Minimum-energy Wavelet Frames Research Articles

A Short Note on Wavelet Frames Based on FMRA on Local Fields

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

The Application of Eigenvectors for the Construction of Minimum-Energy Wavelet Frames Based on FMRA

In 1974, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing. In 1986, Y. Meryer created a real small wave base, and the wavelet analysis began to flourish after a multi scale analysis of the same method of constructing the small wave base with S. Mallat. In order to analyze and deal with non-stationary signals, a series of new signal analysis theories are proposed.: Short Time Fourier Transform, time-frequency analysis, wavelet transform, and fractional Fourier transform and so on. In this paper, an explicit algorithm is given to construct the minimum-energy frames based on frame multiresolution analysis via characteristic vectors of the mask matrix. In section 2, we show the structure of minimum-energy wavelet frames in terms of their masks (Lemma 1) and discuss that we should eliminate the correlation of the rows of the mask matrix by the polyphase decomposition technique. Based on FMRA, an explicit algorithm is given to construct this frames. By this method, all the minimum-energy wavelet frames can be obtained. As an application, several examples are showed to explain this method in section 3. This method can also be applied in other fields of wavelet analysis.

Minimum-Energy Wavelet Frames on Local Fields

To overcome the computational complexity, numerical instability and the complication of change of bases problem, we introduce the concept of minimum-energy wavelet frame on local fields of positive characteristic and present its equivalent characterizations in terms of their wavelet masks. Besides, in view of polyphase representation of the wavelet masks, we give a necessary and sufficient condition for the existence of minimum-energy wavelet frames on local fields. At last, we present the decomposition and reconstruction formulae of minimum-energy wavelet frame for \(L^2(K)\).

Minimum-energy wavelet frames generated by the Walsh polynomials

Drawing inspiration from the construction of tight wavelet frames generated by the Walsh polynomials, we introduce the notion of minimum-energy wavelet frames generated by the Walsh polynomials on positive half-line R+ using unitary extension principles and present its equivalent characterizations in terms of their framelet symbols. Moreover, based on polyphase components of the Walsh polynomials, we obtain a necessary and sufficient condition for the existence of minimum-energy wavelet frames in L2(R+). Finally, we derive the minimum-energy wavelet frame decomposition and reconstruction formulae which are quite similar to those of orthonormal wavelets on local fields of positive characteristic.

The Bivariate Minimum-Energy Tight Wavelet Frames and Applications in Economics and Management

Material science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. Frames have become the focus of active research field, both in the-ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi-resolution resolution are introduced. A precise existence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient condi tion for the existence of tight wavelet frames is obtained by virtue of a generalized multiresolution analysis.

<italic>a</italic>进制最小能量区间小波框架的构造

Based on wavelet multi-resolution features and signal multi-channel theory, results on minimum-energy wavelet frames are presented for intervals with arbitrary integer dilation factors not less than 2. First, we define the concept of minimum-energy wavelet frames on such intervals, give the left and right boundary scaling functions and the left and right boundary wavelet frame functions, and present the necessary and sufficient conditions for the minimum-energy wavelet frame on the interval. Second, the design of the construction algorithm of minimum-energy wavelet frame on the interval with arbitrary integer support is discussed, and the structure of the matrices used in the algorithm is presented constructively. Third, the decomposition and reconstruction formulas of the minimum-energy wavelet frame on the interval are deduced. These are similar to those for orthogonal wavelets on the interval. Finally, based on B-spline functions, several numerical examples are given.

Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

The Research of Bivariate Minimum-Energy Frames and Frames of Subspace and Application in Particle Physics

Materials science is an applied science concerned with the relationship between the struc- ture and properties of materials. Frames have become the focus of active research field, both in the- ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi- resolution are introduced. A precise exist-ence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient con-dition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

The Research of Bivariate Minimum-Energy Frames and Frames of Subspace and Application in Particle Physics

Materials science is an applied science concerned with the relationship between the struc- ture and properties of materials. Frames have become the focus of active research field, both in the- ory and in applications. In the article, the binary minimum-energy wavelet frames and frame multi- resolution are introduced. A precise exist-ence criterion for minimum-energy frames in terms of an ineqity condition on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient con-dition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

A Study of Binary Minimum-Energy Shortly Supported Wavelet Frames Associated with a Scaling Function

Frames have become the focus of active research field, both in theory and in applications. In the article, the binary minimum-energy wavelet frames and frame multiresolution resolution are introduced. A precise existence criterion for minimum-energy frames in terms of an ineqity conditi- -on on the Laurent poly-nomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also established. The sufficient condition for the existence of affine pseudoframes is obtained by virtue of a generalized multiresolution analysis. The pyramid de -composition scheme is established based on such a generalized multiresolution structure.

The Research of Bivariate Minimum-Energy Wavelet Frames and Pseudoframes

Frames have become the focus of active research, both in theory and in applications. In the article, the notion of bivariate minimum-energy wavelet frames is introduced. A precise existence criterion for minimum-energy frames in terms of an inequality condition on the Laurent polynomial symbols of the filter functions is provided. An explicit formula for designing minimum-energy frames is also establish- ed. The sufficient condition for the existence of a class of affine pseudoframes with filter banks is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresol- -ution structure.

Minimum-energy frame of discrete signal space and its de-noising application to rectangular pulse signal

The minimum-energy (wavelet) frame has an extensive application prospect in the field of signal processing, but now it can be applied only to continuous signals. In order to solve the problem, we define a minimum-energy frame in the discrete signal space, and then prove that it has some good properties. In the actual application, we propose a new de-noising algorithm which is special for the binary rectangular pulse signal polluted by the additive gaussian noise and obtain better processing effect by using a minimum-energy frame in the discrete signal space to denoise the sampled sequence of receiving waveform. The simulation results show that if a pre-processing link is used to denoise the receiving waveform through using the algorithm, the receiver can reduce the bit error rate and achieve a 3.4 dB performance gain at 4 dB signal-to-noise ratio.

Minimum-energy wavelet frame on the interval

The construction and properties of interval minimum-energy wavelet frame are systematically studied in this paper. They are as follows: 1) give the definition of interval minimum-energy wavelet frame; 2) give the necessary and sufficient conditions for the minimum-energy frames for L2[0,1]; 3) present the construction algorithm for minimum-energy wavelet frame associated with refinable functions on the interval with any support γ; 4) give the decomposition and reconstruction formulas of the minimum-energy frame on the interval [0,1].