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## 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 ...

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