Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Abstract

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.

On the Diversity of Uncoded OTFS Modulation in Doubly-Dispersive Channels

“…Sampling the spectrum in the frequency domain periodizes the signal in the time domain. Analogously, the discrete symplectic Fourier transform (DSFT) of a sequence is continuous and periodic [20]. Sampling the DSFT of a sequence in the delay-Doppler domain periodizes the signal in the time-frequency domain.…”

The Linear Time Frequency Analysis Toolbox

“…The function gabmulappr provides optimal Gabor multiplier approximations for any linear operator (represented by its matrix), using the algorithm presented in Ref. 20. c The use of this function is exemplified in demo gabmulappr.…”

“…Gabor multipliers can only approximate operators well with localised spreading function, 20 i.e. containing only small time and frequency shifts.…”

A Class of Algorithms for Time-Frequency Multiplier Estimation

“…A more theoretical approach in the case of Gabor Multiplier can be found in [22] in the context of Gabor analysis. It can be shown [23] [24] that underspread operators (i.e. operators that don't involve large time-frequency shifts) can be well approximated by Gabor multipliers, provided the window is suitably chosen.…”

“…When Gabor frames are used to represent signals, the corresponding multipliers (called Gabor Multipliers) have been studied by several authors (see [22], [23] and references therein). A Gabor Multiplier (GM for short) M σ;g,h : x → M σ x is defined by…”