Observer-based Implementation of Discrete Gabor Transform

Abstract

The most commonly used technique in time-frequency analysis is the short-time Fourier transform. It can be used to determine the spectral components as they change over time by computing the Fourier transform of a windowed segment of the signal. A fundamental constraint of this method is that the frequency resolution of the representation in the time-frequency domain will be linear by design. The frequency adaptive nonstationary discrete Gabor transform offers an alternative that does not have this limitation. A Luenberger observer is capable of the implementation of the short-time Fourier transform, and its numerical advantages are already established based on the work of Hostetter and Péceli. Here, we introduce the family of discrete Gabor transforms and its properties along with a constructive method to define such transforms. Furthermore, we show that they are realizable by Luenberger observers which are capable of the error-free reconstruction of the observed signal in finite steps. The dead-beat property is derived for the state variables as well which estimate the transform of the windowed signal.