Digital Filters Design for Signal and Image Processing

10.1. Introduction

A filter of impulse response h(k) is bounded-input, bounded-output (BIBO) stable when a bounded input signal x(k) produces a bounded output y(k).

Let us consider a bounded input signal x(k);

where h*(− k) and |h (−k)| respectively designate the conjugate and the complex modulus of h(−k). With equation (10.1), the output y(k) is written as:

More specifically, equation (10.2) verifies for k = 0:

If the filter is BIBO stable, then it verifies the following condition:

Inversely, if the impulse response h(k) of the filter satisfies equation (10.4) then, for any input signal x(k) bounded by M, the output y(k) verifies:

Let us assume now that the filter admits a transfer function¹:

We write C the open convergence ring (i.e. circular band) represented as:

or it can be represented ...