Adapted and Adaptive Linear Time-Frequency Representations: A Synthesis Point of View

A large variety of techniques exist to display the time and frequency content of a given signal. In this article, we give an overview of linear time-frequency representations, focusing mainly on two fundamental aspects. The first is the introduction of flexibility, more precisely, the construction of time-frequency waveform systems that can be adapted to specific signals or specific signal processing problems. To do this, we base the constructions on frame theory, which allows many options while still ensuring perfect reconstruction. The second aspect is the choice of the synthesis framework rather than the usual analysis framework. Instead of considering the correlation, i.e. the inner product, of the signal with the chosen waveforms, we find appropriate coefficients in a linear combination of those waveforms to synthesize the given signal. We show how this point of view allows the easy introduction of prior information into the representation. We give an overview of methods for transform domain modeling, in particular those based on sparsity and structured sparsity. Finally, we present an illustrative application for these concepts: a denoising scheme.