scholarly journals On Discrete Time Wilson Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Teena Kohli ◽  
Suman Panwar ◽  
S. K. Kaushik
Keyword(s):  
Discrete Time ◽  
Dual Pair ◽  
Gabor Frames ◽  
Sufficient Condition ◽  
Zak Transform ◽  
Frame Operator

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.

scholarly journals Stabilization of a Class of Switched Positive Nonlinear Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Xing Xing ◽  
Zhichun Jia ◽  
Yunfei Yin ◽  
Tingting Wu
Keyword(s):  
Nonlinear Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
Dwell Time ◽  
Average Dwell Time ◽  
Sufficient Condition ◽  
Cooperative System ◽  
Numerical Example ◽  
New Class

The problem of switching stabilization for a class of switched positive nonlinear systems (switched positive homogeneous cooperative system (SPHCS) in the continuous-time context and switched positive homogeneous order-preserving system (SPHOS) in the discrete-time context) is studied by using average dwell time (ADT) approach, where the positive subsystems are possibly all unstable. To tackle this problem, a new class of ADT switching is first defined, which is different from the previous defined ADT switching in the literature. Then, the proposed ADT is designed via analyzing the weightedl∞norm of the considered system’s state. A sufficient condition of stabilization for SPHCSs with unstable positive subsystems is derived in continuous-time context. Furthermore, a sufficient condition for SPHOSs under the assumption that all modes are possibly unstable is also obtained. Finally, a numerical example is given to demonstrate the advantages and effectiveness of our developed results.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

Fourier Analysis in Systems Theory

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Analysis ◽  
Systems Theory ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Homogeneous Systems ◽  
Single Output ◽  
System Operator ◽  
Single Input ◽  
Necessary And Sufficient

In the most general sense, any process wherein a stimulus generates a corresponding response can be dubbed a system. For a temporal system with single input, f (t), and single output, g(t), the relation can be written as . . . g(t) = S{ f (t)} (3.1) . . . where S{·} is the system operator. This is illustrated in Figure 3.1. There exist numerous system types. We define them here in terms of continuous signals. The equivalents in discrete time are given as an exercise. For homogeneous systems, amplifying or attenuating the input likewise amplifying or attenuating the output. For any constant, a,. . . S{a f(t)} = aS{ f (t)} (3.2) If the response of the sum is the sum of the responses, the system is said to be additive. Specifically,. . . S{ f1(t) + f2(t)} = S{ f1(t)} + S{ f2(t)} (3.3) . . . Systems that are both homogeneous and additive are said to be linear. The criteria in (3.2) and (3.3) can be combined into a single necessary and sufficient condition for linearity.. . . S{a f1(t) + bf2(t)} = aS{ f1(t)} + bS{ f2(t)} (3.4) . . . where a and b are constants. All linear systems produce a zero output when the input is zero. . . . S{0} = 0. (3.5). . . To show this, we use (3.4) with a = −b and f1(t) = f2(t). Note that, because of (3.5), the system defined by . . . g(t) = b f(t) + c . . . where b and c¹ 0 are constants, is not linear. It is not homogeneous since . . . S{a f} = b f + c ≠aS{ f} = a (b f + c) .

scholarly journals Stabilisation of Discrete-Time Piecewise Homogeneous Markov Jump Linear System with Imperfect Transition Probabilities

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Ding Zhai ◽  
Liwei An ◽  
Jinghao Li ◽  
Qingling Zhang
Keyword(s):  
Linear System ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Feedback Controller ◽  
Sufficient Condition ◽  
Markov Jump ◽  
Stochastically Stable ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Markov Jump Linear System

This paper is devoted to investigating the stability and stabilisation problems for discrete-time piecewise homogeneous Markov jump linear system with imperfect transition probabilities. A sufficient condition is derived to ensure the considered system to be stochastically stable. Moreover, the corresponding sufficient condition on the existence of a mode-dependent and variation-dependent state feedback controller is derived to guarantee the stochastic stability of the closed-loop system, and a new method is further proposed to design a static output feedback controller by introducing additional slack matrix variables to eliminate the equation constraint on Lyapunov matrix. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed methods.

scholarly journals Cooperative hunting in a discrete predator–prey system

2020 ◽  
Vol 13 (07) ◽  
pp. 2050063
Author(s):  
Yunshyong Chow ◽  
Sophia R.-J. Jang ◽  
Hua-Ming Wang
Keyword(s):  
Growth Rate ◽  
Discrete Time ◽  
Reproductive Number ◽  
Model Parameters ◽  
Predator Population ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Cooperative Hunting ◽  
Prey System ◽  
Dependent Growth

We propose and investigate a discrete-time predator–prey system with cooperative hunting in the predator population. The model is constructed from the classical Nicholson–Bailey host-parasitoid system with density dependent growth rate. A sufficient condition based on the model parameters for which both populations can coexist is derived, namely that the predator’s maximal reproductive number exceeds one. We study existence of interior steady states and their stability in certain parameter regimes. It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small. Large cooperative hunting, however, may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.

scholarly journals Gabor frames with trigonometric spline dual windows

2015 ◽  
Vol 08 (04) ◽  
pp. 1550072 ◽  
Author(s):  
Inmi Kim
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Higher Dimensions ◽  
Gabor Frames ◽  
One Dimension ◽  
Trigonometric Spline

A dual Gabor window pair has two functions that can reconstruct any function in [Formula: see text] using certain systems of their modulated and translated forms. Few explicit examples of dual Gabor window pairs are known. This paper constructs explicit examples with trigonometric form in one dimension as well as higher dimensions. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. The windows have fixed support and arbitrary smoothness.

scholarly journals Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Jinshu Chen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Functional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Convergence Of Operators

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.

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