ear equations with scaling factors as unknown variables. A common technique
to solve the system of linear equations is traditional least square minimization
(^-minimization).
This estimation approach can be improved by incorporating the spatial cor-
relations in our framework. We show that SpatialaCorrelations suggest that there
is a basis in which variations in the scaling factors can be represented sparsely.
We specifically consider wavelet bases that can capture spatial correlation effi-
ciently [25,73]. We experimentally determine a wavelet basis that results in the
sparsest representation for variations. Having a sparse representation for varia-
tions, we use compressive sensing technique to efficiently recover scaling factors.
Here, we regularize the objective function of the optimization problem with an
^ɪ-norm term to impose the sparsity on the solution.
The post-silicon characterization also can be improved by adding spatial con-
straints directly to the optimization. The spatial correlation implies that two
spatially close gates approximately follow similar variations. It is not statisti-
cally expected that two nearby gates follow totaly independent variations. Thus,
in the underlying optimization, we penalize the difference among scaling factors
of the nearby gates. The new formulation results in a better estimation of gates
scaling factors. The approach is based on our paper in ISLPED 2008 confer-
ence [62].
Next, we use path delay analysis to characterize the variations in gate de-