Abstract-We introduce an accurate and robust technique for accessing causality of network transfer functions given in the form of bandlimited discrete frequency responses. These transfer functions are commonly used to represent the electrical response of high speed digital interconnects used on chip and in electronic package assemblies. In some cases small errors in the model development lead to non-causal behavior that does not accurately represent the electrical response and may lead to a lack of convergence in simulations that utilize these models. The approach is based on Hilbert transform relations or KramersKrönig dispersion relations and a construction of causal Fourier continuations using a regularized singular value decomposition (SVD) method. Given a transfer function, non-periodic in general, this procedure constructs highly accurate Fourier series approximations on the given frequency interval by allowing the function to be periodic in an extended domain. The causality dispersion relations are enforced spectrally and exactly. This eliminates the necessity of approximating the transfer function behavior at infinity and explicit computation of the Hilbert transform. We perform the error analysis of the method and take into account a possible presence of a noise or approximation errors in data. The developed error estimates can be used in verifying causality of the given data. The performance of the method is tested on several analytic and simulated examples that demonstrate an excellent accuracy and reliability of the proposed technique in agreement with the obtained error estimates. The method is capable of detecting very small localized causality violations with amplitudes close to the machine precision.
We study the limited data problem of the spherical Radon transform in two and three dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k + 1 2 orders smoother than the generating singularity. Our analysis for three dimensional space contains an important idea of lifting up a space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function might lead to a significant improvement of reconstruction quality.
This study considers the nonlinear dynamics of stratified immiscible fluids when an electric field acts perpendicular to the direction of gravity. A particular setup is investigated in detail, namely, two stratified fluids inside a horizontal channel of infinite extent. The fluids are taken to be perfect dielectrics, and a constant horizontal field is imposed along the channel. The sharp interface separating the two fluids may or may not support surface tension, and the Rayleigh-Taylor instability is typically present when the heavier fluid is on top. A novel system of partial differential equations that describe the interfacial position and the leading order horizontal velocity in the fluid layers is studied analytically and computationally. The system is valid in the asymptotic limit of one layer being asymptotically thin compared to the second fluid layer, and as a result nonlocal electrostatic terms arise due to the multiscale nature of the physical setup. The initial value problem on spatially periodic domains is solved numerically, and it is shown that a sufficiently strong electric field can linearly stabilize the Rayleigh-Taylor instability to produce nonlinear quasiperiodic oscillations in time that are quite close to standing waves. In situations when the instability is present, the system is shown to generically evolve to touch-up singularities with the interface touching the upper wall in finite time while the leading order horizontal velocity blows up. Accurate numerical solutions allied with asymptotic analysis show that the terminal states follow self-similar structures that are different if surface tension is present or absent, but with the electric field present. In the presence of surface tension, the touch-up is found to take place with bounded interfacial gradients but unbounded curvature, with electrostatic effects relegated to higher order. If surface tension is absent, however, the electric field supports touch-up with a local cusp structure so that the interfacial gradients themselves are unbounded. The self-similar solutions are of the second kind and extensive simulations are used to extract the scaling exponents. Distinct and independent methods are described and implemented, and agreement between them is excellent.
We present a method for checking causality of band-limited tabulated frequency responses. The approach is based on Kramers-Krönig relations and construction of periodic polynomial continuations. Kramers-Krönig relations, also known as dispersion relations, represent the fact that real and imaginary parts of a causal function form a Hilbert transform pair. The Hilbert transform is defined on an infinite domain, while, in practice, discrete values of transfer functions that represent high-speed interconnects are available only on a finite frequency interval. Truncating the computational domain or approximating the behavior of the transfer function at infinity causes significant errors at the boundary of the given frequency band. The proposed approach constructs a periodic polynomial continuation of the transfer function that is defined by raw frequency responses on the original frequency interval and by a polynomial in the extended domain, and requires the continuation to be periodic on a wider domain of a finite length and smooth at the boundary. The dispersion relations are computed spectrally using fast Fourier transform and inverse fast Fourier transform routines applied to periodic continuations. The technique does not require the knowledge or approximation of the transfer function behavior at infinity. The method significantly reduces the boundary artifacts that are due to the lack of out-of-band frequency responses, and is capable of detecting small, smooth causality violations. We perform the error analysis of the method and show that its accuracy and sensitivity depend on the smoothness and accuracy of data and a polynomial continuation. The method can be used to verify and enforce causality before the frequency responses are employed for macromodeling. The performance of the method is tested on several analytic and simulated examples.
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