Advanced Symbolic Analysis for VLSI Systems: Methods and by Guoyong Shi, Sheldon X.-D. Tan, Esteban Tlelo Cuautle

By Guoyong Shi, Sheldon X.-D. Tan, Esteban Tlelo Cuautle (auth.)

This publication offers finished assurance of the new advances in symbolic research thoughts for layout automation of nanometer VLSI platforms. The presentation is geared up in elements of basics, easy implementation tools and purposes for VLSI layout. subject matters emphasised contain statistical timing and crosstalk research, statistical and parallel research, functionality certain research and behavioral modeling for analog built-in circuits. one of the contemporary advances, the Binary choice Diagram (BDD) dependent methods are studied intensive. The BDD-based hierarchical symbolic research techniques, have basically damaged the analog circuit measurement barrier.

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That is, det(A(k, b)) , (9) xk = det(A) where A is the coefficient matrix of Eq. (8) and A(k, b) is the matrix resulting from replacing the kth column of A by the vector b, the right-hand side of Eq. (8). In symbolic network analysis, it is usually sufficient to derive one or two unknown components corresponding to the output. Therefore, it is feasible to use the Cramer’s rule for symbolic network analysis. According to Eq. (9), the symbolic solution boils down to finding symbolic expressions for two determinants det(A(k, b)) and det(A), which have all columns equal but one.

The determinant det(Aar,c ) is called the minor of det(A) with respect to ar,c . A determinant also can be expanded along one row or one column, known as Laplace expansions: n det(A) = ar,c (−1)r+c det(Aar,c ), (10) ar,c (−1)r+c det(Aar,c ). 2 Cramer’s Rule Cramer’s rule is the foundation for deriving analytical solution to a system of linear equations. Given an n × n system Ax = b, ⎡ a1,1 a1,2 ⎢ a2,1 a2,2 ⎢ ⎢ .. ⎣ . an,1 an,2 ⎤⎡ ⎤ ⎡ ⎤ b1 x1 . . a1,n ⎢ x2 ⎥ ⎢ b2 ⎥ . . a2,n ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ = ⎢ .

4 Mathematical Concepts and Notation Some basic mathematical concepts and notation, mainly in linear algebra, are summarized in this section for reference. 1 Matrix, Determinant, and Cofactor Let I = {1, . . , n} be a set of integers. Let S = {a1 , . . , am } denote a set of m symbolic parameters or simply symbols, where 1 ≤ m ≤ n2 . When a symbol appears at the rth row and the cth column of an n × n matrix A, where r, c ∈ I, this element is denoted by ar,c . We sometimes use r(a) and c(a) to denote respectively the row and column indices of an element a = ar,c in the matrix A: ⎡ a1,1 ⎢ a2,1 A=⎢ ⎣ ...

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