• 제목/요약/키워드: sum-product convolution

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이산 Convolution 적산의 z변환의 증명을 위한 인과성의 필요에 대한 재고 (A Reconsideration of the Causality Requirement in Proving the z-Transform of a Discrete Convolution Sum)

  • 정태상;이재석
    • 대한전기학회논문지:시스템및제어부문D
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    • 제52권1호
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    • pp.51-54
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    • 2003
  • The z-transform method is a basic mathematical tool in analyzing and designing digital signal processing systems for discrete input and output signals. There are may cases where the output signal is in the form of a discrete convolution sum of an input function and a designed digital processing algorithm function. It is well known that the z-transform of the convolution sum becomes the product of the two z-transforms of the input function and the digital processing function, whose proofs require the causality of the digital signal processing function in the almost all the available references. However, not all of the convolution sum functions are based on the causality. Many digital signal processing systems such as image processing system may depend not on the time information but on the spatial information, which has nothing to do with causality requirement. Thus, the application of the causality-based z-transform theorem on the convolution sum cannot be used without difficulty in this case. This paper proves the z-transform theorem on the discrete convolution sum without causality requirement, and make it possible for the theorem to be used in analysis and desing for any cases.

A STUDY OF COFFICIENTS DERIVED FROM ETA FUNCTIONS

  • SO, JI SUK;HWANG, JIHYUN;KIM, DAEYEOUL
    • Journal of applied mathematics & informatics
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    • 제39권3_4호
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    • pp.359-380
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    • 2021
  • The main purpose and motivation of this work is to investigate and provide some new results for coefficients derived from eta quotients related to 3. The result of this paper involve some restricted divisor numbers and their convolution sums. Also, our results give relation between the coefficients derived from infinite product, infinite sum and the convolution sum of restricted divisor functions.

하이브리드 수의 조건부 기대값 (Conditional Expectation of Hybrid Number)

  • ;최규탁;한성일
    • 한국지능시스템학회:학술대회논문집
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    • 한국퍼지및지능시스템학회 2003년도 춘계 학술대회 학술발표 논문집
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    • pp.18-21
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    • 2003
  • We propose some properties of fuzzy conditional expectation of hybrid number the addition of fuzzy number and random variable using Cartesian product distance for ${\alpha}$-level sets.

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TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL

  • KIM, DAEYEOUL;CHEONG, CHEOLJO;PARK, HWASIN
    • Journal of applied mathematics & informatics
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    • 제34권1_2호
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    • pp.145-156
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    • 2016
  • It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. In this article, we consider relationship between fifth-order combinatoric convolution sums of divisor functions and Bernoulli polynomials. As applications of these identities, we give a concrete interpretation in terms of the procedural modeling method.

CONVOLUTION SUMS AND THEIR RELATIONS TO EISENSTEIN SERIES

  • Kim, Daeyeoul;Kim, Aeran;Sankaranarayanan, Ayyadurai
    • 대한수학회보
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    • 제50권4호
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    • pp.1389-1413
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    • 2013
  • In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.

REMARKS OF CONGRUENT ARITHMETIC SUMS OF THETA FUNCTIONS DERIVED FROM DIVISOR FUNCTIONS

  • Kim, Aeran;Kim, Daeyeoul;Ikikardes, Nazli Yildiz
    • 호남수학학술지
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    • 제35권3호
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    • pp.351-372
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    • 2013
  • In this paper, we study a distinction the two generating functions : ${\varphi}^k(q)=\sum_{n=0}^{\infty}r_k(n)q^n$ and ${\varphi}^{*,k}(q)={\varphi}^k(q)-{\varphi}^k(q^2)$ ($k$ = 2, 4, 6, 8, 10, 12, 16), where $r_k(n)$ is the number of representations of $n$ as the sum of $k$ squares. We also obtain some congruences of representation numbers and divisor function.

CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS OF BOUNDED FUNCTIONS ON AN ANALOGUE OF WIENER SPACE

  • Cho, Dong Hyun
    • 충청수학회지
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    • 제26권2호
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    • pp.323-342
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    • 2013
  • Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $Xn:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ and $X_{n+1}:C[0,t]{\rightarrow}\mathbb{R}^{n+2}$ by $X_n(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$ and $X_{n+1}(x)=(x(t_0),x(t_1),{\cdots},x(t_n),x(t_{n+1}))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t_{n+1}=t$. In the present paper, using simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions which have the form $${\int}_{L_2[0,t]}{{\exp}\{i(v,x)\}d{\sigma}(v)}{{\int}_{\mathbb{R}^r}}\;{\exp}\{i{\sum_{j=1}^{r}z_j(v_j,x)\}dp(z_1,{\cdots},z_r)$$ for $x{\in}C[0,t]$, where $\{v_1,{\cdots},v_r\}$ is an orthonormal subset of $L_2[0,t]$ and ${\sigma}$ and ${\rho}$ are the complex Borel measures of bounded variations on $L_2[0,t]$ and $\mathbb{R}^r$, respectively. We then investigate the inverse transforms of the function with their relationships and finally prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the products of the conditional Fourier-Feynman transforms of each function.

SOME PROPERTIES OF CERTAIN CLASSES OF FUNCTIONS WITH BOUNDED RADIUS ROTATIONS

  • NOOR, KHALIDA INAYAT
    • 호남수학학술지
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    • 제19권1호
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    • pp.97-105
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    • 1997
  • Let $R_k({\alpha})$, $0{\leq}{\alpha}<1$, $k{\geq}2$ denote certain subclasses of analytic functions in the unit disc E with bounded radius rotation. A function f, analytic in E and given by $f(z)=z+{\sum_{m=2}^{\infty}}a_m{z^m}$, is said to be in the family $R_k(n,{\alpha})n{\in}N_o=\{0,1,2,{\cdots}\}$ and * denotes the Hadamard product. The classes $R_k(n,{\alpha})$ are investigated and same properties are given. It is shown that $R_k(n+1,{\alpha}){\subset}R_k(n,{\alpha})$ for each n. Some integral operators defined on $R_k(n,{\alpha})$ are also studied.

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