Discrete-time Fourier transform

About: Discrete-time Fourier transform is a research topic. Over the lifetime, 5072 publications have been published within this topic receiving 144643 citations. The topic is also known as: DTFT.

Estimation of the acoustic field directionality by use of planar and volumetric arrays via the Fourier series method and the Fourier integral method

Abstract: The possibility of estimating the directionality of a stationary homogeneous noise field, directly from the element outputs of a line array, is investigated and found to be feasible for large arrays without encountering ill conditioning This technique, called inverse beamforming for historical reasons, is applicable to line and planar as well as volumetric arrays, and requires no more than two‐dimensional fast Fourier transforms (FFTs) for its realization Derivations and results are presented both for a Fourier series method and a Fourier integral method

Quasi‐Born Fourier migration

Abstract: Summary The Born approximation of the Lippman–Schwinger equation has recently been used to implement a recursive method for seismic migration of pressure wavefields. This Born-based method is stable only when the scattering from heterogeneities within an extrapolation depth interval is weak. To handle strong scattering accurately and efficiently, we propose a quasi-Born approximation of the Lippman–Schwinger equation to extrapolate pressure wavefields downwards recursively. We assume that the scattered wavefield is linearly related to the incident wavefield by a scalar function that varies slowly with lateral position within an extrapolation depth interval. The extrapolation is implemented as a dual-doma in procedure in the frequency–space and frequency–wavenumber domains. Fast Fourier transforms are used to transform data between these two domains. The quasi-Born-based depth-migration algorithm is termed the quasi-Born Fourier method. It can efficiently produce good-quality images of complex structures with strong lateral perturbations of slowness. It is stable for strong scattering and can accurately handle scattering and wave propagation along directions at large angles from the main propagation direction. Image quality obtained using the new method is similar to that of a dual-domain migration method that uses the Rytov approximation within each extrapolation depth interval, but the computational speed of the new method is approximately 27 per cent faster than the latter method for pre-stack migration of an industry standard data set—the Marmousi data set. Compared to the Born-based migration method, the quasi-Born Fourier method is slightly less efficient because it requires an additional multiplication and an additional division for each lateral gridpoint in each step of wavefield extrapolation. For weak scattering, the quasi-Born Fourier method converges to the Born-based method. To improve the efficiency of the quasi-Born Fourier method further without losing its accuracy, we propose a hybrid Born/quasi-Born Fourier method in which the Born-based method is used when the scattering within an extrapolation depth interval is weak, and the quasi-Born Fourier method is used for other cases. This hybrid method is approximately 32 per cent faster than the Rytov-based method for the pre-stack depth migration of the Marmousi data set, while the images obtained using both methods have almost the same quality.

Optimal treatment of diffraction coordinates in wave packet scattering from surfaces

Abstract: In the context of wave packet methodology we show how to take advantage of the diffractive scattering symmetry arising when the incident beam is normal to the surface or to a surface principal axis. This may lead to a reduction in dimensionality being up to a factor of 8. The Fourier transformation is applied to evaluate the translational kinetic energy operator. Two alternative treatments are possible depending on whether the transformation is utilized to calculate the kinetic energy matrix elements in coordinate space, or whether it is applied to the wave function itself to switch between coordinate and momentum representations. The first approach is similar to the discrete variable representation treatment in the spirit of Light and co‐workers whereas the second one enables the use of the fast Fourier transform (FFT) scheme of Kosloff and Kosloff. We provide a detailed comparison between the two approaches as a function of the size of the grid, with and without the presence of symmetry in the diffracti...

Range and error analysis for a fast Fourier transform computed over Z[{omega}]

Abstract: A range and error analysis is developed for a discrete Fourier transform (fast Fourier transform) computed using the ring of cyclotomic integers. Included are derivations of both deterministic and statistical upper bounds for the range of the resulting processor and formulas for the ratio of the mean square error to mean square signal, in terms of the pertinent parameters. Comparisons of theoretical predictions with empirical results are also presented.

Selective deconvolution: A new approach to extrapolation and spectral analysis of discrete signals

Abstract: This paper describes a new algorithm useful for extrapolation and Fourier analysis of discrete signals that are given by a relative small number of samples. The extrapolation is based on the assumption that the discrete Fourier spectrum shows dominant spectral lines. Involving only FFT, the iterative algorithm is not restricted to one-dimensional signals but can also be applied to higher-dimensional problems. Additional knowledge on the signal like band-limitedness or positivity can easily be taken into account.