On the non-negative impulse response of multi-dimensional systems

This paper presents sufficient conditions on one-dimensional (1-D) pole-zero patterns that exhibit a non-negative impulse response (NNIR) for several classes of multi-dimensional (M-D) hyper-planar systems in both the continuous-time and the discrete-time domains. The results provide some intuitive ideas about 1-D to M-D variable substitutions that preserve the non-negativity property of impulse response, as well as the relationship between the impulse response of a transformed M-D system and the impulse response of its 1-D prototype. To the best of our knowledge, this is the first paper in the literature discussing the sufficient conditions on pole-zero patterns that guarantee an NNIR for M-D systems. The sufficient conditions on the pole-zero patterns for the employed 1-D prototype systems represent the most inclusive sufficient conditions that ensure an NNIR. As a result, sufficient conditions on the pole-zero patterns that exhibit an NNIR are revealed for a broad category of M-D systems in this paper. It should be noticed that though there is a significant potential of applications in the area of NNIR M-D systems, NNIR M-D system design has received inadequate attention, which is surprising in view of the abundance of design frameworks developed over decades of research in M-D systems. In response to this inadequacy, this paper provides theoretical background and important guidance in constructing NNIR M-D systems. Specifically, the presented results can be employed to construct NNIR M-D filters using M-D hyper-planar filters as the building blocks. The constructing process is illustrated through a simple design example.