By Sheldon Tan, Lei He
Version order aid (MOR) recommendations decrease the complexity of VLSI designs, paving how to larger working speeds and smaller function sizes. This publication provides a scientific advent to, and remedy of, the major MOR equipment hired more often than not linear circuits, utilizing real-world examples to demonstrate the benefits and downsides of every set of rules. Following a evaluation of conventional projection-based thoughts, assurance progresses to complicated 'state-of-the-art' MOR tools for VLSI layout, together with HMOR, passive truncated balanced awareness (TBR) equipment, effective inductance modeling through the VPEC version, and structure-preserving MOR suggestions. the place attainable, numerical tools are approached from the CAD engineer's point of view, heading off complicated arithmetic and permitting the reader to tackle actual layout difficulties and advance more beneficial instruments. With functional examples and over a hundred illustrations, this booklet is acceptable for researchers and graduate scholars of electric and computing device engineering, in addition to practitioners operating within the VLSI layout undefined.
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Extra resources for Advanced Model Order Reduction Techniques in VLSI Design
2. 75) where B = L. If G + GT ≥ 0 and C = C T ≥ 0 and G + sC is invertible at least at one point s with Re(s) > 0, then the following matrix transfer function is positive real H(s) = B T (G + sC)−1 B. 85) The proof of the theorem can be found in . , W (s)H = G + GT + 2δC at s = δ + jω is positive semideﬁnite. The inverse of the matrix and congruence transformation also do not change the property of positive semideﬁnite. After we perform the model order reduction on both G and C respectively, the reduced system is still passive.
51) where superscript H denotes the Hermitian transpose. We now consider evaluating X by applying numerical quadrature to the above equation, given a quadrature scheme with nodes ωk and weights ωk , and deﬁning zk = (jωk I − A)−1 B. 48) under impulse responses. For general POD methods, we can apply typical inputs to ˆ to X can then be the system and obtain the responses. 53) k which can be viewed as the response matrices’ Gramian. Let Z be a matrix whose columns are zk , and W a diagonal matrix with diagonal √ entries Wkk = wk .
77) where G11 = Ag GATg C11 = Ac CATc C22 = L0 . 78) 28 Projection-based model order reduction algorithms Notice that G11 and C11 are symmetric and positive semideﬁnite. 79) The resulting G and C become positive semideﬁnite, although they are no longer symmetric. 80) where xT1 G11 x1 ≥ 0 means that xT1 G11 x1 is positive semideﬁnite. It will be shown in a later subsection that this property is important for passive model order reduction. 81) with B = L. As a result, G and C are symmetric and positive semideﬁnite.
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