FIRJEW

Finite Impulse Response Filters

A Finite Impurse Response filter (aka FIR filter) is a type of filter often used in digital signal processing and has no feedback loop. Because there is no feedback, the response of an FIR filter to an impulse is finite.

The equation for an FIR filter would look like the following:

$h(mT)=T\int _{-v}^{v}{\hat {H}}(f)e^{j2\pi fmt}df$

where the desired frequencies are in the range from $-v$ to $v$ and ${\hat {H}}(f)$ is the desired response.

Example: Desing an FIR low pass filter to pass between $-{\frac {1}{4T}}<f<{\frac {1}{4T}}$ and reject the rest.

${\hat {H}}(f)={\begin{cases}1,&|f|\leq {\frac {1}{4T}}\\0,&else\end{cases}}$

The FIR filter would then be:

$h(mT)=T\int _{\frac {-1}{4T}}^{\frac {1}{4T}}1e^{j2\pi fmt}df=T{\frac {e^{j2\pi {\frac {1}{4T}}mT}-e^{j2\pi {\frac {-1}{4T}}mT}}{j2\pi mT}}$

$={\frac {e^{j2\pi {\frac {1}{4T}}mT}-e^{j2\pi {\frac {-1}{4T}}mT}}{j2\pi m}}=\left({\frac {1}{2}}\right){\frac {sin\left({\frac {\pi m}{2}}\right)}{\left({\frac {\pi m}{2}}\right)}}$

Principle author: Jeffrey Wonoprabowo