# How does Mathematica's LowpassFilter work?

I don't really understand the documentation for the LowpassFilter[] function in Mathematica. In the course I have about signal processing, low-pass filters come in many different forms and from different kinds of operations, so I was just wondering if someone could explain to me the math behind this specific function.

If you look at the "Properties and Relations" section of the help file for LowpassFilter there is a big hint: it shows how LeastSquaresFilterKernel with a Hamming window gives the same answer as LowpassFilter. So this is a best-fit in the least-squares sense FIR (finite impulse response) filter. Digging a bit deeper into the LeastSquaresFilterKernel function, the calculation is is done by inverse FFT.
Putting this together, we can see that the default values for LowpassFilter calculate the $n$th order linear FIR filter that is closest to the specified frequency response (in a mean-squares sense), assuming preprocessing by a Hamming window.