Variation diminishing linear time-invariant systems

This paper studies the variation diminishing property of k-positive linear time-invariant (LTI) systems, which map inputs with k-1 sign changes to outputs with at most the same variation. We characterize this property for the Toeplitz and Hankel operators of finite-dimensional 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).