On the evaluation of double square integral in the (s1, s2) complex biplane

Author

Jury, E.I. ; Ruridant, T.Y.

Author_Institution

University of Miami, Coral Gables, FL

Volume

Issue

fYear

1986

fDate

6/1/1986 12:00:00 AM

Firstpage

630

Lastpage

632

Abstract

In this correspondence, a method of evaluating the double square integral $\\int_{0}^{\\infty} \\int_{0}^{\\infty} [h(t_{1}, t_{2})]^{2} dt_{1} dt_{2}$ $= {1 \\over (2\\pi j)^{2}} \\int_{-j \\infty}^{j \\infty} \\int_{-j \\infty}^{j \\infty} H(s_{1}, s_{2}) H(-s_{1}, -s_{2}) ds_{1} ds_{2}$ where $h(t_{1}, t_{2}) = L^{-1} [H(s_{1}, s_{2})]$ is discussed. The method of evaluation is based on using the double bilinear transformation (DBT) $s_{k} = (1 - z_{k})/(1 + z_{k}), k = 1, 2$ to reduce the above complex integral to a double integral in the complex ( $z_{1}, z_{2}$ ) bidisk, ${\\infty \\atop {\\sum \\atop {m=0}}} {\\infty \\atop {\\sum \\atop {n=0}}} [a(m, n)]^{2} = {1 \\over (2{\\pi}j)^{2}} {\\oint \\atop {|z1| = 1}} {\\oint \\atop {|z2| = 1}} A(z_{1}, z_{2})$ $\\cdot A(Z_{1}^{-1}, Z_{2}^{-1}) frac{dz_{1}}{z_{1}}frac{dz_{1}}{z_{2}}$ . There exist several methods for evaluating the above double integral. In this correspondence, we mention one promising method. Also, a matrix method is proposed for applying the "DBT" to a quotient of two-dimensional (2-D) polynomials.

Keywords

Digital filters; Energy conservation; Matrices; Polynomials; Quantization; Two dimensional displays;

fLanguage

English

Journal_Title

Acoustics, Speech and Signal Processing, IEEE Transactions on

Publisher

ieee

ISSN

0096-3518

Type

jour

DOI

10.1109/TASSP.1986.1164862

Filename

1164862

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=1108728