SI232 Slide Set #6: Digital Logic (Appendix B) 1 2 Appendix - PowerPoint PPT Presentation

ADMIN • Very different material! • Reading – Appendix: Read B.1, B.2, B.3. Skim B.5. SI232 Slide Set #6: Digital Logic (Appendix B) 1 2 Appendix Goals Logic Design – Digital Signals Establish an understanding of the basics of • Only two valid, stable values – False = logic design for future material – True = • Gates • Vs. voltage levels – Basic building blocks of logic – Low voltage “usually” • Combinational Logic – High voltage “usually” – But for some technologies may be the reverse – Decoders, Multiplexors, PLAs • How can we make a function with these signals? • Clocks 1. Specify equations: • Memory Elements • Finite State Machines 2. Implement with 3 4

Boolean Algebra Gates • One approach to expressing the logic function • Operators: – NOT x = A Output true if – AND: ‘A logical product’ x = A • B = AB Output true if x = A + B – OR : ‘A logical sum’ Output true if – XOR x = A ⊕ B Output true if – NAND x = A • B Output true if – NOR x = A + B Output true if 5 6 Example Truth Tables Part 1 • Alternative way to specify logical functions • List all outputs for all possible inputs A(1) – n inputs, how many entries? B(1) – Inputs usually listed in numerical order G x = A x = A + B C(0) A x A B x D(1) 0 1 0 0 0 1 0 0 1 1 1 0 1 Equation: 1 1 1 7 8

Truth Tables Part 2 Exercise #1 • Show the truth table for NAND and NOR gates • Not just for individual gates • Not just for one output NOR A NAND F B G C A B x A B x A B C F G 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 9 10 Exercise #2 Exercise #3 • A.) Show the truth table for the following logic circuit • Draw a circuit for the following formula: F = ( (A + B) C ) + D A B C y A B y C • B.) Write the Boolean equation for this circuit. 11 12

Exercise #4 – For more thought Laws of Boolean Algebra • Recall – how many entries are in a truth table for a function with n • Identity Law A + 0 = A A • 1 = A inputs? • Consider – how many different truth tables are possible for a function with n inputs? • Zero and One Law + 1 = 1 A A • 0 = 0 + A = 1 A A • A = 0 • Inverse Law • = • A B B A • Commutative Law A + B = B + A 13 14 Laws of Boolean Algebra • Associative Law A + ( B + C ) = ( A + B ) + C • ( • ) = ( • ) • A B C A B C • Distributive Law A • ( B + C ) = ( A • B ) + ( A • C ) + ( • ) = ( + ) • ( + ) A B C A B A C • DeMorgan’s Law A + B = A • B A • B = A + B 15