TABLE OF CONTENTS Chapter written by Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM.
Finite impulse response filters (FIR filters) are filters used in many applications, especially in image processing, as we will see in Chapters 8 and 9.
Their popularity is due to their simplicity; they only use one finite sequence of input signal samples. This step allows FIR filters to easily attain specificities that cannot be obtained with infinite impulse response or IIR filters, especially in the realization of causal linear phase filters.
Moreover, FIR filters have the advantage of always being stable, which makes them very useful for an easy material implantation. Depending on the application, the order of the model usually varies from 25 to 400.
In Chapter 3, difference equations were introduced to characterize linear time-invariant (LTI) digital systems with an input of x(k) and an output y(k). The difference equation deals with digital systems whereas differential equations make it possible to characterize analog systems.
| (5.1) |  |
For finite impulse response filters, equation (5.1) verifies:
| (5.2) |  |
From here, for FIR filters, difference equations also correspond to the convolution between the impulse response and the input of the signal. Indeed, given (5.2), equation (5.1) becomes:
| (5.3) |  |
The output and the input satisfy the following relation:
| (5.4) |  |
from where, by identification:
| (5.5) |  |
Equation (5.3) shows that FIR filters do not present the recursive mode in their implementation.
From the z-transform of equation (5.3), we easily deduce the...
