A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic

A Note on the Induced $${\varvec{l}}_\infty $$ l ∞ Stability of Fixed-Point Digital Filters Without Overflow Oscillations and Instability Due to Finite Wordlength Effects

“…Since the criteria in [5,[7][8][9][10]14,16] do not consider the effect of external interference, as a result, they are not applicable for the situation considered in the present example. …”

“…On the other hand, Theorem 3 utilizes a relatively more precise characterization of the saturation nonlinearities (21), where C is a row diagonally dominant matrix with positive diagonal elements, which has been utilized in [7]. The relation (21) makes use of the structural properties of the multiple saturation nonlinearities in a greater detail (i.e., the so-called passivity property [7,9]) than just the sector information. Further, by restricting the matrix C in (21) to a positive definite diagonal matrix M, (21) is equivalent to (10).…”

“…The criteria appeared in [5,[7][8][9][10]14,16] for the stability of digital filters are established by assuming no external interference in the digital filters. However, when we implement a high-order digital filter in hardware or software, it requires to be broken into several biquad digital filters.…”

“…Several stability criteria for fixed-point state-space digital filters have been reported in [5,[7][8][9][10]14,16], but they are incapable to conclude on stability of digital filters in the existence of external interferences [2,3,11,12]. Theorem 1 in [3] can guarantee to reduce the effect of external disturbance to a prescribed…”

Criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance

“…If the total number of quantization steps is large or, in other words, the internal wordlength is sufficiently long, then the effects of these nonlinearities can be regarded as decoupled or noninteracting and can be investigated separately. The stability properties of digital filters employing saturation overflow arithmetic have attracted the attention of several researchers [1][2][3][4][5][6][7][8]. However, most existing criteria for the stability of digital filters are only available under specific conditions, while in unfavorable environments with parameter uncertainty or external disturbance, these criteria will be of little use.…”