Finite Impulse Response Filter

A Finite Impulse Response Filter is a Linear Time-Invariant System Filter whose Impulse Response eventually becomes zero, or in other words, the duration of response to the Impulse Signal is finite. This property is due to such filters being Feed-Forward.

$$\begin{array}{c}\mathcal{H}:{\mathbb{R}}^{\mathbb{Z}}\times \mathbb{Z}\to \mathbb{R}\\ \mathcal{H}(x,n)=y[n]\\ y[n]={\sum }_{k=0}^M{b}_k\cdot x[n-k]\end{array}$$

• Because FIR Filters are Feed-Forward, they are massively more parallel and belong in CUDA algorithms on the GPU, whereas IIR Filter is inherently iterative and thus is better suited for the CPU

Z Transform

$$\begin{array}{rl}H(z)& =\sum _{k=0}^M{b}_k\cdot z^{-k}\\ & =b_0\prod _{k=1}^M(1-c_k{z}^{-1})\\ & \end{array}$$

• See Z Transform

Impulse Response

$$h[n]=\{\begin{array}{ll}{b}_n& 0\le n\le M\\ 0& \text{else}\end{array}$$

• See Impulse Response