The document discusses the design of combinational logic circuits using Boolean algebra. It covers topics such as converting English descriptions to Boolean equations, using truth tables to derive switching functions, minterm and maxterm expansions, incompletely specified functions, and designing binary adders including ripple carry adders and carry lookahead adders. The goals are to specify circuit behaviors using Boolean algebra and design combinational logic circuits from algebraic expressions or truth tables.

1. Department of Communication Engineering, NCTU 1
Unit 4 Application of Boolean
Algebra

2. Department of Communication Engineering, NCTU 2
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Three main steps in designing a single-output
combinational logic circuit
 Find a switching function that specifies the desired
behavior of the circuit
 Find a simplified algebraic expression for the function
 Realized the simplified function using available logic
elements
 Goals:
 How to specify circuit behaviors
 How to design a combinational logic circuit

3. Department of Communication Engineering, NCTU 3
4.1 Conversion of English
Sentences to Boolean Equations

4. Department of Communication Engineering, NCTU 4
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 For simple problems, go directly from a word description
of the desired circuit behavior to an algebra expression
 Mary watches TV if it is Monday night and she has
finished her homework.
F = A˙B
 The alarm will ring iff
the alarm switch is turned on
and the door is not closed,
or it is after 6 P.M.
and the window is not closed.
Z = AB' + CD'

5. Department of Communication Engineering, NCTU 5
4.2 Combinational Logic Design
Using a Truth Table

6. Department of Communication Engineering, NCTU 6
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 In general, a truth table to design logic circuits
 First, list a true table
 E.g.
 Derive an algebraic expression for f from the table
f = A'BC + AB'C' + AB'C + ABC' + ABC (4-1)
= A+BC

7. Department of Communication Engineering, NCTU 7
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 In stead of writing f in terms of the 1’s of the function,
we may also write f in terms of the 0’s of the function
 E.g.
f ' = (A+B+C)(A+B+C')(A+B'+C) (4-3)
= (A+B)(A+B'+C) = A + BC

8. Department of Communication Engineering, NCTU 8
4.3 Minterm and Maxterm
Expansions

9. Department of Communication Engineering, NCTU 9
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Each term in (4-1) is referred to as a minterm
 f = A'BC + AB'C' + AB'C + ABC' + ABC (4-1)
 A function written as a sum of minterms is referred to
as a minterm expansion or a standard SOP
 Each term in (4-3) is referred to as a maxterm
 f = (A+B+C)(A+B+C')(A+B'+C) (4-3)
 A function written as a product of maxterms is referred
to as a maxterm expansion or a standard POS

10. Department of Communication Engineering, NCTU 10
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 A minterm of n variables is a product of n literals in
which each variable appears exactly once in either true or
complemented form
 The decimal notation of minterm expansion
e.g. f = m (3,4,7)

11. Department of Communication Engineering, NCTU 11
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 A maxterm of n variables is a sum of n literals in which
each variable appears exactly once in either true or
complemented form
 The decimal notation of maxterm expansion
E.g. f = M(0,,2)
 Given the minterm or maxterm expansions for f , the
minterm or maxterm expansions for the complement of f
are easy to obtain
 E.g.
 Or
0 1 2
3 4 5 6 7
(0,1,2)
(3,4,5,6,7)
f m m m m
f M M M M M M
   
 


0 1 2 0 1 2 0 1 2( )f M M M M M M m m m      

12. Department of Communication Engineering, NCTU 12
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 A general switching expansion can be converted to
minterm or maxterm expansion either using a truth table
or algebraically
 For algebraic method, first write the expansion as a sum
of products and then introduce the missing variables in
each term by applying the theorem X + X’=1
 Example f(a,b,c,d) = a’(b’+d) + acd’
 1> SOP: f= a’b’+a’d+acd’
 Introduce missing variables
f= a’b’(c+c’)(d+d’)+a’(b+b’)(c+c’)d’+ a(b+b’)cd’
= a’b’c’d’+a’b’c’d+a’b’cd’+a’b’cd+a’b’c’d+a’b’cd
+ a’b’cd + a’bcd + abcd’+ ab’cd’
= m (0,1,2,3,5,7,10,14)

13. Department of Communication Engineering, NCTU 13
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 General minterm and maxterm expansions
 A general minterm expansion
f = a0m0 + a1m1+  + a7m7 = ai mi
ai = 0 or 1
 mi is not present if ai = 0
 A general maxterm expansion
f = (a0 + m0)(a1 + m1)  (a7 + m7) = (ai + mi)
ai = 0 or 1
 mi is not present if ai = 1
 Equality ai mi = (ai + mi)

