Timing analysis with crosstalk is a fixpoint on a complete lattice

Increasing delay variation due to capacitive and inductive crosstalk has a dramatic impact on deep submicron technologies. It is now impossible to exclude crosstalk from timing analysis. However, timing analysis with crosstalk is a mutual dependence problem since the crosstalk effect in turn depends on the timing behavior of a circuit. In this paper, we establish a theoretical foundation for timing analysis with crosstalk. We show that solutions to the problem are fixpoints on a complete lattice. Based on that, we prove in general the convergence of any iterative approach. We also show that, starting from different initial solutions, an iterative approach will reach different fixpoints. The current prevailing practice, which starts from the worst case solution, will always reach the greatest fixpoint, which is the loosest solution. In order to reach the least fixpoint, we need to start from the best case solution. The convergence rates for both discrete and continuous models are discussed. Based on chaotic iteration and heterogeneous structures of coupled circuits, techniques to speed up iterations are also provided.