Digital Filters Design for Signal and Image Processing

11.1. Recursive filters

In this chapter we present the practical results of a study of the stability of two-dimensional (2-D) filters that are both digital and recursive. We begin by introducing the concept of the transfer function of a 2-D filter and then go on to define the class of recursive filters. As with one dimensional filters, 2-D filters are used in compression and data analysis applications. However, a recursive filter must be stable to be usable. This means that a small perturbation, applied to an input signal, must be transformed by a small perturbation on the output signal. This is why a reliable and rapid algorithm is crucial for testing the stability of any recursive filter for applications. An algorithm that tests the stability of 2-D recursive filters is the translation into programming language (possibly virtual) of a necessary and sufficient condition that assures stability. We call the stability criterion this condition. First we will present several stability criteria, then several algorithms based on these criteria.

11.1.1. Transfer functions

Let us consider a two-dimensional digital filter of impulse response (h(m, n))(m,n)Z². We saw in Chapter 8 that when it is affected by the signal (x(m, n))(m,n) Z² it admits the signal (y (m, n))(m,n) Z² as output, which ...