Transient Simulation of Complex High-Speed Channels via Waveform Relaxation

Macromodel-Based Iterative Solvers for Simulation of High-Speed Links With Nonlinear Terminations

“…The precise definition of the coefficients in (6) in terms of the coefficients of (5) is available in [49]. The DRM formulation (6) is competitive with respect to the direct convolution (4) when the total number of coefficients {α ,m dn , β ,m dn , γ ,m dn } is less than P 2 L. 3) Discrete-Time Channel Operator: Both the direct convolution formulation (4) and the delayed recursive convolution (6) can be cast in the compact operator form…”

“…Earlier WR formulation approaches, as summarized in [49] setup fixed-point iterative schemes to compute the solution of this system by simple forward evaluations of the channel and termination operator. In particular, longitudinal partitioning (LP) approaches start at some initial guess a 0 and apply the two operators sequentially as…”

“…where the last two rows are solved concurrently at the μth iteration, e.g., by an inner WR-LP iteration, leading to the so-called WR-LPTP scheme [49]. In all above cases, the WR scheme can be interpreted as a fixed point iteration that, starting from some initial condition a 0 , produces a sequence of estimates a 0 → a 1 → a 2 → · · · that hopefully converges to the correct solution a * .…”

“…WR-based transient analysis of complex channels (including irregular routing through packages, cards, boards, connectors, and via fields) rather than simpler uniform transmission lines has been proposed in [48] and [49]. Unfortunately, the analysis in [49] shows that when transverse partitioning (TP) is introduced to improve simulation efficiency, the convergence of the WR scheme is only conditional and cannot be guaranteed in all cases, despite some improvements based on possibly frequency-dependent over-relaxation have been documented [50], [51].…”

“…Unfortunately, the analysis in [49] shows that when transverse partitioning (TP) is introduced to improve simulation efficiency, the convergence of the WR scheme is only conditional and cannot be guaranteed in all cases, despite some improvements based on possibly frequency-dependent over-relaxation have been documented [50], [51]. This consideration motivated the approach in [52] and [53], where more robust iterative solvers have been suggested to replace the fixed point iteration typical of basic WR implementations.…”