Variance and Covariance Computations for 2-D ARMA Processes |
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| Authors: | Jyh-Haur Hwang Sun-Yuan Tsay Chyi Hwang |
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| Affiliation: | (1) Department of Chemical Engineering, National Cheng Kung University, Tainan, 701, Taiwan |
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| Abstract: | An algorithm is presented to compute the variance of the output of a two-dimensional (2-D) stable auto-regressive moving-average (ARMA) process driven by a white noise bi-sequence with unity variance. Actually, the algorithm is dedicated to the evaluation of a complex integral of the form
, where
and G(z1,z2) = B(z1, z2) / A(z1, z2) is stable (z1,z2)-transferfunction. Like other existing methods, the proposed algorithmis based on the partial-fraction decomposition G(z1,z2)G(z
1
-1
, z
2
-1
) = X(z1, z1) / A(z1,z2)+ X(z
1
-1
, z
2
-1
) / A(z
1
-1
, z
2
-1
). However,the general and systematic partial-fraction decomposition schemeof Gorecki and Popek 1] is extended to determine X(z1,z2).The key to the extension is that of bilinearly transforming thediscrete (z1, z2)-transfer function G(z1,z2)into a mixed continuous-discrete (s1, z2)-transferfunction
. As a result, the partial-fraction decomposition involves only efficient DFT computations for the inversion of a matrix polynomial, and the value of I is finally determined by the residue method with finding the roots of a 1-D polynomial. The algorithm is very easy to implement and it can be extended to the covariance computation for two 2-D ARMA processes. |
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| Keywords: | Complex double integrals Two-dimensional systems Variance and Covariance |
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