Evaluation of quantization error in two-dimensional digital filters

Author

Agathoklis, P. ; Jury, E.I. ; Mansour, M.

Author_Institution

Swiss Federal Institute of Techonolgy, Zurich, Switzerland

Volume

Issue

fYear

1980

fDate

6/1/1980 12:00:00 AM

Firstpage

273

Lastpage

279

Abstract

In the evaluation of the quantization error in two-dimensional (2-D) digital filters, a procedure for computing ${\\infty \\atop {\\sum \\atop m=0}} {\\infty \\atop {\\sum \\atop n=0}} y^{2}(m,n) = {1 \\over (2{\\pi}j)^{2}} \\oint \\oint Y(z_{1},z_{2}) Y(z_{1}^{-1},z_{2}^{-1}) {dz_{1}dz_{2} \\over z_{1}z_{2}}$ $T^{2} = {(z_{1},z_{2}): |z_{1}|=1, |z_{2}|=1}$ is required. In this paper a condition for a finite quantization error is given and a discussion on the evaluation of the integral based on the residue method is presented. Examples for such an evaluation are given. Furthermore, the salient differences between the one-dimensional (1-D) complex integral evaluation and the two-dimensional one are discussed. Notation: We note with $\\bar{U}^{2} = \\{(z_{1}, z_{2}): \\mid z_{1} \\mid \\leq 1, \\mid z_{2} \\mid \\leq 1 \\}$ } the closed unit bidisk, with $u^{2} = {(z_{1}, z_{2}): |z_{1}| < 1, |z_{2}| \\leq 1}$ the open unit bidisk, and with $T_{2} = {(z_{1}, z_{2}): |z_{1}| = 1, |z_{2}| = 1}$ the distinguished boundary of the unit bidisk. The 2-D $z$ -transform is defined as $Y(z_{1}, z_{2}) = \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} y (m,n)z_{1}^{m}z_{2}^{n}$ .

Keywords

Acoustics; Automatic control; Digital filters; Industrial electronics; Integral equations; Polynomials; Quantization; Sufficient conditions; Transfer functions; 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.1980.1163403

Filename

1163403

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=1093996