Download presentation

Presentation is loading. Please wait.

Published byCayden Milem Modified about 1 year ago

1

2
Gate-Level Minimization Chapter 3

3
The Map Method Two-variable map

4

5
Three-variable map Adjacent Adjacent when minterms differ by one variable

6
Graphical view of adjacency Taking the difference between adjacent squares gives:

7
Example 3.1 Simplify the Boolean function

8
Example 3.2 Simplify the Boolean Function

9
Three variable map

10
Three-variable map The number of adjacent squares that may be combined must always represent a number that is a power of two: – One square represents one minterm which in this case gives a term with three literals – Two adjacent squares represent a term with two literals – Four adjacent squares represent a term with one literal – Eight adjacent squares encompass the entire map and produce a function equal to 1

11
Three-variable map

12
Three-variable map

13
Three-variable map

14
Example 3.3

15
Example 3.4 Let the Boolean Function (a) Express this function as a sum of minterms

16
Example 3.4 (b) Find the minimal sum-of-products expression Find adjacent squares

17
Problem 3.3 (a) Simplify the following Boolean function, using three-variable maps Find adjacent squares

18
Problem 3.3 (a) Another solution (without a three-variable map) Factorize the expression Voila!

19
Four-Variable Map

20
Four-variable map The number of adjacent squares that may be combined must always represent a number that is a power of two: – One square represents one minterm which in this case gives a term with four literals – Two adjacent squares represent a term with three literals – Four adjacent squares represent a term with two literals – Eight adjacent squares represent a term with one literal – Sixteen adjacent squares encompass the entire map and produce a function equal to 1

21
Adjacency in a four-variable map

22

23
Example 3.5 Simplify the Boolean Function

24
Example 3.6 Simplify the Boolean Function

25
Prime implicants 1.All the minterms are covered when combining the squares 2.The number of terms in the expression is minimized 3.There are no redundant terms (minterms already covered by other terms) When choosing adjacent squares in a map, make sure that:

26
Example 3.5 (revisited) Simplify the Boolean Function

27
Example 3.6 (revisited) Simplify the Boolean Function

28
Prime implicants Simplify the function

29
Prime implicants Simplify the function

30
Prime implicants Simplify the function

31
Prime implicants Simplify the function

32
Five-Variable Map

33
How can adjacency be visualized in a five-variable map?

34
Five-Variable Map

35
Simplify the Boolean function

36
Product-Of-Sums Simplification Take the squares with zeros and obtain the simplified complemented function Complement the above expression and use DeMorgan’s

37
Product-Of-Sums Simplification Sum-of-productsProduct-of-sums

38
Don’t-Care Conditions Used for incompletely specified functions, e.g. BCD code in which six combinations are not used (1010, 1011, 1100, 1101, 1110, and 1111) Those unspecified minterms are neither 1’s nor 0’s Unspecified terms are referred to as “don’t care” and are marked as X In choosing adjacent squares, don’t care squares can be chosen either as 1’s or 0’s to give the simplest expression

39
Don’t-Care Conditions Simplify the Boolean function which has the don’t care conditions

40
NAND and NOR Implementation

41

42
Two-level implementation

43
NAND and NOR Implementation Implement the following Boolean function with NAND gates

44
Multilevel NAND circuits

45

46
NOR Implementation

47

48

49

50
Exclusive OR Function Exclusive-OR or XOR performs the following logical operation Exclusive-NOR or equivalence performs the following logical operation Identities of the XOR operation XOR is commutative and associative

51
Exclusive-OR implementations

52
Parity generation and checking

53

54

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google