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 \(I = \frac{1}{{(2\pi i)^2 }}\oint_{\left| {z_1 } \right| = 1} {\oint_{\left| {z_2 } \right| = 1} {G(z_1 ,z_2 )} } {\text{ }}G(z_1^{ - 1} ,z_2^{ - 1} )\frac{{dz_2 dz_1 }}{{z_2 z_1 }}\), where \(i = \sqrt { - 1} \) 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 -11 , z -12 ) = X(z1, z1) / A(z1,z2)+ X(z -11 , z -12 ) / A(z -11 , z -12 ). 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 \(\hat G(s_1 ,z_2 )\). 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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Hwang, JH., Tsay, SY. & Hwang, C. Variance and Covariance Computations for 2-D ARMA Processes. Multidimensional Systems and Signal Processing 10, 137–160 (1999). https://doi.org/10.1023/A:1008498312387
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DOI: https://doi.org/10.1023/A:1008498312387