Stability criterion for<tex>N</tex>-dimensional digital filters

Synthesis of Nonseparable 3-D Spatiotemporal Bandpass Filters on Analog Networks

“…For example, we analyze a velocity selective filter based on linear CNNs [4] with a trivial feedforward template and a feedback template given by (8) The quantities , for , are real constants. The velocity-selective filter implements the following mixed-domain STTF: (9) where is the parallel capacitance at each CNN cell as described in Section I, with and relating to the spatial frequencies and the temporal frequency as follows:…”

On Analog Networks and Mixed-Domain Spatio-Temporal Frequency Response

“…Specifically, we treat the denominator of (10) as a polynomial of the temporal Laplace variable and solve for the poles in terms of . The condition for BIBO stability of a spatial general support and causal time network implementing (10) is given by [13]-there are no poles in the space of (17) which reduces to having all poles lying on the dual unit circles to be on the left half temporal-Laplace plane.…”

“…This is a reasonable assumption since any terms in the form can have the -domain variables eliminated by multiplying the whole transfer function by . However, this affects the stability criterion of the filter [13]. In the case for which an unstable transfer function results after the multiplication of , the filter is not suitable for the integrator-based implementation outlined here.…”

“…In a previous publication [17], we included a first-order velocity selective filter [8], [21] as an example. The authors would like to take the opportunity to point out that the example used in [17] is marginally stable [13].…”

“…Therefore, we aim to express in (66) as a function of such that an increase in causes a decrease in and hence an increase in radius of the passband boundary of the spatial filter (65). In this way, the desired temporal frequency- To ensure filter stability [7], [13], we introduce an extra constant term in the denominator (69)…”