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Lecture 5 Circuits Simplification Cont.

Lecture 5 Circuits Simplification Cont. CSCE 211 Digital Design. Topics Karnaugh Maps Sums-of-Products Form Products-of-Sums Form Readings 4.2-4.3. September 4, 2003. Overview. Last Time Basic Gates Adders: Half-adder, full-adder, ripple-carry adder Boolean Algebra: Axioms, Theorems,

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Lecture 5 Circuits Simplification Cont.

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  1. Lecture 5Circuits Simplification Cont. CSCE 211 Digital Design • Topics • Karnaugh Maps • Sums-of-Products Form • Products-of-Sums Form • Readings 4.2-4.3 September 4, 2003

  2. Overview • Last Time • Basic Gates • Adders: Half-adder, full-adder, ripple-carry adder • Boolean Algebra: Axioms, Theorems, • On last Time’s Slides(what we didn’t get to) • Principle of Duality • N-variable Theorems • New • Combinational Circuit Analysis • Algebraic analysis, Truth tables, Logic Diagrams • Sums-of-Products and Products-of-Sums • Circuit Simplification: Karnuagh Maps • VHDL – Half-Adder

  3. Karnaugh Maps • Tabular technique for simplifying circuits • two variable maps three variable map XY X 0 1 0 1 00 01 11 10 Z Y 0 1 X XY 0 1 Y 0 1 00 01 11 10 Z 0 1

  4. Karnaugh Map Simplification • F(X,Y,Z) = XY 00 01 11 10 Z 0 1 Z Sum of minterms form F(X,Y,Z)= Minimize ? Fewer gates, fewer inputs F(X,Y,Z)= F(X,Y,Z)=

  5. Karnaugh Map Terminology • F(X,Y,Z) = XY 00 01 11 10 Z 0 1 Z Implicant set - rectangular group of size 2i of adjacent containing ones Each implicant set of size 2i of corresponds to a product term in which i variables are true and the rest false Implicant Sets:

  6. Karnaugh Map Terminology • F(X,Y,Z) = XY 00 01 11 10 Z 0 1 Z Prime implicant – an implicant set that is as large as possible Implies – We say P implies F if everytime P(X1, X2, … Xn) is true then F (X1, X2, … Xn) is true also. If P(X1, X2, … Xn) is a prime implicant then P implies F

  7. Karnaugh Map Terminology • F(X,Y,Z) = XY 00 01 11 10 Z 0 1 Z Prime implicants – If P(X1, X2, … Xn) is a prime implicant then P implies F and if we delete any variable from P this does not imply F.

  8. Karnaugh Map Simplification • F(X,Y,Z) = XY 00 01 10 11 Z 0 1 Z F(X,Y,Z) =

  9. Karnaugh Map Simplification • F(X,Y,Z) = XY 00 01 10 11 Z 0 1 Z F(X,Y,Z) =

  10. 4 Variable Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  11. 4 Variable Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  12. Karnaugh Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  13. Karnaugh Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  14. Karnaugh Map Simplification • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  15. Larger Maps • Five variable maps - Figure X4.72 page 307 • Six variable maps - Figure X4.74 page 308 • But who cares, use Quine-McKluskey section 4.5

  16. Don’t Care Conditions • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  17. Don’t Care Conditions • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  18. Don’t Care Conditions • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  19. Don’t Care Conditions • F(W,X,Y,Z) = X WX 00 01 11 10 YZ 00 01 11 10 Z Y W

  20. Summary • Homework • 4.6b • 4.31 • 4.39b • 4.40 • 4.55 • 4.13d • 4.19c,e

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