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chap3(4_in_1) - ELEC151 Digital Circuits and Systems...

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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, 10-3, 10-5 – AND-OR-Inverter (AOI) gate Exclusive-OR/NOR functions 3-8 – odd/even function, equality/in-equality Hardware Description Language (HDL) postpone Reading Assignments: – Chapter 3, Section 10-3, 10-5 ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 3-2 Logic Minimization without Boolean Algebra 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’ Logic Minimization » READ the simplified Boolean expressions from the graphs – 4.Digital Circuit » Map the simplified Boolean expressions to digital circuits – 5.Waveforms » CAD tool simulation and 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 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 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 K-map
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