This document discusses digital logic design and analysis of logic circuits. It provides two examples of evaluating Boolean expressions, constructing truth tables, simplifying expressions, and deriving simplified logic circuits. Standard forms such as SOP and POS are discussed along with their relationship to minterms, maxterms, and truth tables. Conversion between different representations is also covered.

1. Digital logic design
lecture 07
By-Farhat Ullah

2. Analysis of Logic Circuits
Example 1
3
6
4
5
1
2
A
B
C
D
B
BA.
DC.
DCBA ...
A
DCBABA .... 

3. Evaluating Boolean Expression
 The expression
 Assume and
 Expression
 Conditions for output = 1 X=0 & Y=0
 Since X=0 when A=0 or B=1
 Since
Y=0 when A=0, B=0, C=1 and D=1
DCBABA .... 
BAX . DCBAY ...
YX 
BAX .
DCBAY ...

4. Evaluating Boolean Expression
& Truth Table
 Conditions for o/p =1
 A=0, B=0, C=1 & D=1
Input Output
A B C D F
0 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 0
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0

5. Simplifying Boolean Expression
 Simplifying by applying Demorgan’s theorem
=DCBABA ....  )...).(.( DCBABA
)...).(( DCBABA 
)...).(( DCBABA 
DCBBADCBAA ....)....( 
DCBA ...

6. Truth Table of Simplified
expression
Input Output
A B C D F
0 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 0
1 1 0 0 0
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0

7. Simplified Logic Circuit
7
3
4
A
B
C
D

8. Simplified Logic Circuit
 Simplified expression is in SOP
form
 Simplified circuit
DCBA ...

9. Second Example
 Evaluating Boolean Expression
 Representing results in a Truth Table
 Simplification of Boolean Expression
results in POS form and requires 3
variables instead of the original 4
 Representing results in a Truth Table
 Verifying two expressions through truth
tables

10. Analysis of Logic Circuits
Example 2
4
5
1
2 6
A
B
C
D
A
C
CBA ..
DC
)).(..( DCCBA 

11. Evaluating Boolean Expression
 The expression
 Assume and
 Expression
 Conditions for output = 1 X=0 OR Y=0
 Since
X=0 when A=1,B=0 or C=1
 Since
Y=0 when C=1 and D=0
)).(..( DCCBA 
CBAX .. DCY 
YX.
CBAX ..
DCY 

12. Evaluating Boolean Expression
& Truth Table
 Conditions for o/p =1
 (A=1,B=0 OR C=1)
OR (C=1 AND D=0)
Input Output
A B C D F
0 0 0 0 1
0 0 0 1 1
0 0 1 0 1
0 0 1 1 1
0 1 0 0 0
0 1 0 1 0
0 1 1 0 1
0 1 1 1 1
1 0 0 0 1
1 0 0 1 1
1 0 1 0 1
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1

13. Rewriting the Truth Table
 Conditions for o/p =1
 (A=1,B=0 OR C=1)
OR (C=1 AND D=0)
Input Output
A B C F
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1

14. Simplifying Boolean Expression
 Simplifying by applying Demorgan’s theorem
= )()..( DCCBA )).(..( DCCBA 
).()( DCCBA 
).()( DCCBA 
)1( DCBA 
CBA 

15. Truth Table of Simplified
expression
Input Output
A B C F
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1

16. Simplified Logic Circuit
73
A
B
C

17. Simplified Logic Circuit
 Simplified expression is in POS
form representing a single Sum term
 Simplified circuit
CBA 

18. Standard SOP and POS form
 Standard SOP and POS form has all the
variables in all the terms
 A non-standard SOP is converted into
standard SOP by using the rule
 A non-standard POS is converted into
standard POS by using the rule 0AA
1 AA

19. Standard SOP form
CBCA 
CBAABBCA )()( 
CBACABCBACAB 
CBACBACAB 

20. Standard POS form
))()(( DCBADBACBA 
))(( DCBADCBA 
))()(( DCBADCBADCBA 

21. Why Standard SOP and POS
forms?
 Minimal Circuit implementation by
switching between Standard SOP or POS
 Alternate Mapping method for
simplification of expressions
 PLD based function implementation

22. Minterms and Maxterms
 Minterms: Product terms in Standard SOP
form
 Maxterms: Sum terms in Standard POS
form
 Binary representation of Standard SOP
product terms
 Binary representation of Standard POS
sum terms

23. Minterms and Maxterms &
Binary representations
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
CBA .. CBA 
A B C Min-
terms
Max-
terms
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

24. SOP-POS Conversion
 Minterm values present in SOP expression
not present in corresponding POS
expression
 Maxterm values present in POS
expression not present in corresponding
SOP expression

25.  Canonical Sum
 Canonical Product
 =
SOP-POS Conversion
ABCCBABCACBACBA 
)CBA)(CBA)(CBA( 
)7,5,3,2,0(,, CBA
)6,4,1(,, CBA
)7,5,3,2,0(,, CBA )6,4,1(,, CBA

26. Boolean Expressions and Truth
Tables
 Standard SOP & POS expressions
converted to truth table form
 Standard SOP & POS expressions
determined from truth table

27. SOP-Truth Table Conversion
BCBA 
ABCCBACBABCA )7,5,4,3(,, CBA
Input Output
A B C F
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1

28. POS-Truth Table Conversion
))(( CBBA  )5,3,2,1(,, CBA
))()()(( CBACBACBACBA 
Input Output
A B C F
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1