New PDF release: Advanced Model Order Reduction Techniques in VLSI Design

By Sheldon Tan, Lei He

ISBN-10: 0521865816

ISBN-13: 9780521865814

ISBN-10: 1601197292

ISBN-13: 9781601197290

Version order relief (MOR) concepts decrease the complexity of VLSI designs, paving how one can larger working speeds and smaller function sizes. This booklet offers a scientific creation to, and remedy of, the major MOR tools hired normally linear circuits, utilizing real-world examples to demonstrate the benefits and drawbacks of every set of rules. Following a overview of conventional projection-based options, insurance progresses to complicated 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced recognition (TBR) equipment, effective inductance modeling through the VPEC version, and structure-preserving MOR strategies. the place attainable, numerical equipment are approached from the CAD engineer's point of view, warding off advanced arithmetic and permitting the reader to tackle actual layout difficulties and advance more suitable instruments. With sensible examples and over a hundred illustrations, this e-book is acceptable for researchers and graduate scholars of electric and machine engineering, in addition to practitioners operating within the VLSI layout undefined.

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Example text

Next we perform SVD on X . 7) where Σ and Σk are diagonal matrices with singular values in a decreasing order along the diagonal and Σk is a k × k diagonal matrix. As a result, we have the eigendecomposition of the response Gramian as X X T = U ΣΣT U T ≈ Uk Σk ΣTk UkT . 8) Let x(t) ≈ Uk η(t), η(t) ∈ Rk ; then we have the following reduced system in terms of η(t), η(t) ˙ = Uk T f (Uk η(t), u(t)), y(t) = h(Uk η(t), u(t)). 10) Thus the approximation of the state x(t) uses a low-dimensional space that is spanned by the k leading columns of U , which essentially are the dominant eigen vectors of the response Gramian X X T .

The projection matrix is then used to reduce the original circuit matrices by congruence transformation, which ensures that the reduced system is passive (thus stable) if original matrix are formulated properly [63, 85]. By using this method, we only require q moments to find q poles. Such a method guarantees that all the poles computed are stable (less than zero in their real part) [63, 85] owing to the nature of congruence transformation and the MNA formulation of the original RLC circuit matrices.

2 The singular value decomposition (SVD) In this section, we briefly review the singular value decomposition method, as it is the key reduction step used in this chapter. 1) T T where Um×m and Vn×n are orthogonal matrices, Um×m Um×m = I and Vn×n Vn×n = I. Σ = diag(σ1 , σ2 , . . , σmin(m,n) ), σi is called singular values: σi = λi (AT A) ≥ σi+1 and σ1 ≥ σ2 ≥ · · · ≥ σmin(m,n) . 1 (Schmidt–Mirsky, Eckart–Young). 3) where {ui } and {vi } are the left and right singular vectors respectively. 3) reflects the fact that the rank-k approximation matrix Ak is just σk+1 away from original matrix A in terms of the 2-norm distance.

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Advanced Model Order Reduction Techniques in VLSI Design by Sheldon Tan, Lei He


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