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.

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

The paper considers filters described by linear shift-variant difference (LSV) equations. We present the notion of a generalized transfer function and discuss the frequency characteristic of a shift-variant digital filter in terms of the generalized transfer function. A method is presented for determining LSV difference equations from a certain class of impulse responses, and vice versa. In addition, some properties of the impulse response and the generalized transfer function of a shift-variant system are investigated in the present work.

On linear shift-variant digital filters

Frequency-time analysis (FTAN) results are usually presented as an array of amplitude values associated with discrete intervals group velocity and log period. The display-equalized filter, a modified form of the Gaussian frequency-time filter, produces amplitude estimates for which the frequency-time “averaging region” is of approximately constant shape, although varying in size, throughout the display. The shape of the averaging region can be adjusted to permit greater frequency resolution, with a corresponding loss of time resolution, and vice versa. An improved time-variable filter effectively utilizes the optimum frequency-time resolution of the Gaussian filter.

The display-equalized filter for frequency-time analysis

In this paper, problems associated with the synthesis and implementation of recursive linear shift-variant digital filters are investigated. We propose two techniques to approximate a given impulse response as a degenerate sequence that is realizable as a recursive difference equation. Both techniques use a least squares error criterion to minimize the difference between the given and the approximated impulse responses. Numerical examples illustrating and comparing the results of these techniques are included. In addition, we present several recursive structures for the implementation of both causal and noncausal degenerate impulse responses.

Synthesis and implementation of recursive linear shift-variant digital filters

An opto-electronic oscillator (OEO) scheme which operates at "chirp oscillation" mode and generates low-phase-noise, frequency-swept microwave is proposed and experimentally demonstrated. This frequency-swept OEO is achieved by embedding a rapidly frequency-scanning microwave filter in an opto-electronic cavity. The filter has fixed passband while its center frequency scans rapidly and periodically at cavity round-trip time, covering a large frequency range (~GHz). Experimentally, the generated frequency-swept microwave is linear frequency-modulated continuous wave (FMCW) which centers at 7 GHz with 1-GHz bandwidth. Its instantaneous frequency varies linearly from 6.5 GHz to 7.5 GHz, back and forth, in a period of 12.8 μs, resulting in a frequency scanning rate of ~156 MHz/μs. The single-side-band (SSB) noise of the generated FMCW is -104 dBc/Hz at 10 kHz offset frequency, which is much lower than that from a commercial electronic arbitrary waveform generator (E-AWG). Improvement as large as 23 dB is experimentally reported.

Rapidly and continuously frequency-scanning opto-electronic oscillator.

A linear time‐varying filtering process usually is realized by applying a number of time‐invariant filters to overlapping time regions of a trace and transitionally merging these different regions. We describe the design of a different type of linear time‐varying filter where the frequency spectrum of the filter is shifted along the frequency axis as a function of time without appreciable change in the spectrum shape. The design is based on a given time‐invariant filter with the desired spectrum shape. For long filter length, the design procedure is rather complicated. Furthermore, only a constant rate of frequency shift is possible. However, for many practical situations where the frequency shift over the filter length is much less than the bandwidth of the filter, the process of time‐varying filtering can be further simplified without appreciable frequency error. In fact, time‐varying filtering is achieved through modifying the complex signal representation of the original time invariant filter by a pre...

Keh Pann

Papers

a Class of Convolutional Time-Varying Filters