## 5.1. Introduction to finite impulse response filters

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.

### 5.1.1. *Difference equations and FIR filters*

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.

For finite impulse response filters, equation (5.1) verifies:

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:

The output ...