Behavior of Attenuation for Systems Having Monotonic Step and Impulse Responses

Abstract

If a system has a monotonically increasing step response, the magnitude of the system function cannot attentuate too rapidly. This well-known fact is given greater precision in this paper by the establishment of a set of lower bounds on the magnitude function, these results being an improvement over some previously published ones. More precisely, if at some\omegathe value of|W(j\omega)/W(0)|is known, thereby determining\deltathrough|W(j\omega)|^2 = |W(0)|^2 (1-\delta), then lower bounds on|W(j\eta\omega)/W(0)|^2are determined for\eta = 2, 3,4 \cdotsand for\delta> 1. Stronger results are then established for system functions whose impulse responses are monotonically decreasing. The strengthening of these results resides in the fact that\etaassumes continuous values with\eta\gt; 1 rather than the discrete integer values of the previous case.