Theory of nonstationary linear filtering in the Fourier domain with application to time variant filtering
Gary F. Margrave
A general linear theory is presented which describes the extension of the convolutional method to nonstationary processes. Two alternate extensions are explored. 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 show 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.
The theory is then recast into the Fourier domain where it is shown that stationary filters correspond to a multiplication of the input signal spectrum by a diagonal filter matrix while nonstationary filters generate off-diagonal terms in the filter matrix. The width of significant off-diagonal power is directly proportional to the degree of nonstationarity. Both nonstationary convolution or combination may be applied in the Fourier domain, and for quasi-stationary filters, efficiency is improved by using sparse matrix methods.
Unlike stationary theory, a third domain which combines time and frequency is also possible. Here, nonstationary convolution expresses as a generalized forward Fourier integral of the product of the nonstationary filter and the time domain input signal. The result is the spectrum of the filtered signal. Nonstationary combination reformulates as a generalized inverse Fourier integral of the product of the spectrum of the input trace and the nonstationary filter which results in the time domain output signal. The mixed domain is an ideal domain for filter design which proceeds by specifying the filter as an arbitrary complex function on a time-frequency grid. Explicit formulae are given to move nonstationary filters expressed in any one of the three domains into any other.