Lecture 3

# Lecture 3 - ECE52 Spring 11 Lecture 3 1/19/11 HW2 posted...

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1 ECE52 Spring 11 Lecture 3 1/19/11 HW2 posted – due Monday Website protected area – user: ece52 password: digital PC Viruses: 25 years old!

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2 We said “one set of axioms” Unique choice? NO! For example, Huntington’s postulates take the Commutative and Distributive properties as axioms, along with x·0=0, x+1=1, x·x’=0,x+x’=1; the rest can be derived from these A Modern Algebra course in math gets into this! 2-valued Boolean algebra is the near trivial case of an entire family of n-valued algebras
3 Synthesis Building a circuit to realize a function preferably a low-cost circuit, but definitely a correct one! many many ways to implement a given function

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4 Figure 2.15. A function to be synthesized. In canonical sum-of-products form , we could (and indeed must) write the expression f = x 1 ’x 2 ’ + x 1 ’x 2 + x 1 x 2 Sum-of-products : writing down those combinations for inputs for which f = 1 as product terms – summing all combinations that should be 1 Canonical : every input combination causing f=1 appears exactly once! Can be a universal reference expression – every truth table has exactly one canonical sum-of-products expression associated with it
5 f (a) Canonical sum-of-products f (b) Minimal-cost realization x 2 x 1 x 1 x 2 Figure 2.16. Two implementations of a function in Figure 2.15. Cost: number of gates + number of inputs a) 17 b) 5

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6 How to minimize the function? f = x1’x2’ + x1’x2 + x1x2 canonical SOP = x1’(x2’+x2) + x1x2 distributive = x1’ + x1x2 x2’+x2 = 1 = x1’+x2 Great Unnamed Simplification Theorem!
7 Minterms SOP form is so powerful we have shorthand ways of working with it –

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## This note was uploaded on 07/05/2011 for the course ECE 52 taught by Professor Dr.jonathanboard during the Spring '11 term at Duke.

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Lecture 3 - ECE52 Spring 11 Lecture 3 1/19/11 HW2 posted...

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