Discrete Convolution Operators Research Articles

Approximation and spectral properties of periodic spline operators

We consider discrete convolution operators whose range is the k-dimensional space spanned by the translates of a single function. Examples of include the space of trigonometric polynomials, periodic polynomial splines and trigonometric splines. The eigenfunctions of these operators corresponding to the nonzero eigenvalues are independent of α, and they form an orthogonal basis for . The limiting behaviour of as α, k→∞, is also considered. The corresponding limiting semigroups are computed explicitly.

The discrete Fourier transform method of solving differential-integral equations in scattering theory

An accurate and efficient numerical method is presented for solving many differential-integral equations arising from electromagnetic scattering theory. It uses the discrete Fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations. As a consequence, this method is extremely simple to implement, uses less computer memory than comparable methods, and yields accurate predictions. The differential-integral equation is recast into a periodic form conducive to application of the discrete Fourier convolution theorem. The differential operators are approximated by appropriate finite-difference and discrete-convolution operators. All these quantities are computed by using the fast Fourier transform. An approximate solution is obtained by using the conjugate gradient method. Results are compared to experimental data or analytical solutions for a 3 lambda *3 lambda metal plate (where lambda is the wavelength), a homogeneous and a layered infinite circular dielectric cylinder, and a dielectric sphere. The accuracy of the method is further illustrated by comparing predictions with independent measurements by R.A. Ross (1966) on a 2 lambda *1 lambda metal plate at grazing incidence. In all cases, agreement is excellent.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of MMIC structures using an efficient iterative approach

A class of two-dimensional and three-dimensional monolithic microwave integrated circuit (MMIC) structures is analyzed theoretically using an efficient iterative technique. The transfer-matrix approach to constructing the spectral Green's functions of multilayered structures is adopted. Discretization of the continuous functions and utilization of the periodicity of the MMIC structures enclosed by sidewalls lead to a discrete convolution operation, which is carried out efficiently using the FET (fast Fourier transform) algorithm. The spectral Green's functions are modified for both the periodic and aperiodic structures to improve the numerical efficiency of the iterative algorithm. Numerical results are presented and compared with available data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Discrete convolution operators with almost-stabilized coefficients

For a convolution operator of the form\(K = \sum\nolimits_{j = 0}^{N - 1} {P_j A_j } \) discrete Wiener—Hopf operators and Pj are coordinate projections whose indices are congruent to j modulo N, necessary and sufficient conditions for it to be Notherian inlp+(i ⩽p ⩽ ∞),c + 0 and formulas for the index are obtained.

An&amp;lt;tex&amp;gt;l&amp;lt;/tex&amp;gt;&amp;lt;inf&amp;gt;2&amp;lt;/inf&amp;gt;stability criterion with frequency domain interpretation for a class of nonlinear discrete systems

Using the notation and methods of functional analysis, a stability criterion is derived for a class of nonlinear discrete systems. The class of systems investigated consists of a nonlinear static operator satisfying a sector condition, followed by a bounded linear causal operator that satisfies an inner product inequality. A simple graphical means of obtaining stability information similar to the Popov criterion is obtained when the bounded causal linear operator is constrained to be a discrete convolution operator.

Popov-like stability criterion for a class of time-varying discrete systems

A Popov-like stability criterion is derived for use in the analysis of a class of time-varying discrete control systems. The class of systems consists of a time-varying gain followed by a causal time-invariant discrete convolution operator. The stability criterion provides a simple graphical means of obtaining stability information which complements the existing information for this class of systems.