Spectrally Accurate Causality Enforcement Using SVD-Based Fourier Continuations for High-Speed Digital Interconnects

Abstract

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.

Time Delay Extraction from Frequency Domain Data Using Causal Fourier Continuations for High-Speed Interconnects

“…For a causal periodic continuation, according to (5), we need Im C(H)(x) to be the Hilbert transform of − Re C(H)(x). It can be shown [4] that this implies α k = 0 for k ≤ 0 in (6). Hence, a causal Fourier continuation has the form…”

“…The following result is true [4]. Theorem : Consider a rescaled transfer function H(x) defined by symmetry on Ω = [−0.5, −a] ∪ [a, 0.5], where a = 0.5 wmin wmax , whose values are available at points x j ∈ Ω, j = 1, .…”

“…is the error due to a causal Fourier series approximation and it decays at least as fast as O(M −k+1 ), k is the smoothness order of H(x), which can be estimated numerically using reconstruction errors with different resolution (see [4]).…”

Causality Enforcement of High-Speed Interconnects via Periodic Continuations

“…The dispersion relations are very important in many areas of physics, science and engineering. Applications in electronics include reconstruction and correction [9] of measured data, delay extraction [6], estimation of optimal bandwidth and data density using causality checking [14] and causality enforcement techniques [10], [13], [1], [3], [2] that is the subject of the present study.…”

“…To improve the smoothness of periodic continuations and, thus, significantly reduce or completely eliminate boundary artifacts caused by using bandwidth limited frequency responses, an idea of approximating the transfer function by Fourier series in an extended domain was proposed in [3]. The approach allows one to obtain extremely accurate approximations of the given function on the original frequency interval.…”