Finite impulse response

In system theory, a causal finite impulse response (FIR) system is a DT LTI system that has the form:

$$y[n]=\sum _{k=0}^M{b}_{k}x[n-k]=b_{0}x[n]+b_{1}x[n-1]+\cdots +b_{M}x[n-M]$$

where $\{b_k\}$ are constants and $M$ is the order of the system. In plain English, it says that the current output value is a finite linear combination of the past and present input values. FIR systems are used in signal processing and filtering.

The impulse response of an FIR system is given by:

$$h[n]=\sum _{k=0}^M{b}_k\delta [n-k]=\{\begin{array}{llc}{b}_n& & \text{if}0\le n\le M\\ 0& & \text{otherwise}\end{array}$$

This system is also BIBO stable.