Variation diminishing linear time-invariant systems

This paper studies the variation diminishing property of k-positive linear time-invariant (LTI) systems, which diminish the number of sign changes (variation) from input to output, if the input variation is at most k−1. We characterize this property for the discrete-time Toeplitz and Hankel operators of finite-dimensional causal systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k first order positive systems. This is shown by expressing the k-positivity of a LTI system as the external positivity (that is, 1-positivity) of k compound LTI systems. Our characterization generalizes well known properties of externally positive systems (k=1) and totally positive systems (k=∞; also known as relaxation systems in case of the Hankel operator). All results readily extend to continuous-time systems by considering sampled impulse responses.