Page 239 - System on Package_ Miniaturization of the Entire System

Mixed-Signal (SOP) Design 213

Probabilistic Diagnosis
As a result of the statistical variations in the process parameters during batch fabrication,
some embedded circuits may display unacceptable variations in performance. For
parametric defects, the information extracted from the aforementioned statistical
analysis can be utilized as a diagnosis tool. Using the diagnosis methodology, the most
probable layout parameters causing unacceptable variations in performance can be
systematically searched.
For explaining the diagnosis approach, let [X] and [Y] be the random vectors for m
layout parameters and n performance metrics, respectively. The functional relation
m
n
between [X] and [Y] (R → R ) is obtained by sensitivity simulations explained in the
previous section. If n is less than m, then a unique solution of [X] does not exist for a
measured set of unacceptable performance [Y] . Hence, the real parameter(s) causing
the failure cannot be decided. However since all design parameters are associated with
PDFs, the most probable solution can be searched. The conditional PDF of the parameter
vector [X] for measured performance y is defined as

fX Y)
(,
fX Y = y) = (4.25)
(|
fY ()
where f(X, Y) is the joint PDF of the random vector of the design parameters and
T T
performance measures [X Y ] . In Equation (4.25), f(y) is the joint PDF of the performance
T
which is computed in Equation (4.24). Then, the expected value of f(X|Y = y) is the most
probable parameter set causing the failure.
Let P P 2 ... P ] be the set of unacceptable performance metrics. Equations for
1
nT
Y = [

the performance metrics can be written by subtracting the intercept term from Y as
shown [60g]:
Y = β X + ε
(4.26)
where X is the parameter vector, Y is defined as the performance vector, and b is the
sensitivity coefficient matrix without the intercept terms. The error column e is a
gaussian random vector with a zero mean computed from the approximation errors.
Since X and Y are gaussian random vectors, a new random vector z can be defined as
T T
T
Z = [X Y ] . The PDF of Z is equivalent to the joint PDF of X and Y,
mx1
The expected value of the conditional PDF can be computed as
EX Y = y] = μ + Cov ( X Y ,)[Cov ( Y Y ,)] ( − μ ) (4.27)
−1
Y
[ |
X Y
The diagnosis technique does not give the exact statistical variation of the layout
parameters during batch fabrication, but it captures the dominant variations.
As an example, consider the filter variations in Table 4.9. Let’s assume that after
manufacturing, the filter parameters are measured as min_attn = 1.9933 dB, ripple =
0.6513 dB, and F2 (higher side of 1-dB cutoff frequency) = 2.53 GHz. For this filter, the
center frequency has shifted to a higher frequency and hence does not pass the intended
band. The measured results can now be applied as described in this section to capture
the significant parameter variations. Table 4.10 shows the estimated (from diagnosis
methodology) manufacturing variations based on the measurements. The dominant
parameter variations that cause the shift in the filter response have been captured. The
diagnosis can be improved through additional measurements on the filter response that
have little correlation with each other.