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CONVOLUTIONAL INTEGRAL APPROXIMATION VIA TRAPIZOIDAL RULE

Platform : DSP

AIM : Implementation of Convolution via Trapizoidal Abstract We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + j=1 m µj x(t – t j) + 0 k(t – s)x(s) ds = g(t), 0 t , where t j (– , ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions. Fredholm equations - displacement kernel - Toeplitz matrices - quadrature rules LEARNING OBJECTIVE : To understand a numerical scheme to approximate the integral equation such that the discretization matrix with Convolution integral INPUT: Matrix length of four and Matrix length of three. OUTPUT: Simulated wave form of multiplied version of Matrix four and matrix three APPLICATIONS: Speech Communication SOFTWARE USED: MATLAB2007A

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