FIR Filter

A FIR filter is a type of digital interpolating filter. We will be looking at its use to both distort and interpolate a digital signal as in a CD Player. Mathematically this this is done by the following convolution:

$\sum _{l=-\infty }^{\infty }h\left({\frac {lT}{2}}\right)\delta \left(t-{\frac {lT}{2}}\right)*\sum _{k=-\infty }^{\infty }x(kT)\delta (t-kT)$

Where the term on the left is ther filter and the term of the right is you data. So how do we do an infinite sum in a computer? Well we can't because we can't fit an infinite amount of data in a computer, so we have to trim it down. The above sum for the filter is the impulse responce of the filter. This is an infinite sum, and for your filter to work exactly as described you would need all of the terms. Since that is not possible you want to keep as many of the most important terms as you can. The most important terms are the ones centered around zero. So if you wanted three terms they would be $l=-1,0,1$ and if you wanted 5 terms $l=-2,-1,0,1,2$ etc. Once you have your filter limited to a certain number of terms you can proceed as follows with the convolution.

$\sum _{l=-M}^{M}h\left({\frac {lT}{2}}\right)\delta \left(t-{\frac {lT}{2}}\right)*\sum _{k=-\infty }^{\infty }x(kT)\delta (t-kT)$

$\int _{-\infty }^{\infty }\sum _{l=-M}^{M}h\left({\frac {lT}{2}}\right)\delta \left(\tau -{\frac {lT}{2}}\right)\sum _{k=-\infty }^{\infty }x(kT)\delta (t-\tau -kT)\,d\tau$

$\sum _{k=-\infty }^{\infty }\sum _{l=-M}^{M}h\left({\frac {lT}{2}}\right)x(kT)\int _{-\infty }^{\infty }\delta \left(\tau -{\frac {lT}{2}}\right)\delta (t-\tau -kT)\,d\tau$

$\sum _{k=-\infty }^{\infty }\sum _{l=-M}^{M}h\left({\frac {lT}{2}}\right)x(kT)\delta \left(t-{\frac {lT}{2}}-kT\right)$

Let $n=2k+l\,$ so $k={\frac {n-l}{2}}$

$\sum _{n=-\infty }^{\infty }\sum _{l=-M}^{M}h\left({\frac {lT}{2}}\right)x\left({\frac {n-l}{2}}T\right)\delta \left(t-{\frac {nT}{2}}\right)$