On Generating Tight Gabor Frames at Critical Density

We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori, (b) the construction of tight Gabor frames. The present investigation relies an interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel's condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in noncommutative tori. In particular, the projection associated with a Gabor frame generated by a Gaussian turns out to be Boca's projection. Our approach to Boca's projection allows us to characterize the range of existence of Boca's projection. The presentation of our main result provides a natural approach to the Wexler-Raz biorthogonality relations in terms of Hilbert C*-modules over noncommutative tori.

The subject of this study is paraunitary modulated filter banks. A factorization of the polyphase matrices of these filter banks, which is described, gives complete characterization of tight Gabor frames in l/sup 2/(Z), with arbitrary rational oversampling ratios. Tight Gabor frames, being less constrained than orthogonal bases, allow for filter bank designs with good localization in both time and frequency.

This paper presents a coarse quantization algorithm (TFΣΔ-II) for tight Gabor frame expansions of certain functions in L2(ℝ), an alternative to the TFΣΔ of Ref. 11. By using some a priori information about the function to be quantized and compromising the translation invariance of the TFΣΔ, TFΣΔ-II produces an approximation in L2(ℝ), as opposed to the weak type approximations of TFΣΔ. In particular, for a tight Gabor frame with frame bound A, we prove that the L2-approximation error corresponding to a kth-order TFΣΔ-II quantizer is of order O(A-k). Furthermore, motivated by TFΣΔ-II, we construct an algorithm to coarsely quantize the Fourier coefficients of certain compactly supported functions.

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.

Gabor systems play an important role in various practical problems related to signal and image processing.In this paper, we study the computability of Gabor frames in L 2 .Computable version of some existing results related to Gabor systems are obtained.Also, computability of operators associated with Gabor frames is discussed and the computability of Zak transform is defined.Finally, we discuss the existence of the computable dual frames associated with the Gabor Systems.

We consider the difficult problem of deciding whether a triple(g,a,b),with window g ∈ L 2(ℝ) and time shift parameter a and frequency shift parameterbis a Gabor frame from two different points of view. We first identify two classes of non-negative windows g ∈ L 2(ℝ) such that their Zak transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, non-negative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, non-negative, continuous, integrable windowsg that satisfy on [0, ∞) a condition slightly stronger than strict convexity (superconvexity). Accordingly, the members of these two classes generate Gabor frames for integer oversampling factor (ab)-1 ≥ 1 and ≥ 2, respectively. When we weaken the condition of superconvexity into strict convexity, the Zak transformsZg may have as many zeros as one wants, but in all cases (g, a, b) is still a Gabor frame when(ab) -1 is an integer ≥ 2. As a second issue we consider the question for which a, b > 0 the triple (g, a, b) is a Gabor frame, where gis the characteristic function of an interval [0, ∞) with c0 > 0 fixed. It turns out that the answer to the latter question is quite complicated, where irrationality or rationality of abgives rise to quite different situations. A pictorial display, in which the various cases are indicated in the positive (a, b)-quadrant, shows a remarkable resemblance to the design of a low-budget tie.

Invertibility of the Gabor frame operator on the Wiener amalgam space

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

In this paper we identified a collection of unitary operators which maps Gabor frames in \(L^2(\mathbb{R})\) to Gabor frames in \(l^2(\mathbb{Z})\). This is very important in construction of Gabor frames in \(l^2(\mathbb{Z})\) from Gabor frames in \(L^2(\mathbb{R})\) other than which obtained from Gabor frames in \(L^2(\mathbb{R})\) through sampling.

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

In this paper, we propose a new type of a neural network which is inspired by Gabor systems from harmonic analysis. In this regard, we construct a class of sparsely connected neural networks utilizing the concept of time–frequency shifts, and we show that their approximation error rates can be tied to the number of modulations in the corresponding Gabor frame and to the smoothness of the input function. Furthermore, we show that such networks are easily implementable and we illustrate their performance with some numerical examples.

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Construction of unimodular tight frames for compressed sensing using majorization-minimization

A tight frame in Rn is a redundant system which has a reconstruction formula similar to that of an orthonormal basis. For a unit-norm frame F = {f i }k i=1 , a scaling is a vector c = (c(l),…, c(k)) e Rk≥0 such that {c(i)f i }k i=1 is a tight frame in Rn. If a scaling c exists, we say that F is a scalable frame. A scaling c is a minimal scaling if {fi : c{i) > 0} has no proper scalable subframes. In this paper, we present the uniqueness of the orthogonal partitioning property of any set of minimal scalings and provide a construction of scalable frames by extending the standard orthonormal basis of Rn.