Representation of the inverse of a frame multiplier

This page provides resources for the visualizations and the algorithms in the research manuscript:

"Representation of the inverse of a frame multiplier"

Peter Balazs and Diana T. Stoeva

Abstract

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. They are not only interesting mathematical objects, but also important for applications, for example for the realization of time-varying filters. In this paper we show a surprising result about the inverse of such operators, if existing, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with dual frames and reciprocal symbol. Furthermore, one of those dual frames is uniquely defined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in l². We investigate invertible Gabor multipliers; we show that the inverse of every invertible lattice-invariant operator (in particular, every invertible Gabor frame multiplier with a constant symbol (1)) can be represented as a Gabor frame multiplier with a constant symbol (1). Finally we give a numerical example for the invertibility of multipliers in the Gabor case.

Figures:

Figure 1 in the manuscript (an illustrative example to visualize actions of a multiplier on a musical signal)

In Figure 1 we show a visualization of a multiplier M_m,Φ,Ψ in the time-frequency plane. We consider a music signal f and the action of a multiplier M_m,Φ,Ψ on f. For f we use a 2 seconds long excerpt of the "Jump" from Van Halen (click here to listen the signal). For a time-frequency representation of the musical signal f (TOP LEFT) we use a 'painless' Gabor frame Ψ (a 80 ms Hanning window with 12,5% overlap). By manual estimation, we determine the symbol m that should describe the time-frequency region of the singer's voice. This region is then multiplied by 0.01, the rest by 1 (TOP RIGHT) (see the symbol here). Finally, we show a time-frequency representation of the modified signal M_m,Φ,Ψf (BOTTOM). To listen the modified signal, click here.

Clearly, the naive approach has strong artifacts. The error is especially big at the boundaries of the constant region of the symbols. The chosen atoms are well localized in time-frequency, so that within the interior of the constant regions, this inversion works well. This could be expected as we have shown in the manuscript that constant symbols allow this kind of inversion for equivalent frames.

The iterative inversion worked well with an error of 3%. This could, naturally, be decreased by investing more calculation time. But also in the chosen setting for the iterative inversion (100 iterations in iframemul [2]) no difference can be seen in the time-frequency representation, as well as no audible difference can be detected.