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FIR Filters

What is an FIR Filter? FIR stands for Finite Impulse Response, which means that the response to an individual impulse is going to stop at some point in time, where and Infinite Impulse Response is going to have a response that, at least in theory, never stops. The FIR Filter is almost always more useful than the IIR filter, so FIR filters are more widely used.

What is it used for?

An FIR Filter is used in Digital Signal Processing. A digital signal is really a series of impulse functions that were sampled from the orignal sound/radio/whatever wave. From these impulse functions the computer can re-create the original waveform. This all works great, but what if you want to change something about the wave, like it's frequency response, or maybe the phase delay? This is where FIR filters come into play.

How does it work?

An FIR Filter is basically a convolution of two series of impulse functions. It takes the impulse functions stored digitally and convolves them with another series of impulse functions which represent a certain frequency response. This convolution in time multiplies the original frequency response with the frequency response of the FIR Filter to give the desired output from the filter. Ideally you would use an infinite convolution like this: $\sum _{n=-\infty }^{\infty }\sum _{m=-M}^{M}x(nT)h\left({\frac {mT}{K}}\right)\delta \left(t-nT-{\frac {mT}{K}}\right)$ where K is a variable for the amount of interpolation desired in the filter. Unfortunately, we can't do an infinite convolution like this so we end up doing a discrete convolution where we let n in the outer sum be as big or as small as we like, as long as we get the response that we think is good enough (the bigger N is, the better the response.) It looks like this: $\sum _{n=-N}^{N}\sum _{m=-M}^{M}x(nT)h\left({\frac {mT}{K}}\right)\delta \left(t-nT-{\frac {mT}{K}}\right)$