chap3 - ELEC151 Digital Circuits and Systems ELEC151...

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ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-1 Lecture Note #3 Gate-Level Minimization • Karnaugh map or K-map 3-1, 3-2, 3-3 – 2-6 variables • Two-level combinational logic 3-4, 3-5 – Sum of product, product of sum • Multi-level combinational logic 3-6 – NAND or NOR network • Other TTL building blocks 3-7 – AND-OR-Inverter (AOI) gate • Exclusive-OR/NOR functions 3-8 – odd/even function, equality/in-equality • Reading Assignments: – Chapter 3 ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-2 Logic Minimization • Design Sequence: – 1. Truth Table (input and output relations) » Derive truth table from project specifications 2. Boolean Expressions » Derived from the truth table 3.Logic Minimization » Use Laws of Boolean Algebra to reduce the complexity of Boolean expressions while maintaining the same function –2 Graphic Method (K-Map) (Chapter 3 ) » Re-express the truth table in graphs (K-Map) – 3’ Simplified Boolean Expressions » READ the simplified Boolean expressions from the graphs – 4.Digital Circuit » Map the simplified Boolean expressions to digital circuits – 5.Waveforms » CAD tool simulation » Experiments ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-3 A B 0 1 0 1 0 1 2 3 0 1 2 3 6 7 4 5 AB C A B 00 01 11 10 0 1 2- variable K-map 3- variable K-map Karnaugh Map Method • Truth table and Karnaugh Map (K-Map) – Truth table: input/output relation in one-dimensional table – K map: input/output relation in two-dimensional graph • K-map helps visualize adjacencies in up to 6 variables – but it is hard to draw cubes of more than 4 variables • Beyond that, computer-based methods are needed • Adjacency is arranged in 2-bit Gray code sequence – 00, 01, 11, 10 – only one-bit change from one code word to the next code word In the K-map, adjacency wraps from left to right and from top to bottom In the K-map, adjacency wraps from left to right and from top to bottom ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-4 Alternative K-Map Expressions m 1 m 2 m 3 m 4 m 5 m 6 m 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 Minterms m 0 0 1 2 3 6 7 4 5 AB C A B 00 01 11 10 0 1 0 4 1 5 3 7 2 6 BC A B C 00 01 11 10 0 1 3-variable truth table and K-Maps There is an equal opportunity to use either expression. The textbook uses the right one There is an equal opportunity to use either expression. The textbook uses the right one 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 AB CD A 00 01 11 10 00 01 11 10 C B D 0 4 12 8 1 5 13 9 3 7 15 11 2 6 14 10 CD AB A 00 01 11 10 00 01 11 10 C B D 4- variable
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ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-5 K-Map Examples F = A A=1 or A column B varies, B + B’ = 1 G = B' B=0 or B’ row Cout = A B + B Cin + A Cin carry output of the full adder F(A,B,C) = A A B 0 1 0 1 0 1 0 1 A B 0 1 1 1 0 0 0 1 A B A B Cin 00 01 11 10 0 1 0
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This note was uploaded on 10/16/2011 for the course ELEC 308,315,10 taught by Professor Prof.shenghuisong during the Spring '11 term at CUHK.

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chap3 - ELEC151 Digital Circuits and Systems ELEC151...

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