A generalized expression for the impulse responses of recursive linear time-variant digital filters

Digital filters described by linear time-variant (LTV) difference equations, especially for the general case of time-variant order, are considered. Three notions, i.e. the linearly dependent length for a group of sequences, the lowest time-variant order and the standard form for LTV difference equations, are presented. It is shown that the lowest time-variant order of an LTV difference equation is equal to the linearly dependent length of its linearly independent homogeneous solutions. The properties of an LTV homogeneous difference equation and its homogeneous solutions are investigated. A method to determine one from the other is given. Based on this, LTV homogeneous difference equations have been studied and the general expression for the impulse responses of all the forward recursive LTV digital filters with a finite time-variant order has been determined