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Chapter 2 - Part 1 - PPT - Mano & Kime - 2nd EdSlide 1 Lecture 9 Multilevel OptimizationLecture 9 Multilevel Optimization
Slide 2 OutlineOutline Multi-level Optimization Transformations Factoring Decomposition Substitution Elimination Extraction
Slide 3 TransformationsTransformations Factoring - finding a factored form from SOP or POS expression Algebraic - No use of axioms specific to Boolean algebra such as complements or idempotence Boolean - Uses axioms unique to Boolean algebra Decomposition - expression of a function as a set of new functions
Slide 4 Transformations (continued)Transformations (continued) Substitution of G into F - expression function F as a function of G and some or all of its original variables Elimination - Inverse of substitution Extraction - decomposition applied to multiple functions simultaneously
Slide 5 Transformation ExamplesTransformation Examples Algebraic Factoring F = + B + ABC + AC G = 16 Factoring: F = ( + B ) + A (BC + C ) G = 16 Factoring again: F = ( B + ) + AC (B + ) G = 12 Factoring again: F = ( + AC) (B + ) G = 10 D C A A C D C C A D D A C D D A C D
Slide 6 Transformation ExamplesTransformation Examples Decomposition F = A C D + A BC + ABC + ACD G = 16 The terms A C + AC and B + D can be defined as new functions H and E respectively, decomposing F = (A C + AC)(B + D ): F = H E, H = A C + AC, E = B + D G = 10 This series of transformations has reduced G from 16 to 10, a substantial savings. The resulting circuit has three levels plus input inverters.
Slide 7 Transformation ExamplesTransformation Examples Substitution of E into F Returning to F just before the final factoring step: F = ( B + ) + AC (B + ) G = 12 Defining E = B + , and substituting in F: F = E + ACE G = 10 This substitution has resulted in the same cost as the decomposition A C D D A C D
Slide 8 Transformation ExamplesTransformation Examples Elimination Beginning with a new set of functions: X = B + C Y = A + B Z = X + C Y G = 10 Eliminating X and Y from Z: Z = (B + C) + C (A + B) G = 10 Flattening (Converting to SOP expression): Z = B + C + AC + BC G = 12 This has increased the cost, but has provided an new SOP expression for two-level optimization. A A A A
Slide 9 Transformation ExamplesTransformation Examples The result of 2-level optimization is: Z = B + C G = 4 This example illustrates that: Optimization can begin with any set of equations, not just with minterms or a truth table Increasing gate input count G temporarily during a series of transformations can result in a final solution with a smaller G A
Slide 10 Transformation ExamplesTransformation Examples Extraction Beginning with two functions: E = + BD H = C + BCD G = 16 Finding a common factor and defining it as a function: F = + BD We perform extraction by expressing E and H as the three functions: F = + BD, E = F, H = CF G = 10 The reduced cost G results from the sharing of logic between the two output functions B A A B D D B D A B D
Slide 11 SummarySummary Multi-level Optimization Transformations Factoring - find a factored form from SOP or POS expression Decomposition - express a function as a set of new functions Substitution - express function F as a function of G and some or all of its original variables Elimination - inverse of substitution Extraction - decompose multiple functions simultaneously
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