02 combinational logic
- 2. 2 This Lecture Binary Storage Registers Binary Logic Boolean Algebra Boolean Algebra Functions Boolean Function Implementation Canonical & Standard Forms Gate Level Minimization
- 3. 3 Binary storage & registers How do we store binary information? Binary cell : place to store one bit of information. 0 or 1. Register: a group of binary cells. Register transfer: An operation in a digital system
- 4. 4 Binary storage & registers
- 5. 5 Binary information processing Example: Add two 10-bit binary numbers
- 6. 6 Binary logic Binary logic deals with variables that take on two discrete values and operations that assume logical meaning. Logic gates: electronic circuits that operate on one or more input signals to produce an output signal. Example x y x AND y 0 0 0 0 1 0 1 0 0 1 1 1
- 7. 7 Electrical signals Two values: 0 or 1
- 8. 8 Symbols for digital logic circuits
- 9. 9 Input-Output signals for gates
- 10. 10 Gates with multiple inputs
- 13. 13 Boolean Algebra Functions examples: F1=x+y’.z F2=x’.y’.z+x’.y.z+x.y’ =x’.z(y’+y)+x.y’ F2=x’.z+x.y’ A Boolean Function can be represented in many algebraic forms We look for the most simple form
- 14. 14 Boolean Function: Example Truth table x y z F1 F2 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 A Boolean Function can be represented in only one truth table forms
- 17. 17 Canonical & Standard Forms Consider two binary variables x, y and the AND operation four combinations are possible: x.y, x’.y, x.y’, x’.y’ each AND term is called a minterm or standard products for n variables we have 2n minterms Consider two binary variables x, y and the OR operation four combinations are possible: x+y, x’+y, x+y’, x’+y’ each OR term is called a maxterm or standard sums for n variables we have 2n maxterms Canonical Forms: Boolean functions expressed as a sum of minterms or product of maxterms.
- 18. 18 Minterms x y z Terms Designation 0 0 0 x’.y’.z’ m0 0 0 1 x’.y’.z m1 0 1 0 x’.y.z’ m2 0 1 1 x’.y.z m3 1 0 0 x.y’.z’ m4 1 0 1 x.y’.z m5 1 1 0 x.y.z’ m6 1 1 1 x.y.z m7
- 19. 19 Maxterms x y z Designation Terms 0 0 0 M0 x+y+z 0 0 1 M1 x+y+z’ 0 1 0 M2 x+y’+z 0 1 1 M3 x+y’+z’ 1 0 0 M4 x’+y+z 1 0 1 M5 x’+y+z’ 1 1 0 M6 x’+y’+z 1 1 1 M7 x’+y’+z’
- 20. 20 How to express algebraically Question: How do we find the function using the truth table? Truth table example: x y z F1 F2 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0
- 21. 21 How to express algebraically 1.Form a minterm for each combination forming a 1 2.OR all of those terms Truth table example: x y z F1 minterm 0 0 0 0 0 0 1 1 x’.y’.z m1 0 1 0 0 0 1 1 0 1 0 0 1 x.y’.z’ m4 1 0 1 0 1 1 0 0 1 1 1 1 x.y.z m7 F1=m1+m4+m7=x’.y’.z+x.y’.z’+x.y.z=Σ(1,4,7)
- 22. 22 How to express algebraically Truth table example: x y z F2 minterm 0 0 0 0 m0 0 0 1 0 m1 0 1 0 0 m2 0 1 1 1 m3 1 0 0 0 m4 1 0 1 1 m5 1 1 0 1 m6 1 1 1 1 m7 F2=m3+m5+m6+m7=x’.y.z+x.y’.z+x.y.z’+x.y.z=Σ(3,5,6,7)
- 23. 23 How to express algebraically 1.Form a maxterm for each combination forming a 0 2.AND all of those terms Truth table example: x y z F1 maxterm 0 0 0 0 x+y+z M0 0 0 1 1 0 1 0 0 x+y’+z M2 0 1 1 0 x+y’+z’ M3 1 0 0 1 1 0 1 0 x’+y+z’ M5 1 1 0 0 x’+y’+z M6 1 1 1 1 F1=M0.M2.M3.M5.M6 = л(0,2,3,5,6)
- 24. 24 How to express algebraically Truth table example: x y z F2 maxterm 0 0 0 0 x+y+z M0 0 0 1 0 x+y+z’ M1 0 1 0 0 x+y’+z M2 0 1 1 1 1 0 0 0 x’+y+z M4 1 0 1 1 1 1 0 1 1 1 1 1 F=M0.M1.M2.M4=л(0,1,2,4)=(x+y+z).(x+y+z’).(x+y’+z).(x’+y+z)
- 25. 25 Maxterms & Minterms: Intuitions Minterms: If a function is expressed as SUM of PRODUCTS, then if a single product is 1 the function would be 1. Maxterms: If a function is expressed as PRODUCT of SUMS, then if a single product is 0 the function would be 0. Canonical Forms: Boolean functions expressed as a sum of minterms or product of maxterms.
- 26. 26 Standard Forms Standard From: Sum of Product or Product of Sum
- 27. 27 Nonstandard Forms Nonstandard From: Neither a Sum of Product nor Product of Sum
- 28. 28 Implementations Three-level implementation vs. two-level implementation Two-level implementation normally preferred due to delay importance.
- 30. 30 Integrated Circuits (ICs) Levels of Integration SSI: fewer than 10 gates on chip MSI:10 to 1000 gates on chip LSI: thousands of gates on chip VLSI:Millions of gates on chip Digital Logic Families TTL transistor-transistor logic ECL emitter-coupled logic MOS metal-oxide semiconductor CMOS complementary metal-oxide semiconductor
- 31. 31 Digital Logic Parameters Fan-out: maximum number of output signals Fan-in : number of inputs Power dissipation Propagation delay Noise margin: maximum noise
- 32. 32 Gate-Level Minimization The Map Method: A simple method for minimizing Boolean functions Map: diagram made up of squares Each square represents a minterm
- 34. 34 Two-Variable Map Maps representing x.y and x+y
- 35. 35 Three-Variable Map Minterms are not arranged in a binary sequence Minterms arranged in gray code: Only one bit changes from one column to the next
- 36. 36 Three-Variable Map-example 1 Sum of two adjacent minterms can be simplified to a single AND term consisting of two literals
- 43. 43 Graphic symbols for NAND gates
- 44. 44 Summary Logic Gates Combinational Circuit Integrated Circuits Gate Level Minimization
- 45. 45 Recommended Reading • Read textbook & readings • Solve exercises • Digital Design Fifth edition, by Morris Mano, Prentice Hall Publishers