Generalization of the Concept of Bandwidth

Abstract

In the sciences and engineering, there is no generally accepted definition of bandwidth beyond those used to characterize low-loss linear time-invariant systems of low order operating at steady-state. In fact, the concept of bandwidth is often subject to interpretation depending upon context and the requirements of a specific community. The focus of this work is to formulate this concept for a general class of passive oscillatory dynamical systems, including but not limited to mechanical, structural, acoustic, electrical, and optical. Typically, the (linearized) bandwidth of these systems is determined by the half-power (-3 dB) method, and the result is often referred to as “half-power bandwidth.” The fundamental assumption underlying this definition is that the system performance degrades once its power decreases by 50%; moreover, there are restrictive conditions, rarely met, that render a system amenable to the use of this approach, such as linearity, low-dimensionality, low-loss, and stationary output. Here the concept of root mean square (RMS) bandwidth is considered, justified by the Fourier uncertainty principle, to generalize the definition of bandwidth to encompass linear/nonlinear, single/multi-mode, low/high loss and time-varying/invariant oscillating systems. By tying the bandwidth of an oscillatory dynamical system directly to its dissipative capacity, one can formulate a definition based solely on its transient energy evolution, effectively circumventing the previous restrictions. Further, applications are given that highlight the limitations of the traditional half-power bandwidth; these include a Duffing oscillator with hardening nonlinearity, and a bi-stable, geometrically nonlinear oscillator with tunable hardening or softening nonlinearity. The resulting energy-dependent bandwidth computations are compatible with the nonlinear dynamics of these systems, since at low energies they recover the (linearized) half-power bandwidth, whereas at high energies they accurately capture the nonlinear physics. Moreover, the bandwidth computation is directly tied to nonlinear harmonic generation in the transient dynamics, so that the contributions to the bandwidth of the individual harmonics and of inter-harmonic targeted energy transfers can be directly quantified and studied. The new bandwidth definition proposed in this work has broad applicability and can be regarded as a generalization of the traditional linear half-power bandwidth which is used widely in the sciences and engineering.