1. EEE2243 Digital System Design Chapter 1: Combinational Logic Recap by Muhazam Mustapha, January 2011

2. Learning Outcome By the end of this chapter, students are expected to refresh their knowledge on combinatorial logic related to HDL

3. Chapter Content Boolean Logic Representation of Boolean Function

4. Boolean Logic

5. Definition Boolean algebra has 2 similar but different definitions for mathematician and engineers Mathematician: Boolean algebra is part of the so called abstract mathematics that involves structures and lattice –Wikipedia: http://en.wikipedia.org/wiki/Boolean_algebra_(structure) Engineers: Boolean algebra is calculus of truth values that is used as a means to facilitate the design of logic and digital systems –Wikipedia: http://en.wikipedia.org/wiki/Boolean_algebra_(logic) We’ll refresh our boolean algebra / logic enough for use with HDL (Verilog)

6. Boolean Values Pairs of 2 distinct values 10 HIGHLOW TrueFalse ONOFF

7. Boolean Operations OR (binary operation) –symbol: ‘+’ (plus) –gives true ( 1 ) if any of the parameters is true AND (binary operation) –symbol: ‘·’ (dot) – optional –gives true ( 1 ) if both parameters are true NOT (unary operation) –symbol: ‘‾’ (bar / overline) –gives true ( 1 ) if the parameter is false –gives false ( 0 ) if the parameter is true –also called ‘inverter’

8. Boolean Operations NOR –OR followed by NOT –A NOR B = A+B NAND –AND followed by NOT –A NAND B = AB XOR –true if either one, but not both, is true –symbol: ‘ ⊕ ’ XNOR –true if both parameters are the same –XOR followed by NOT –symbol: ‘ ⊙ ’

9. Boolean Gates AND NAND NOT OR NOR XOR XNOR

10. Basic Truth Table ABAB 000 010 100 111 ABA+B 000 011 101 111 ABAB 001 011 101 110 ABA+B 001 010 100 110 AB 000 011 101 110 AB 001 010 100 111 AA 01 10 AND NAND NOT OR NOR XOR XNOR

11. General Truth Table ABCACBF 000011 001011 010000 011000 100011 101111 110000 111101 Example of step by step construction of truth table of function:

12. DeMorgan’s Theorem Used to split groups of large inversion ABABABA+B 0011101 0110010 1001010 1100010 ABAB AB 0011101 0110101 1001101 1100010

13. DeMorgan’s Theorem Reflections of DeMorgan’s Theorem on logic gates NAND NOR A B A B Negative OR A B Negative AND A B

14. Representation of Boolean Function

15. Boolean Function Representation There are 3 ways of representing boolean functions that are convenient for HDL: –Truth Table –Minterm (Maxterm is less useful for HDL) –AOI (AND-OR-INVERTER) Gate Tree (OAI (OR-AND-INVERTER) Gate Tree is less popular)

16. Boolean Function Representation ABCF 0001 0011 0100 0110 1001 1011 1100 1111 Truth Table Sum of Product (SOP)

17. Boolean Function Representation A B C F Sum of Product Gate Tree in FPGA ××× × ××× ××× ××× ××× × × × ×

18. Boolean Function Representation ABCF 0001 0011 0100 0110 1001 1011 1100 1111 Truth Table Product of Sum (POS) POS is not so useful for HDL as most of FPGA (programmable logic) hardware are build with SOP structure

19. Simplification In introductory courses of digital electronics we spent so much on boolean function simplification Thank God! In intermediate digital electronics we don’t have to do that for FPGA programming This is due to the fact that FPGA structure is made of minterms This means we are not going to cover function simplification