Abstract
Unstable modes, caused by finite wordlength effects are investigated for the case of 2-D digital filters. An extension of the term '2-D periodic' is introduced as well as chaotic types of instabilities, which are studied through a proposed state diagram. Two especially chosen examples illustrate the major differences between instabilities in 1-D and m-D filters.Stability conditions for the case of multidimensional direct realizations with nonlinearities of the finite wordlength type are developed, which prove to be a direct extension of results for linear shift invariant 1-D systems, obtained by earlier authors. This method is then extended to m-D discrete state space realizations, providing more general results than those obtained by applying norm tests for the system matrix. All these methods do not assume the instability to be of the periodic type, as the analysis is conducted in the space rather than in the frequency domain. These tests are not restricted by the order or the dimensionality of the system and any number of nonlinearities can be handled. The use of these techniques are demonstrated by numerous examples.Finally a new method, based on the effect of $L\sb1$-norm reduction through computer generated extreme matrix products, is introduced for the above mentioned class of realizations. The problem of computational feasibility and required computational complexity for this approach is analyzed. The effectiveness of this approach is demonstrated through two types of nonlinear systems.For some cases, the topic of non-zero input systems under the influence of nonlinearities is also addressed. The importance of these results for systems other than nonlinear filters, such as shift-varying or robust systems, is discussed.