Wave Packet Frames Research Articles

Wave packet frames in linear canonical domains: construction and perturbation

Wave packet frames in linear canonical domains: construction and perturbation

Duality for matrix-valued wave packet frames in L2(ℝd, ℂs×r)

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space [Formula: see text] from dual pairs of atomic wave packet frames in [Formula: see text] is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of [Formula: see text].

A note on discrete wave packet frames in ℂN

We show that for every nonzero vector [Formula: see text] in [Formula: see text], the discrete wave packet system [Formula: see text] constitutes a frame for the unitary space [Formula: see text]. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in [Formula: see text]. The canonical dual of discrete wave packet frame is also discussed.

On matrix-valued wave packet frames in $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$

In this paper, we study matrix-valued wave packet frames for the matrix-valued function space $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$ . An interplay between matrix-valued wave packet frames and its associated atomic wave packet frames is discussed. This is inspired by examples which show that frame properties cannot be carried from matrix-valued wave packet scaling functions to its associated atomic wave packet scaling functions and vice versa. Construction of matrix-valued wave packet frames for $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$ from corresponding atomic wave packet frames for $$L^2({\mathbb {R}}^d)$$ (and conversely) are given. Some special classes of matrix-valued scaling functions are given. A characterization of tight matrix-valued wave packet frames in terms of orthogonality of Bessel sequences has been obtained. Further, we provide a characterization of superframes which can generate matrix-valued frames. Finally, a Paley-Wiener type perturbation result with respect to matrix-valued wave packet scaling functions is given.

On Some Characterization of Generalized Representation Wave-Packet Frames Based on Some Dilation Group

In this paper we consider (extended) metaplectic representation of the semidirect product $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$ where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Finally, we rewrite the dual conditions, by using the Wigner distribution and obtain more reconstruction formulas.

K $\mathcal {K}$ -Matrix-valued Wave Packet Frames in L 2 ( ℝ d , ℂ s × r ) $L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$

We study frame properties of a matrix-valued wave packet system in the matrix-valued function space $L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$ , where the lower frame condition is controlled by a bounded linear operator $\mathcal {K}$ on $L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$ (lower $\mathcal {K}$ -frame condition, in short). There are many differences between ordinary frames and $\mathcal {K}$ -frames. The lower $\mathcal {K}$ -frame condition for matrix-valued wave packet Bessel sequences in $L^{2}(\mathbb {R}^{d},\mathbb {C}^{s\times r})$ in terms of operators; a trace functional associated with a bounded linear operator on $L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$ ; and a series associated with a matrix-valued Bessel sequence is presented. It is shown that matrix-valued wave packet frames are stable under small perturbation with respect to wave packet window functions.

Wave Packet Frame in Two Dimensions in General Lattices

Frame theory is a useful tool for many engineering applications such as in signal processing, image processing, data compression and sampling theory. In this paper, symmetric wave packet frames with several generators about general shift lattices in two dimensions are constructed from any frames given, which generalizes the existing results to the case of general shift lattices.

Fractional wave packet systems in L2(R)

Wave packet systems have been used as a potential tool for solving dynamic systems and have recently been studied using various Fourier-type transforms. In this article, we introduce wave packet systems of fractional order in L2(R) and investigate their orthogonal properties by means of two basic equations in the frequency domain. We also provide a sufficient condition for such a system to be orthonormal in L2(R). Finally, we establish the necessary and sufficient conditions for fractional wave packet frames in L2(R), which include the corresponding results of wave packet, wavelet analysis, and Gabor theory as the special cases.

On WH-packets of matrix-valued wave packet frames in L2(ℝd, ℂs×r)

A WH-packet is a system of vectors which is analogous to Aldroubi’s model for explicit expression of vectors (including frame vectors) in terms of a series associated with a given frame. In this paper, we study frame properties of WH-packet type system for matrix-valued wave packet frames in the function space [Formula: see text]. A necessary and sufficient condition for WH-packets of matrix-valued wave packet frames in terms of a bounded below operator is given. We present sufficient conditions for both lower and upper frame conditions on scalars associated with WH-packet of matrix-valued wave packet frames. Finally, a Paley–Wiener type perturbation theorem for WH-packet of matrix-valued wave packet frames is given. Several examples are given to illustrate the results.

