A counterexample to 2-D Lyapunov equations by Mertzios

In [1], Mertzios introduced two-dimensional (2-D) Lyapunov equations for the covariance and noise matrices to analyze the output roundoff noise under
l_{2}
-norm scaling in 2-D digital filters described by the local state-space model. However, his 2-D Lyapunov equations include some crucial errors. This paper presents a counterexample to his 2-D Lyapunov equations, and shows that his 2-D Lyapunov equations cannot give the covariance and noise matrices in 2-D state-space digital filters.