Nonstationary filtering: review and update

Gary F. Margrave

A nonstationary generalization of the convolution integral is presented. Called nonstationary convolution, it retains the interpretation of forming the scaled superposition of impulse responses while allowing those impulse responses to become arbitrary functions of time or position. A similar, alternate formulation is also given which also has stationary convolution as a limiting form but does not have the immediate interpretation of forming the superposition of impulse responses. Called nonstationary combination, this alternate form is closely approximated by the commonpractice of forming a nonstationary result by interpolation between a set of stationary filtered results. Both filter forms can be re-expressed in a dual time-frequency domainwhere nonstationary convolution becomes a generalized forward Fourier integral and combination is a generalized inverse Fourier integral. It is shown that pseudodifferential operators can be considered as a nonstationary combination filter whose filter form is a spectral polynomial. It is then argued that nonstationary convolution can be inverted by inverting the dual-domain filter function and applying it as a nonstationary combination and vice-versa. Finally both nonstationary filter forms are re-expressed in the full Fourier domain in a result which generalizes the convolution theorem. The possible applications of this methodology are illustrated with examples from wave propagation and deconvolution.