Gabor Frames in Finite Dimensions
Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequ
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Gabor Frames in Finite Dimensions Götz E. Pfander
Abstract Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setting, we give elementary proofs of the central results on Gabor frames in the greatest possible generality; that is, we consider Gabor frames corresponding to lattices in arbitrary finite Abelian groups. In the second half of this chapter, we review recent results on the geometry of Gabor systems in finite dimensions: the linear independence of subsets of its members, their mutual coherence, and the restricted isometry property for such systems. We apply these results to the recovery of sparse signals, and discuss open questions on the geometry of finite-dimensional Gabor systems. Keywords Gabor analysis on finite Abelian groups · Linear independence · Coherence · Restricted isometry constants of Gabor frames · Applications to compressed sensing · Erasure channel error correction · Channel identification
6.1 Introduction In his seminal 1946 paper “Theory of Communication,” Dennis Gabor suggested the decomposition of the time-frequency information area of a communications channel into the smallest possible boxes that allow exactly one information-carrying coefficient to be transmitted per box [41]. He refers to Heisenberg’s uncertainty principle to argue that the smallest time-frequency boxes are achieved using time-frequency shifted copies of probability functions, that is, of Gaussians. In summary, he proposes transmitting the information-carrying complex-valued sequence {cnk } in the
G.E. Pfander () School of Engineering and Science, Jacobs University, 28759 Bremen, Germany e-mail: [email protected] P.G. Casazza, G. Kutyniok (eds.), Finite Frames, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_6, © Springer Science+Business Media New York 2013
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G.E. Pfander
form of the signal ∞
ψ(t) =
∞
cnk e
−π (t−nΔt)2 2(Δt)
2
kt
e2πi Δt ,
n=−∞ k=−∞
where the parameter Δt > 0 can be chosen depending on physical consideration and the application at hand. Denoting modulation operators by Mν g(t) = e2πiνt g(t),
ν ∈ R,
and translation operators by Tτ g(t) = g(t − τ ),
τ ∈ R,
Gabor proposed to transmit on the carriers {Mk/Δt TnΔt g0 }n,k∈Z , where g0 is the −π
t2
Gaussian window function g0 (t) = e 2(Δt)2 . In the second half of the twentieth century, the suggestion of Gabor, and in general the interplay of information density in time and in frequency, was studied extensively; see, for example, [24, 25, 33, 38, 61–63, 88]. This line of work focuses on functional analytic properties of function systems such as the ones suggested by Gabor. Apart from the following historical remarks, functional analysis will not play a
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