IIR filters are awesome, but they can have some crazy behavior. If you want a neat mathematical framework to understand them check out the z-transform, which ma

IIR filters are awesome, but they can have some crazy behavior. If you want a neat mathematical framework to understand them check out the z-transform, which maps IIR filters to the space of rational functions (a polynomial divided by another polynomial). You can extract filter's behavior from this function. For example the complex zeros of the polynomials are crucial and have names ("poles" for zeros of the denominator, "zeros" for zeros of the numerator). The special case when the denominator disappears corresponds to FIR filters. IIR filters have decades of research behind them, and are used in all radio equipment. IIR filters cannot compute in the Turing sense, though. They're linear operators which is particularly evident in the Fourier basis. Complex exponentials (i.e. simple rotations) are their eigenvectors. Much like with neural networks you also need a non-linear operator to get the "real deal".