By Ruijing Shen

Since technique edition and chip functionality uncertainties became extra suggested as applied sciences diminish into the nanometer regime, actual and effective modeling or characterization of adaptations from the gadget to the structure point became valuable for the winning layout of VLSI chips.

This e-book offers readers with instruments for variation-aware layout methodologies and computer-aided layout (CAD) of VLSI structures, within the presence of method adaptations on the nanometer scale. It provides the most recent advancements for modeling and research, with a spotlight on statistical interconnect modeling, statistical parasitic extractions, statistical full-chip leakage and dynamic strength research contemplating spatial correlations, statistical research and modeling for giant international interconnects and analog/mixed-signal circuits.

  • Provides readers with well timed, systematic and finished remedies of statistical modeling and research of VLSI structures with a spotlight on interconnects, on-chip energy grids and clock networks, and analog/mixed-signal circuits;
  • Helps chip designers comprehend the aptitude and obstacles in their layout instruments, bettering their layout productivity;
  • Presents research of every set of rules with useful functions within the context of actual circuit design;
  • Includes numerical examples for the quantitative research and assessment of algorithms awarded.
  • Provides readers with well timed, systematic and accomplished remedies of statistical modeling and research of VLSI platforms with a spotlight on interconnects, on-chip energy grids and clock networks, and analog/mixed-signal circuits;
  • Helps chip designers comprehend the capability and barriers in their layout instruments, enhancing their layout productivity;
  • Presents research of every set of rules with useful purposes within the context of genuine circuit design;
  • Includes numerical examples for the quantitative research and review of algorithms offered.

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15. X2 ; Xn / :: ::: : When X1 ; : : : ; Xn are mutually independent random variables, it can be shown by induction that # " X X Var Xi D VarŒXi : i i And the covariance matrix is a diagonal matrix in this case. 1 Components of Covariance in Process Variation In general, process variation can be classified into two categories [13]: inter-die and intra-die. Inter-die variations are variations from die to die, while intra-die variations correspond to variability within a single chip. Inter-die variations are global variables and, hence, affect all the devices on a chip in the similar fashion.

W1 ; w2 ; :::; wn / is a diagonal matrix of weights. As a result, the covariance matrix of , ˝. / now contains the weight information, and performing PFA on ˝. / leads to the weighted variable reduction. Specifically, we have ˝. W /T D W ˝. 16) and denote its eigenvalues and eigenvectors by i and ei . 14) but uses the weighted eigenvalues i . 5 Principal Component Analysis Technique We first briefly review the concept of principal component analysis (PCA), which is used here to transform the random variables with correlation to uncorrelated random variables [75].

42) Â Ã n 1 is the combinatorial number and w is the weight for the n C P jij corresponding quadrature points. n 1/ ; where NP is the number of quadrature points and r is the order of the maximum derivative that exists Á for the delay function. P /Š . It can be shown that a sparse grid at least with level P is required for an order P representation. The reason is that the approximation contains order P polynomials for both y. / and Hj . /. Thus, there exists y. /Hj . / with order 2P , which requires a sparse grid of at least level P with an exactness degree of 2P C 1.

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