable STFT as the space of test functions, L 2 as the Hilbert space and the dual space S 0 * as space of mild distributions.
The abstract concept of weak*-convergence corresponds to uniform
approximation of the signal over a finite rectangle in phase space. A good CD thus provides an excellent reproduction of a piece of music, within its full duration and in the full audible range from zero to 20kHz.
Participante: Professor Hans Georg Feichtinger
Institución: Universidad de Viena
Fecha y hora: Este evento terminó el Lunes, 30 de Septiembre de 2019