Mathematical definitions and tools

Analog filters (cf. ) built with resistors, capacitors and self-induction coils, with or without active components, work with signals that are continuous functions of time in the mathematical sense. The relationship between input and output signals is linear and obeys the principle of superposition: mathematically, it takes the form of a linear differential equation with constant coefficients. It follows that the transfer function, i.e. the ratio between the Fourier or Laplace transforms of the output and input signals, is a rational fraction in p (for the Laplace transform) or in $\text{j}\omega$ (for the Fourier transform) (cf. ).

With the advent of computers and, more generally, all programmable circuits, the filtering...