Finite impulse response

An important and useful result is that the behavior of a linear filter can be fully characterized in terms of its finite impulse response (FIR). The filter in (11.3) is often called an FIR filter. A finite impulse is a signal for which $x[0] = 1$ and $x[k] = 0$ for all $k > 0$. Any other signal can be expressed as a linear combination of time-shifted finite impulses. If a finite impulse is shifted, for example $x[2] = 1$, with $x[k] = 0$ for all other $k \not = 2$, then a linear filter produces the same result, but it is just delayed two steps later. A finite impulse can be rescaled due to filter linearity, with the output simply being rescaled. The results of sending scaled and shifted impulses through the filter are also obtained directly due to linearity.