Criterion for non‐existence of limit cycles in 2D state‐space digital filters described by the Fornasini–Marchesini second model with finite wordlength non‐linearities

Abstract

This study investigates the problem of global asymptotic stability of fixed-point two-dimensional (2D) digital filters described by the Fornasini-Marchesini second local state-space model with the combined effects of quantisation and overflow non-linearities. Utilising a more precise characterisation of the non-linearities, an improved criterion for the limit cycle-free realisation of 2D digital filters is brought out. The criterion is compared with several previously reported criteria. Nomenclatureset of integer numbers Z + set of non-negative integers I n identity matrix of order n 1 n n × 1 vector with all elements equal to 1 0 null matrix or null vector of appropriate dimension f null set x i ith entry of a vector x J t list of coordinate indices, which we regard as a subset of {1, 2, …, n}; the subscript t refers to the iteration number n t number of indices in J t J c t complement set of J t such that J t < J c t = 1, 2, . . . , n { } ; the superscript c stands for the complementmaximum normalised quantisation error N max maximum representable number for a given wordlength T superscript 'T' refers to the transpose R > 0 R is positive definite symmetric matrix l min (P) minimum eigenvalue of matrix P l max (P) maximum eigenvalue of matrix P . 2 Euclidean norm

Analysis of Memory Effects in Digital Filters with Overflow Arithmetic

“…Due to the presence of such nonlinearities, the implemented filter may exhibit undesired zero-input oscillations [2][3]. The problem of stability of digital filters with finite wordlength nonlinearities has been studied extensively [2][3][4][5][6][7][8][9][10][11][12][13][14]. Several stability results to reduce the effects of nonlinear phenomenon have appeared recently [10][11][12][13][14][15][16].…”

“…The analysis of undesired memory effects in digital filters with finite wordlength nonlinearities is an important problem. The criteria in [3][4][5][6][7][8][9][10][11][12][13][14][15] cannot be used to measure the memory effects in digital filters. Recently, the Hankel norm performance criterion for the reduction of memory effects of digital filters with saturation arithmetic has been established in [16].…”