A generalized spline framework for mixed mode signal processing

In the context of mixed-mode signal processing applications, a system-level framework is presented for best approximation of “analog (continuous-time) functions” by “digital (discrete-time) functions” based on dynamic source models. Specifically, let x(t), a⩽t⩽b, denote the input to an analog filter F, and y(t), a⩽t⩽b the corresponding output of F. We assume that the input x belongs to a class C of signals generated by a linear dynamical system, characterized by an m^th order linear differential operator L(D), D=d/dt, driven by inputs of bounded energy. On this basis, we obtain the best discrete-time approximation F* to the analog filter F by interpolating the samples x(t₁),...,x(t _n) of x(t) by a generalized L-spline S(t) (corresponding to the differential operator L(D)), and then convolving S(t) with the impulse response of the analog filter F. This provides the optimal mapping F* (in min-max error sense) of the samples x(t₁),...,x(t_n) into the best approximation y*(t ₁),...,y*(t_a) of the samples y(t₁),...,y(t_n) of F