Theory of nonstationary linear filtering in the Fourier domain with application to time‐variant filtering

Abstract

A general linear theory describes the extension of the convolutional method to nonstationary processes. This theory can apply any linear, nonstationary filter, with arbitrary time and frequency variation, in the time, Fourier, or mixed domains. The filter application equations and the expressions to move the filter between domains are all ordinary Fourier transforms or generalized convolutional integrals. Nonstationary transforms such as the wavelet transform are not required. There are many possible applications of this theory including: the one‐way propagation of waves through complex media, time migration, normal moveout removal, time‐variant filtering, and forward and inverse Q filtering. Two complementary nonstationary filters are developed by generalizing the stationary convolution integral. The first, called nonstationary convolution, corresponds to the linear superposition of scaled impulse responses of a nonstationary filter. The second, called nonstationary combination, does not correspond to such a superposition but is shown to be a linear process capable of achieving arbitrarily abrupt temporal variations in the output frequency spectrum. Both extensions have stationary convolution as a limiting form and, in the discrete case, can be formulated as matrix operations. Fourier transformation shows that both filter types are nonstationary filter integrals in the Fourier domain as well. This result is a generalization of the convolution theorem for stationary signals because, as the filter becomes stationary in one domain, the integral in the other domain collapses to a scalar multiplication. For discrete signals, stationary filters are a matrix multiplication of the input signal spectrum by a diagonal spectral matrix, while nonstationary filters require off‐diagonal terms. For quasi‐stationary filters, a computational advantage is obtained by computing only the significant terms near the diagonal. Unlike stationary theory, a mixed domain of time and frequency is also possible. In this context, the nonstationary filter is applied simultaneously with the transform from time to frequency or the reverse. Nonstationary convolution becomes a generalized forward Fourier integral and nonstationary combination is a generalized inverse Fourier integral.