1. CS 1110 Digital Logic Design
Chapter 3-2
Gate-Level Minimization

2. Outline 3.1 Introduction 3.2 The Map Method 3.3 Four-Variable Map
3.4 Product of sums simplification
3.5 Don‘t Care Conditions
3.7 NAND and NOR Implementaion
3.8 Other Two-Level Implementaion
3.9 Exclusive-OR function

3. 3.7 NAND and NOR Implementation (1-15)
Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gates.

4. 3.7 NAND and NOR Implementation (2-15)
NAND Implementation
NAND gate: a universal gate.
Any digital system can be implemented with it.

5. 3.7 NAND and NOR Implementation (3-15)
NAND Implementation
To facilitate the conversion to NAND logic, there are alternative graphic symbol for it.

6. 3.7 NAND and NOR Implementation (4-15)
NAND Implementation
Two-Level Implementation
Procedures of Implementation with two levels of NAND gates:
Express simplified function in sum of products form.
Draw a NAND gate for each product term that has at least two literals to constitute a group of first-level gates
Draw a single gate using AND-invert or invert-OR in the second level
A term with a single literal requires an inverter in the first level.

7. 3.7 NAND and NOR Implementation (5-15)
NAND Implementation
Two-Level Implementation
F = AB + CD
= [(AB + CD)’]’
= [(AB)’*(CD)’]’

8. 3.7 NAND and NOR Implementation (6-15)
NAND Implementation
Two-Level Implementation
Example (3.10):
F(X,Y,Z) = ∑ (1,2,3,4,5,7)
X’Y
F = XY’ + X’Y + Z
y z 00 01 11 10 x 1 1 1 1 1 1 1 Z
XY’

9. 3.7 NAND and NOR Implementation (7-15)
NAND Implementation
Multilevel Implementation
Procedures of Implementation with multilevel of NAND gates:
Convert all AND gates to NAND gates with AND-invert graphic symbols
Convert all OR gates to NAND gates with invert-OR graphic symbols
Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal

10. 3.7 NAND and NOR Implementation (8-15)
NAND Implementation
Multilevel Implementation
EXAMPLE 1:
F = A(CD + B) + BC’

11. 3.7 NAND and NOR Implementation (9-15)
NAND Implementation
Multilevel Implementation
EXAMPLE 2:
F = (AB’ + A’B).(C + D’)

12. 3.7 NAND and NOR Implementation (10-15)
The NOR operation is the dual of the NAND operation.
The NOR gate is anothar universal gate to implement any Boolean function.

13. 3.7 NAND and NOR Implementation (11-15)
To facilitate the conversion to NOR logic, there are alternative graphic symbol for it.

14. 3.7 NAND and NOR Implementation (12-15)
Two-Level Implementation
Procedures of Implementation with two levels of NOR gates:
Express simplified function in product of sums form.
Draw a NOR gate for each product term that has at least two literals to constitute a group of first-level gates
Draw a single gate using OR-invert or invert-AND in the second level
A term with a single literal requires an inverter in the first level.

15. 3.7 NAND and NOR Implementation (13-15)
Two-Level Implementation
Example :
F = (A+B).(C+D).E E

16. 3.7 NAND and NOR Implementation (14-15)
Multilevel Implementation
Procedures of Implementation with multilevel of NOR gates:
Convert all OR gates to NOR gates with OR-invert graphic symbols
Convert all AND gates to NOR gates with invert-AND graphic symbols
Check all the bubbles in the diagrams. For a single bubble, invert aninverter (one-input NAND gate) or complement the input literal

17. 3.7 NAND and NOR Implementation (15-15)
Multi-Level Implementation
Example :
F = (A B’ + A’B).(C+D’) A B’ A’ B

18. 3.9 Exclusive-OR Function (1-7)
Exclusive-OR (XOR) denoted by the symbol :
x  y = xy‘ + x‘y
Exclusive-OR is equal to 1, when the values of x and y are diffrent.
Exclusive-NOR (XNOR):
(x  y )‘ = xy + x‘y‘
Exclusive-NOR is equal to 1, when the values of x and y are same.
Only a limited number of Boolean functions can be expressed in terms of XOR operations, but it is particularly useful in arithmetic operations and error-detection and correction circuits.

19. 3.9 Exclusive-OR Function (2-7)
Exclusive-OR principles:
x  0 = x
x  1 = x‘
x  x = 0
x  x‘ = 1
x  y‘ = x‘ y = (x  y)‘
x  y = y  x
(x  y)  z = x  (y  z)

20. 3.9 Exclusive-OR Function (3-7)
Implementaion Exclusive-OR with AND-OR-NOT:
x  y = xy‘ + x‘y
Implementaion Exclusive-OR with NAND:
= x (x‘+y‘) + y (x‘+y‘)
= x (xy)‘ + y (xy)‘
= [ (x(xy)‘ + y(xy)‘)‘]‘
= [ (x(xy)‘)‘ + (y(xy)‘)‘ ]‘

21. 3.9 Exclusive-OR Function (7-7)
Parity Generation and Checking:
Exclusive-OR function is useful in systems requiring error-detection and correction circuits.
A parity bit is used for purpose of detection errors during transmission.
Parity bit : an extra bit included with a binary message to make the number of 1’s either odd or even.
The circuit generates the parity bit in transmitter is called parity generator.
The circuit checks the parity bit in receiver is called parity checker.

22. 3.9 Exclusive-OR Function (8-7)
Parity Generation and Checking:
Example : Three-bit message with even parity
Three-bit Massage
Parity bit X Y Z P 1
From the truth table , P constitutes an odd function.
It is equal 1 when numerical value of 1’s in a minterm is odd
P = x  y  z

23. 3.9 Exclusive-OR Function (8-7)
Parity Generation and Checking:
Example : Three-bit message with even parity
From the truth table , C constitutes an odd function.
It is equal 1 when numerical value of 1’s in a minterm is odd
C = x  y  z  P