Correlation of WID Variations for Intratier Buffers

In this appendix, the uncorrelated WID variations and multilevel correlation model are used to characterize the skew distribution for several 3-D clock trees. The correlation between WID variations of buffers i and j within one tier (intratier buffers) is characterized by corr(i, j). Two types of correlations are considered, uncorrelated and multi-level spatial correlation. In the former type, WID variations of the buffers within one tier are considered, independent of the other tiers. For any pair of buffers i and j, corr(i,j) = 0. Consequently, from (17–37), the standard deviation of the delay of sink u due to WID variations is

${\sigma }_{D_u^{\mathrm{WID}}}^2=\sum _{j=1}^{n_u}{\sigma }_{d_i^{\mathrm{WID}}}^2.$ (E.1)

(E.1)

Alternatively, the standard deviation is described differently if spatial correlations are considered. Based on the multilevel correlation model described in [640], a multi-level quad-tree partition is used, and the intra-die variations of a device are divided into l levels, as illustrated in Fig. E.1. At the l^th level, there are 4^l−1 regions. An independent variable is assigned to each region to represent a component of the WID variations of a device. The overall WID variations of buffer k are composed of the sum of these independent components at different levels,

$\mathrm{\Delta }{L}_{\mathrm{WID},k}=\sum _{\mathrm{region}r\mathrm{intersects}k}^{1\le i\le l}\mathrm{\Delta }{L}_{i,r},$ (E.2)

(E.2)

where ΔL_i,r is the random variable associated with the quad-tree at level i, region (i,r). The distribution of ΔL_WID,k is captured by the elementary circuit shown in Fig. 17.6. This distribution is obtained by assigning the same probability distribution to all of the random variables associated with a particular level, and by dividing the total intra-die variability among the different levels. Consequently, the spatial correlation among devices in the same tier is modeled to ensure that devices located close to each other are highly correlated, while those devices located at a large horizontal distance from each other exhibit low correlation (Fig. E.1).

The correlation between the WID variations of buffers i and j is described by the sum of the correlations at all of the levels,

$\mathrm{corr}\left(i,j\right)=\frac{1}{l}\sum _{k=1}^l{\mathrm{corr}}_k\left(i,j\right),$ (E.3)

(E.3)

where corr_k(i,j) is the correlation between buffers i and j at the k^th level. As illustrated in Fig. E.1, assuming buffers i and j are located, respectively, in zones (k, region_i) and (k, region_j),

${\mathrm{corr}}_k\left(i,j\right)=\{\begin{array}{ll}1,\hfill & \mathrm{if}\left(k,{\mathrm{region}}_i\right)=\left(k,{\mathrm{region}}_j\right)\hfill \\ 0,\hfill & \mathrm{if}\left(k,{\mathrm{region}}_i\right)\ne \left(k,{\mathrm{region}}_j\right).\hfill \end{array}$ (E.4a and b)

(E.4a and b)