Optimization in the construction of cardinal and symmetric wavelets on the line

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

High-concentration time–frequency analysis for multi-component nonstationary signals based on combined multi-window Gabor transform

PurposeThis study aims to reveal the essential characteristics of nonstationary signals and explore the high-concentration representation in the joint time–frequency (TF) plane.Design/methodology/approachIn this paper, the authors consider the effective TF analysis for nonstationary signals consisting of multiple components.FindingsTo make it, the authors propose the combined multi-window Gabor transform (CMGT) under the scheme of multi-window Gabor transform by introducing the combination operator. The authors establish the completeness utilizing the discrete piecewise Zak transform and provide the perfect-reconstruction conditions with respect to combined TF coefficients. The high-concentration is achieved by optimization. The authors establish the optimization function with considerations of TF concentration and computational complexity. Based on Bergman formulation, the iteration process is further analyzed to obtain the optimal solution.Originality/valueWith numerical experiments, it is verified that the proposed CMGT performs better in TF analysis for multi-component nonstationary signals.

On Discrete Time Wilson Systems

In this paper, we define the discrete time Wilson frame (DTW frame) for l 2 ℤ and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.

The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

We characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

A quantitative Balian–Low theorem for higher dimensions

We extend the quantitative Balian–Low theorem of Nitzan and Olsen to higher dimensions. We use Zak transform methods and dimension reduction. The characterization of the Gabor–Riesz bases by the Zak transform allows us to reduce the problem to the quasiperiodicity and the boundedness from below of the Zak transforms of the Gabor–Riesz basis generators, two properties for which dimension reduction is possible.

Some Properties of Fundamental Linear Canonical Zak Transform

In this paper, we introduce some fundamentally properties of Zak linear canonical transform (LCZT) such as linearity and translation properties. LCZT is developing of Zak transform ( ZT) and linear canonical transform (LCT). Keywords: linear canonical Zak transform; Zak transform; linear canonical transform; linearity property; translation property. AbstrakDalam jurnal ini akan diungkapkan beberapa sifat fundamental dari transformasi Zak linear kanonik (LCZT), yaitu sifat linear dan sifat translasi. LCZT merupakan hasil pengembangan dari dua buah transformasi yaitu transformasi Zak (ZT) dan transformasi linear kanonik (LCT). Kata kunci:Transformasi Zak linear kanonik; transformasi Zak; transformasi linear kanonik; sifat linear; sifattranslasi.

A class of vector-valued subspace weak Gabor duals of type II

The theory of vector-valued frames (also called superframes) has important applications in signal multiplexing, and has interested many mathematicians in recent years. In this paper, we introduce the notion of weak Gabor duals of type II under the setting of subspaces [Formula: see text], where [Formula: see text] is a periodic subset of [Formula: see text]. Using the Zak transform matrix method, we characterize weak Gabor duals of type II and their uniqueness. An example is also presented to illustrate the generality of our results.