Lecture05

# Lecture05 - Announcements ECE 2300 Introduction to Digital...

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Lecture 5: Spring 2010 1 ECE 2300 Introduction to Digital Logic Design More Minimization Logic Functions Lecture 5: 2 • Lab 2 is on Blackboard Announcements Lecture 5: 3 Combinational Logic Outputs depend only on current inputs – Example: Detect if the input is odd and less than ten Lecture 5: Karnaugh Map • Idea : Use combining and covering theorems visually to simplify canonical forms • Multidimensional representation of a truth table • Adjacent cells represent minterms (or maxterms) that differ by one variable – Cyclic encoding along each dimension • At most two variables per dimension 4 Lecture 5: 5 1 1 1 00 01 11 10 00 01 10 WX YZ XZ Combining Theorem in Action W’XYZ W’XY’Z WXY’Z WXYZ + + + W’XZ WXZ + Lecture 5: 6 1 1 1 1 00 01 10 00 01 10 WX YZ X’Z’ Combining Theorem in Action W’X’YZ’ W’X’Y’Z’ WX’Y’Z’ WX’YZ’ + + + W’X’Z’ WX’Z’ + Lecture 5: Tabular Method Also called the Quine-McCluskey method Iterative alternative to Karnaugh Map Can be programmed in software Procedure – Group the minterms according to the number of 1’s – Combine minterms in adjacent groups that differ by only one bit position – Create new groups and repeat until no more combining is possible – Use Prime Implicant Chart to select Prime Implicants 7 Lecture 5: Tabular Method Example Prime number detector Row N 3 N 2 N 1 N 0 F 0 0 0 0 0 0 1 0 0 0 1 1 2 0 0 1 0 1 3 0 0 1 1 1 4 0 1 0 0 0 5 0 1 0 1 1 6 0 1 1 0 0 7 0 1 1 1 1 8 1 0 0 0 0 9 1 0 0 1 0 10 1 0 1 0 0 1 0 1 1 1 12 1 1 0 0 0 13 1 1 0 1 1 14 1 1 1 0 0 15 1 1 1 1 0 F = ! N3N2N1N0 ( 1,2,3,5,7,11,13 ) 8 Lecture 5: Tabular Method Example Minterm 0-cube 2-term 1-cube 4-term 2-cube m1 0001 * m(1,3) 00-1 * m(1,3,5,7) 0--1 m2 0010 * m(1,5) 0-01 * m3 0011 * m(2,3) 001- m5 0101 * m(3,7) 0-11 * m7 0111 * m(3,11) -011 m11 1011 * m(5,7) 01-1 * m13 1101 * m(5,13) -101 Terms with ‘-’ can be combined only if all dashes are in the same position * indicates that the term has been combined with another term Prime implicants are those without an * 9

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Lecture05 - Announcements ECE 2300 Introduction to Digital...

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