1. Combinational Logic: Other Gate Types

2. Gate classifications
Primitive gate - a gate that can be described using a single primitive operation type (AND or OR) plus an optional inversion(s).
Complex gate - a gate that requires more than one primitive operation type for its description

3. Fall 2005
primitive gates
NAND
NOR

4. NAND Gates

5. NOR
NOT OR
Also common

6. NAND is Universal
Universal gate : Can express any Boolean Function using only this type of gate
Equivalents below

7. Sum of Products with NAND
Easy to think of bubbles as canceling

8. AND-OR Circuit Easy to Convert

9. NOR Also Universal
Dual of NAND

10. Buffer No inversion No change, except in power or voltage
Used to enable driving more inputs

11. Parity Function How does parity work ?
Fall 2005
Parity Function
How does parity work ?
Given 7- bit ASCII code for A ( )
What is the ASCII code for A with even parity ?
Write truth table for two input even parity generator
What needs to be generated for parity bit?
What function of two inputs gives you this?
This is called: Exclusive OR function
In Chapter 1, a parity bit added to n-bit code to produce an n + 1 bit code
Even parity – set bit to make number of 1’s even
Examples
A ( ) with even parity is
C ( ) with even parity is
X Y P
0 0 0
0 1 1
1 0 1
1 1 0
Exclusive OR

12. Example Complex Digital Logic Gates: Exclusive OR/ Exclusive NOR
Fall 2005
Example Complex Digital Logic Gates: Exclusive OR/ Exclusive NOR
The Exclusive OR (XOR) function is defined as:
The eXclusive NOR (XNOR) function, otherwise known as equivalence is: Y X + = Å Y X + = Å

13. Symbols For XOR and XNOR
XOR symbol:
XNOR symbol:
Symbols exist only for two inputs

14. Truth Tables for XOR/XNOR
Fall 2005
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 ? X Y Å 1 X Y 1
or X (X Å Y)
Because it is defined as X Y + X’ Y’ that equals 1 if and only if X = Y implying X is equivalent to Y.

15. XOR Implementations
The simple SOP implementation uses the following structure:
A NAND only implementation is: X Y X + = Å X Y Y X X Y Y

16. XOR/XNOR (Continued) Z Y X Å Å = + + + = Y Z ) ( X 1 Å = =
Fall 2005
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: Z Y X Å Å = + + + X 1 Å = Y Z ) ( = =
For an odd function to be 1, with three or more variables an odd number of variables must be = to 1

17. Question
Draw the K-map of a 4 variable odd function 1 B C D A

18. Example: Odd Function Implementation
Design a 3-input odd function F = X Y Z with 2-input XOR gates +

19. Example: Odd Function Implementation
Design a 3-input odd function F = X Y Z with 2-input XOR gates
Factoring, F = (X Y) Z + +

20. Example: Odd Function Implementation
Design a 3-input odd function F = X Y Z with 2-input XOR gates
Factoring, F = (X Y) Z
The circuit: + + X Y Z F

21. Example: Odd Function Implementation
Design a 3-input odd function F = X Y Z with 2-input XOR gates
Factoring, F = (X Y) Z
The circuit:
Based on the above, given (X,Y,Z,F), then F would be the even parity bit for the three bits X,Y,Z. Hence, the circuit is an even parity generator. + + X Y Z F

22. Even Parity Generators and Checkers
An even parity bit could be added to n-bit code to produce an n + 1 bit code:
Use an odd function to produce codes with even parity
Use odd function circuit to check code words with even parity
Example: n = 3. Generate even parity code words of length 4 with an odd function circuit (parity generator):
Check even parity code words of length 4 with odd function circuit
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 between generator and checker, then E = 1 indicates an error. X Y Z P E

23. Odd Parity Generators and Checkers
Similarly, an odd parity bit could be added to n-bit code to produce an n + 1 bit code
Use an even function to produce codes with odd parity
Use even function circuit to check code words with odd parity

24. Tri-State Output w/ 3 states: H, L, and Hi-Z High impedance
Behaves like no output connection if in Hi-Z (Hi Impedance) state
Allows connecting multiple outputs

25. Data Selection If s = 0, OL = IN0, else OL = IN1 IN0 OL IN1 s
Data Selector
(2 to 1 Multiplexer)
IN0
IN1 OL s

26. Data Selection Function Implementation with 3-State Logic
Data Selection Function: If s = 0, OL = IN0, else OL = IN1
Performing data selection with 3-state buffers:
Since EN0 = S and EN1 = S, one of the two buffer outputs is always Hi-Z plus the last row of the table never occurs.
EN0
IN0
EN1
IN1 OL X 1
IN0
IN1
EN0
EN1 S OL