Once the worst case conditions of the selected performance objective
have been calculated, the inverse problem that consists of determining
the input variables values that will produce these conditions needs to
be solved. As an example, for the 
corner the statistical input variables
must be determined such as the current ratio satisfies the equation:
This is an under-determined problem that could have an infinite number of solutions. In [24], a method to resolve this uncertainty is proposed. It is suggested that the most probable input vector with minimum probability distance be selected. This corresponds to a maximum value of the multinormal jpdf of the input variables which is reached when the argument of the exponential in (2.63) is at its minimum:
where X is the vector of statistical variables with mean 
and
covariance matrix C:
The function 
denotes the empirical relationship represented in
Equation 6.7. The above problem can be solved by nonlinear programming
to determine the values of the statistical input variables at the corners.
For example, Table 6.3 lists the input vectors for the and 
corners as determined by the maximum probability method.
Figure 6.2:
Simulated 
and 
distributions for a 
m CMOS
technology.
Table 6.3: Values of the statistical input variables at the FS and SF corners.
Using the above procedure, the values of the principal statistical
variables that will result in the performance function limits can be
determined. A complete specification of the design process variation space
becomes possible. This is illustrated for and
in Fig. 6.2 where the center and corners are superimposed on the
simulated distribution.
It is important to note that an identical approach can be applied to other circuit level performance functions in case the use of the P- and N-channel saturation currents is not an adequate objective function.
