Evaluation of quantization error in two-dimensional digital filters
P. Agathoklis, E.I. Jury, M. Mansour · IEEE Transactions on Acoustics Speech and Signal Processing · 1980
In the evaluation of the quantization error in two-dimensional (2-D) digital filters, a procedure for computing\sum\min{m=0}\max{\infin} \sum\min{n=0}\max{\infin} y^{2}(m,n)= \frac{1}{(2\pij)^{2}}\oint\oint Y(z_{1},z_{2})Y(z_{1}^{-1},z_{2}^{-1}) \frac{dz_{1}dz_{2}}{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}): |z_{1}| \leq 1, |z_{2}| \leq 1} the closed unit bidisk, withu^{2} = {(z_{1}, z_{2}): |z_{1}| < 1, |z_{2}| \leq 1}the open unit bidisk, and withT_{2} = {(z_{1}, z_{2}): |z_{1}| = 1, |z_{2}| = 1}the distinguished boundary of the unit bidisk. The 2-Dz-transform is defined asY(z_{1}, z_{2} = \sum\min{m=0}\max{\infin}\sum\min{n=0}\max{\infin} y (m,n)z_{1}^{m}z_{2}^{n}.