CS 151: Introduction to Digital Design

1. CS 151: Introduction to Digital Design Chapter 2-9 Exclusive-OR Gates

2. Complex Digital Logic Gates CS 151

3. Exclusive OR/ Exclusive NOR • The eXclusive OR (XOR) function is an important Boolean function used extensively in logic circuits. • The XOR function may be; • implemented directly as an electronic circuit (truly a gate) or • implemented by interconnecting other gate types (used as a convenient representation) • The eXclusive NOR function is the complement of the XOR function • By our definition, XOR and XNOR gates are complex gates. CS 151

4. Å = + X Y X Y X Y Exclusive OR/ Exclusive NOR • Uses for the XOR and XNORs gate include: • Adders/subtractors/multipliers • Counters/incrementers/decrementers • Parity generators/checkers • Definitions • The XOR function is: • The eXclusive NOR (XNOR) function, otherwise known as equivalence is: • Strictly speaking, XOR and XNOR gates do no exist for more that two inputs. Instead, they are replaced by odd and even functions. Å = + X Y X Y X Y CS 151

5. Å Å X Y X Y X Y (X Y) º or X Y 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 Truth Tables for XOR/XNOR • Operator Rules: XOR XNOR • The XOR function means: X OR Y, but NOT BOTH • Why is the XNOR function also known as the equivalence function, denoted by the operator ? CS 151

6. X Y Z X Y Z X Y Z X Y Z X Y Z Å X 0 X X 1 X Å = = Å X X 0 X X 1 Å = X Y Y X Å Å ) = = Å Å Å Å Å Å ( X Y Z X ( Y Z ) X Y Z XOR/XNOR (Continued) • The XOR function can be extended to 3 or more variables. For more than 2 variables, it is called an odd function or modulo 2 sum (Mod 2 sum), not an XOR: • The complement of the odd function is the even function. • The XOR identities: Å Å = + + + = = CS 151

7. Symbols For XOR and XNOR • XOR symbol: • XNOR symbol: • Symbols exist only for two inputs CS 151

8. X Y X Y Y X X Y XOR Implementations • The simple SOP implementation uses the following structure: • A NAND only implementation is: CS 151

9. 010 100 111 001 Odd and Even Functions • What about the case of more than 2 variables? • A  B  C = (AB’ + A’B) C’ + (AB + A’B’) C = AB’C’ + A’BC’ + ABC + A’B’C • Then, F is the logical sum of the four minterms having an index with an odd number of 1’s F is called an odd function • The other four minterms not included are 000, 011, 101 and 110, they have an index with even number of 1’s  F’ is called an even function. • In general, an n-variable exclusive-OR function is an odd function defined as the logical sum of the 2n/2 minterms whose binary index have an odd number of 1’s. CS 151

10. Odd and Even Functions CS 151

11. X Y F Z Example: Odd Function Implementation • Design a 3-input odd function F = X  Y  Zwith 2-input XOR gates • Factoring, F = (X  Y)  Z • The circuit: CS 151

12. W X F Y Z Example: Even Function Implementation • Design a 4-input even function F = (W  X  Y  Z)’with 2-input XOR and XNOR gates • Factoring, F = ( (W  X)  (Y  Z) )’ • The circuit: CS 151

13. X Y P Z Parity Generators and Checkers • Example: n = 3. Generate even parity code words of length 4: P=X  Y  Z CS 151

14. X Y E Z P Parity Generators and Checkers-Cont. • Check even parity code words of length 4 with odd parity checker: E should indicate an error (be true) when (X,Y,Z,P) have an odd number of 1’s. •  E = X  Y  Z P • Example: operation (X,Y,Z) = (0,0,1) gives(X,Y,Z,P) = (0,0,1,1) and E = 0.If Y changes from 0 to 1 betweengenerator and checker, then E = 1 indicates an error. CS 151