Finite Impulse Response (FIR) Digital Filters

Abstract

Digital filters are typically used to modify or alter the attributes of a signal in the time or frequency domain. The most common digital filter is the linear time-invariant (LTI) filter. An LTI interacts with its input signal through a process called linear convolution, denoted by y = f * x where f is the filter’s impulse response, x is the input signal, and y is the convolved output. The linear convolution process is formally defined by:

$$ y\left[ n \right] = x\left[ n \right] * f\left[ n \right] = \sum\limits_k {x\left[ k \right]f\left[ {n - k} \right] = } \sum\limits_k {f\left[ k \right]x\left[ {n - k} \right].} $$

(3.1)