GABOR ANALYSIS AND TIME-FREQUENCY METHODS

harmonic analysis explains how to describe the (global) Fourier Tranasform (FT) of signals even over general LCA (locally compact Abelian) groups, but typically requires square integrability or periodicity. For the analysis of time-variant signals an alternative is needed, the so-called sliding window FT or the STFT, the shorttime Fourier transform, defined over phase space, the Cartesian the product of the time domain with the frequency domain. Starting from a signal f it is obtained by first localizing f in time using a (typically bump-like) window function g followed by a Fourier analysis of the localized part [1]. Another important application of timefrequency analysis is in wireless communication where it helps to design reliable mobile communication systems. This article presents the key ideas of Gabor Analysis as a subfield of time-frequency analysis, as inaugurated by Denis Gabor’s work [21]. There are two equivalent views: either focus on redundancy reduction of the STFT by sampling it along some lattice (Gabor himself suggested to use the integer lattice in phase space) requiring stable linear reconstruction, or to emphasize the representation of f as superposition of timefrequency shifted atoms as building blocks. Thinking in terms of real-time signal processing engineers also use the concept of filter banks to describe the situation. Each frequency channel contains all the Gabor coefficients corresponding to a fixed frequency [4]. From a mathematical perspective Gabor analysis can be considered as a modern branch of harmonic analysis over the Heisenberg group. The most useful description of the Heisenberg group for time-frequency analysis is the one, where R × R × R is endowed with the group law (x, ω, s)⊗ (y, η, t) = (x+ y, ω+ η, s+ t+ y ·ω−x · η). The representation theory of the Heisenberg group constitutes the mathematical framework of time-frequency analysis [22]. Gabor analysis is a branch of time-frequency analysis, which has turned out to have applications in audio-mining, music, wireless communication, pseudodifferential operators, function spaces, Schrödinger equations, non-commutative geometry, approximation theory, or the Kadison-Singer conjecture [33, 7, 8, 23, 10, 11, 32, 34, 35, 5]. Recent progress in wireless communication relies on modeling the transmission channels as pseudodifferential operators, whose symbol belongs to a Sjöstrand’s class [43, 29]. 1991 Mathematics Subject Classification. Primary 42C15, 42B35.