An optimal order structure for 2-D digital filters

Abstract

E ma i 1: IN % "zi 1 ouch i @a c c .fa U. ed u IfIn this paper, the model approximation as well as implementation of 2-D Recursive Sep arable-in-Denomina tor (RSD) digital filters with fixed-point arithmetic are investigated. A minimum-order structure is proposed based on optimality condition of an error criteria d u e to b o t h approximation and implementation. In order to reduce the number of multiplications for an optimal order realization, a class of second-order structures are proposed. This class of realizations presents a good compromise between low round-off noise, order reduction, number of multipliers and hardware complexity.During the last two decades, the study of 2-D discrete systems has emerged as one of the most important area of research within the field of controls, estimation theory and digital signal processing. This attention is due to usefulness of 2-D digital filters in many applications which require the utilization of two-dimensional signal processing techniques[I], [2]. Moreover, with the introduction of VLSI a n d r a p i d d e v e l o p m e n t of microprocessor circuits, digital filtering applications are increasing at an expanding rate [3], [4]. In general, the design of 2-D digital filters usually involves two closely related phases. In t h e first p h a s e , t h e mathematicaI representation of a filter is obtained from a set of specifications or given data either in statespace or transfer function form(approximati0n phase). In this phase it is particularly desirable to obtain a simplified model which greatly reduces the computation efforts or the complexity of the filter[5],[6],[9],[10]. On the other hand, given a transfer function of a filter, there are theoretically an infinite number of realizations to implement such filter (implementation phase). This freedom can be used to optimized specific criteria associated with actual implementation of a filter such as the effect of finite word length registers[7]-[8 1,[12]- [16]. Since the registers can only store a limited number of bits, the parameters and data variables within a filter model must be quantized to a finite set of allowable values resulting in quantization. Without these quantization errors, the discrete time filters could be realized exactly and operated as described by the linear model. In general, the change in the filter characteristic due to the input data and coefficient quantization is deterministic, and therefore can be computed easily. However, the quantization error due to the internal data or various points in a statespace digital filter are non-determinestic and strongly effected by the filter structure. In many publications such as [12], [14] the optimal realization of 2-D digital filters has been addressed in order to minimize the round-off noise under L2 scaling to prevent overflow [8]. However, t h e a p p r o x i m a t i o n a n d implementation steps (phase I and 11) of 2-D digital filters should not necessarily need to be considered independently[ 151. In this paper, the combination of these two phas...