A.J.E.M. Janssen, Peter L. Søndergaard
Iterative Algorithms to Approximate Canonical Gabor Windows: Computational Aspects
Preprint series: ESI preprints

MSC:

42C15 Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions

41A25 Rate of convergence, degree of approximation

47A58 Operator approximation theory

94A12 Signal theory (characterization, reconstruction, etc.)

Abstract: In this paper we investigate the computational aspects of some recently
proposed iterative methods for approximating the canonical tight and
canonical dual window of a Gabor frame $\left(g,a,b\right)$. The
iterations start with the window $g$ while the iteration steps comprise
the window $g$, the $k^{th}$ iterand $\gamma_{k}$, the frame operators
$S$ and $S_{k}$ corresponding to $\left(g,a,b\right)$ and $\left(\gamma_{k},a,b\right)$,
respectively, and a number of scalars. The structure of the iteration
step of the method is determined by the envisaged convergence order
$m$ of the method. We consider two strategies for scaling the terms
in the iteration step: norm scaling, where in each step the windows
are normalized, and initial scaling where we only scale in the very
beginning. Norm scaling leads to fast, but conditionally convergent
methods, while initial scaling leads to unconditionally convergent
methods, but with possibly suboptimal convergence constants. The iterations,
initially formulated for time-continuous Gabor systems, are considered
and tested in a discrete setting in which one passes to the appropriately
sampled-and-periodized windows and frame operators. Furthermore, they
are compared with respect to accuracy and efficiency with other methods
to approximate canonical windows associated with Gabor frames.

Keywords: Gabor frame, tight window, dual window, iterative method, scaling, adjoint orbit, Zak transform