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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.

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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.

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