Symmetric Wave Packet Frame in Higher Dimensions

Frame with redundancy properties has an important role in signal processing, image processing, and so on. In this paper, symmetric wave packet frames with several generators about origin in higher dimensions are constructed from any frames given, which generalizes the existing results to the case of higher dimensions. Also, this way increases tremendously the amount of wavelets.

Construction of -stage discrete periodic wave packet frames

In this paper, our main goal is to introduce the construction of the -stage periodic wave packet frames in a discrete setting. We first establish a general characterization of wave packet systems to be Parseval frames in using discrete Fourier transforms, and provide a sufficient condition for the system to be a first-stage discrete periodic wave packet frame for . And then, we construct a class of -stage discrete periodic wave packet frame by iterating the filter sequence, and establish the associated decomposition and reconstruction algorithms for these wave packet frames, which include the corresponding results of wavelet analysis and Gabor theory as the special cases. Finally, we give an illustrative example to demonstrate the validity of the proposed scheme.

Stability of multivariate wave packet frames for $$L^2(\mathbb {R}^n)$$ L 2 ( R n )

In this paper we study stability of multivariate wave packet frames. Necessary and sufficient conditions for a certain system to be multivariate wave packet frames are obtained.

Nonuniform Wave Packet Frames in L2(R)

Nonuniform Wave Packet Frames in L2(R)

Wave packet frames on local fields of positive characteristic

The objective of this paper is to construct wave packet frames on local fields of positive characteristic. A necessary and sufficient condition for the wave packet system DpjTu(n)aEu(m)bψj∈Z,m,n∈N0 to be a frame for L2(K) is given by means of the Fourier transform.

Necessary Conditions of the Wave Packet Frames with Several Generators

Necessary Conditions of the Wave Packet Frames with Several Generators

Necessary Conditions of the Wave Packet Frames with Several Generators

The main goal of this paper is to consider the necessary conditions of wave packet systems to be frames in higher dimensions. The necessary conditions of wave packet frames in higher dimensions with several generators are established, which include the corresponding results of wavelet analysis and Gabor theory as the special cases. The existing results are generalized to the case of several generators and general lattices.

Irregular Weyl–Heisenberg wave packet frames in [formula omitted

Let {fk} be a frame for a Hilbert H and let {αj,k} be a sequence of scalars. Aldroubi (1995) [1] considered a linear combination of the form gj=∑k=1∞αj,kfk (j∈N) and proved sufficient conditions on {αj,k} such that {gj} constitutes a frame for H. In this paper we discuss linear combination of the irregular Weyl–Heisenberg wave packet frames for L2(R). Necessary and sufficient conditions for a finite sum of irregular Weyl–Heisenberg wave packet frames to be a frame for L2(R) are given.

Symmetric Wave Packet Frames

Frames play an important role in field of the engineering and technology. In this paper, symmetric wave packet frames are constructed, which generalizes the existing result to the general case. This way makes the amount of wavelets largely increase.

Tight wave packet frames for [formula omitted] and [formula omitted

In this paper we establish a characterization of tight wave packet frames for L2(R) and we also prove that it is possible to construct frames in H2(R) which are given by dilation, translation and modulation of a single function ψ, where ψ, as well as ψˆ, belongs to the Schwartz class.

The geometry of sets of parameters of wave packet frames

We study wave packet systems WP ( ψ , M ) ; that is, countable collections of dilations, translations, and modulations of a single function ψ ∈ L 2 ( R ) . The parameters of these unitary actions form a discrete subset M ⊂ R + × R × R . We introduce analogues of the notion of Beurling density, adapted to the geometry of discrete subsets of R + × R × R , and notions of lower and upper dimensions associated with these densities. Our goal is to describe completeness properties of wave packet systems via geometric properties of the sets of their parameters. In particular, we show necessary conditions for WP ( ψ , M ) to be a Bessel system, and we construct multiple examples of non-standard wave packet frames with prescribed dimensions.