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ITEC 352

ITEC 352. Lecture 4 Boolean logic / Karnaugh Maps. Review. Truth tables Conversion between Multiplexers. Outline. Multiplexers Math Karnaugh Maps Minterm / Maxterm Relationship to gates. Multiplexers. Purpose? Construction using gates. Demultiplexer.

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ITEC 352

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  1. ITEC 352 Lecture 4 Boolean logic / Karnaugh Maps

  2. Review • Truth tables • Conversion between • Multiplexers

  3. Outline • Multiplexers • Math • Karnaugh Maps • Minterm / Maxterm • Relationship to gates

  4. Multiplexers • Purpose? • Construction using gates

  5. Demultiplexer • Only passes a 1 through if the input is 1

  6. Combination • Why would you combine a multiplexer and a de-multiplexer?

  7. Canonical forms • A Boolean expression can be expressed in a number of ways. • E.g., • S = X’+Y*Z can also be represented as • S = X’ + X’ + Y*Z • Hence,we use canonical forms (or normalized forms) that are unique in representing in expressions. • Note, they need not be minimal – they are just unique. • Two popular canonical forms: • Sum of products and Product of sums.

  8. Minterms • Minterm: Boolean function that is 1 in only one row of the truth table. • Example: what are the minterms of function s. • S=X’ +Y*Z • Represent the sum (s) function using minterms.

  9. Maxterms • Maxterm: Boolean function that is 0 in only one row of the truth table. For example, the function: • Example maxterms for the sum function. • S=X’ +Y*Z • Exercise: express F = X + Y' * Z as product of sums (in terms of maxterms)

  10. Class exercise • Implement a majority function: • Given three inputs, the functions equals 1, if the majority of inputs are 1’s else it is a 0. • Steps: • (a) Draw the truth table. • (b) Write down the function in a canonicalized way. • (c) Draw the circuit.

  11. K-Maps • 2 Level Maximum

  12. Construction • Variant of a truth table • Grid format • Allows you to see patterns quickly

  13. Terms • Implicant: a single minterm or group of minterms that can be combined together on the K-map • Prime-implicants: Implicant that can not be combined with another one to remove a literal • Essential prime implicants: prime implicant that includes a minterm not covered by any other prime implicant

  14. Exercise • Reduce the following Boolean expression using a Karnaugh Map: • F = A’BC + AB’C + ABC + ABC’

  15. Review • A few mathematical terms • Minterm • Maxterm • Visualization technique for Boolean expressions

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