Civil Engineering Reference
In-Depth Information
rewritten using Equation (8.25) to depend purely on the
D
variables. In ad-
dition‚ topological information must be incorporated: specifically‚ the slacks
in the network are organized into a graph in which slacks may be allocated
to gates. This results in a network flow formulation that can be solved to
find the delay budgets.
The W-phase
computes the optimal transistor sizes (“W”) corresponding
to these delay budgets by formulating the problem as a simple monotonic
program [Pap98].
The CPU times provided by this method are‚ at the time of writing this topic‚
the fastest demonstrated results for the transistor sizing problem.
8.7
GENERALIZED POSYNOMIAL DELAY MODELS FOR
TRANSISTOR SIZING
8.7.1 Generalized posynomials
A major problem with the use of Elmore-based delay models is the limited ac-
curacy that can be achieved by these models. Particularly in deep-submicron
technologies and later‚ Elmore models leave much to be desired in terms of
accuracy‚ although their convexity properties are well-beloved. As a result‚ the
transistor sizing techniques described in Section 8.5.3 willingly abandoned the
convexity properties of the Elmore model in favor of accurate‚ but nonconvex
models. The idea of generalized posynomials for delay modeling was proposed
in [KKS00] in an effort to provide accurate convex gate delay models for tran-
sistor sizing. The philosophy behind the approach is that posynomials and
convex functions are a rich class of functions‚ and as described in Section 8.3.2‚
the posynomials that are used in Elmore-based delay models constitute a very
restricted set where the exponents for each term of the posynomial belong to
the set {
-
1‚ 0‚ 1}. Therefore‚ much of the space of posynomials and convex
functions remains unexploited when Elmore models are used.
A generalized posynomial is defined recursively in [KKS00] as follows.
A generalized posynomial of order 0 is simply a posynomial‚ as defined earlier
in Equation (8.12).
A generalized posynomial of order
is defined as
where
is a generalized posynomial of order less than or equal to
each
R‚ and each
