# Finite impulse response (FIR) systems

Consider an LTI system*H*:[

*Integers*→

*Reals*] → [

*Integers*→

*Reals*] with impulse response

*h*:

*Integers*→

*Reals*that has the properties

*h*(

*n*) = 0 for all

*n*< 0, and

*h*(

*n*) = 0 for all

*n*>

*M*

where *M* is some positive integer. This is called a **finite
impulse response (FIR) **system because the interesting part (the
non-zero part) of the impulse response is finite in extent. Because of
that property, the convolution sum becomes a finite sum,

This sum, since it is finite, it much more convenient to work with than anything we have seen yet. It can be used to define a computer procedure for computing the output of an FIR system given its input.

The continuous-time version could be defined, but in practice, useful continuous-time systems almost never have finite impulse response, so there is not much motivation for doing so. Moreover, even though the convolution integral acquires finite limits, this does not make it any more computable. Computing integrals on a computer is a difficult proposition whether the limits are finite or not.