COMP541 Combinational Logic - II

1. COMP541Combinational Logic - II Montek Singh Aug 27, 2014

2. Today • Digital Circuits (review) • Basics of Boolean Algebra (review) • Identities and Simplification • Basics of Logic Implementation • Minterms and maxterms • Going from truth table to logic implementation

3. Digital Circuits • Digital Circuit = network that processes binary variables • one or more binary inputs • one or more binary outputs • inputs and outputs are called “terminals” • a functional specification • relationship between inputs and outputs • a timing specification • describes delay from inputs changing to outputs responding

4. Digital Circuits • Inside the black box • subcircuits or components or elements • connected by wires • wires and terminals often called “nodes” • each node has a binary value • each node is an input, an output, or “internal” • Example: • E1, E2, E3 are elements • A, B, C are input nodes • Y, Z are output nodes • n1 is an internal node

5. Types of circuits • Two types: with memory and without • Combinational Circuit • output depends only on the current values of the inputs • provided enough time is given for output to respond • output does not depend on past inputs or outputs • called “memoryless” • example: AND gate • Sequential Circuit • anything not combinational is sequential • output depends on not only current inputs, but also past behavior • previous inputs and/or outputs affect behavior • has “memory”, or is “stateful” • example: counter

6. Combinational Circuits: Examples OR Adder Multi-output example Slash notation

7. Combinational Circuits • Theorem: A circuit is combinational if: • every element is itself combinational • every node is either designated as an input, or connects to exactly one output terminal of an element • outputs of two elements are never “shorted together” • ensures that each node has a unique/unambiguous value • contains no cyclic paths • every path through the circuit visits each node at most once • no “feedback” • Conditions above ensure that output is only a function of inputs • Proof: By induction

8. Combinational Circuits: Examples • Which meet the conditions for combinational logic?

9. Identities in Boolean Algebra • Use identities to manipulate functions • You can use distributive law … … to transform to

10. Table of Identities

11. Duals • Left and right columns are duals • Replace ANDs and ORs, 0s and 1s

12. Single Variable Identities

13. Commutativity • Operation is independent of order of variables

14. Associativity • Independent of order in which we group • So can also be written asand

15. Distributivity

16. Substitution • Can substitute arbitrarily large algebraic expressions for the variables • Distribute an operation over the entire expression • Example: X + YZ = (X+Y)(X+Z) Substitute ABC for X ABC + YZ = (ABC + Y)(ABC + Z)

17. DeMorgan’sTheorem • Used a lot • NOR  invert, then AND • NAND  invert, then OR

18. Truth Tables for DeMorgan’s

19. Algebraic/Boolean Manipulation • Apply algebraic and Boolean identities to simplify expression • example:

20. Simplification Example Apply Apply Apply

21. Fewer Gates

22. Consensus Theorem • The third term is redundant • Can just drop third term (consensus term) • Proof summary (for first version): • For third term to be true, Y & Z both must be 1 • Then one of the first two terms is already 1! • Exercise: Provide a similar proof for the 2nd version

23. Complement of a Function • Definition: 1s & 0s swapped in truth table • Mechanical way to derive algebraic form • Take the dual • Recall: Interchange AND and OR, and 1s & 0s • Complement each literal • x becomes x’ (x’ means complement of x)

24. Next Lecture • Next Class: More on combinational logic • Commonly-used combinational building blocks