14. Department of Communication Engineering, NCTU 14
4.5 Incompletely Specified
Functions

15. Department of Communication Engineering, NCTU 15
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 A large system is usually divided into many subcircuits.
The output of module 1 may not generate all possible
combinations for the input variables of module 2.
 In this case, we don’t care these specific combinations
when designing the switch circuit for B

16. Department of Communication Engineering, NCTU 16
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 When realizing the function, the don‘t care terms can be
assigned 0’s or 1’s
 If both X’s are assigned 0
F = A'B'C' + A'BC +ABC = A'B'C' + BC
 If first X is assigned 1 and the second 0
F = A'B'C' + A'B'C + A'BC +ABC = A'B' + BC
 If we assign 1 to both X’s
F = A'B'C' + A'B'C + A'BC + ABC' + ABC
= A'B' + BC + AB

17. Department of Communication Engineering, NCTU 17
4.5 Examples of Truth Table
Construction

18. Department of Communication Engineering, NCTU 18
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu

19. Department of Communication Engineering, NCTU 19
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Error detector for 6-3-1-1 binary-coded-decimal digits

20. Department of Communication Engineering, NCTU 20
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Switching Expression


21. Department of Communication Engineering, NCTU 21
4.5 Design of Binary Adders

22. Department of Communication Engineering, NCTU 22
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Design a 4-bit binary ripple carry adder
 Approach 1:
construct a truth table
 Approach 2: cascade 4 1-bit Full Adders

23. Department of Communication Engineering, NCTU 23
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Construct the true table for
1-bit full adder
 Find the switching expressions
( ) ( )
( ) ( )
in in in in
in in in in
in in in
Sum X Y C X YC XY C XYC
X Y C YC X Y C YC
X Y C X Y C X Y C
      
      
       
( ) ( ) ( )
out in in in in
in in in in in in
in in
C X YC XY C XYC XYC
X YC XYC XY C XYC XYC XYC
YC XC XY
     
       
  

24. Department of Communication Engineering, NCTU 24
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Implement the functions with logic gates
 Overflow occurs if adding two positive numbers gives a
negative result, or adding two negative numbers results in
a positive number
3 3 3 3 3 3V A B S A B S    

25. Department of Communication Engineering, NCTU 25
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 The pros and cons of ripple carry adder
 Simple in concept
 The carry output at stage i+1
Ci+1 = XiYi + (Xi + Yi) Ci
 The carries propagate like a ripple and introduce circuit
delays : C0 C1  C2   Ci+1
 Ci+1 = f (Xi,Yi, Ci) = f (Xi,Yi,Xi-1,Yi-1,Ci-1) = 
 Alternative: Carry lookahead adder
 To avoid circuit delays due to the propagation of
carries
 Express Ci+1 in terms of C0 and {X0,Yi  Xi,Yi} only

26. Department of Communication Engineering, NCTU 26
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Re-write the output carry at the ith stage as
 Ci+1 = gi + pi Ci
 The carry-generate function: gi = XiYi
 The carry-propagate function pi = Xi + Yi
 Expression the carry bit in terms of gi and pi
 C1 = g0 + p0 C0
 C2 = g1 + p1 C1 = g1 + p1 g0 + p1 p0 C0
 C3 = g2 + p2 C2 = g2 + p2 g1 + p2 p1 g0 + p2 p1 p0 C0
 
 Ci = gi + pi gi-1 + pi pi-1gi-2 +  + pi pi-1 pi-2  g0
+ pi pi-1 pi-2  p0C0

27. Department of Communication Engineering, NCTU 27
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 The circuit implementation of 4-bit carry lookahead adder
block
Carry lookahead network

28. Department of Communication Engineering, NCTU 28
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 For adders with higher number of bits, the carry
lookahead network can get quite large in terms of gates
and gate inputs. This also presents a limitation in the
realization of a large high speed adder
 How to circumvent this problem?
 Cascade 4-bit carry lookahead adders to form a lager adder

29. Department of Communication Engineering, NCTU 29
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Partition the operands into blocks
E.g.
C8 = g7 + p7 g6 + p7 p6g5 + p7 p6p5g4 + p7 p6p5p4g3 +
p7 p6p5p4p3g2 + p7 p6p5p4p3p2g1 + p7 p6p5p4p3p2p1g0+
p7 p6p5p4p3p2p1p0C0
= g7 + p7 g6 + p7 p6g5 + p7 p6p5g4 +
p7 p6p5p4 (g3 + p3g2 + p3p2g1 + p3p2p1g0) +
p7 p6p5p4(p3p2p1p0C0)
= G1 + P1G0 + P1P0C0
G1= g7 + p7 g6 + p7 p6g5 + p7 p6p5g4
P1 = p7 p6p5p4
G0= g3 + p3g2 + p3p2g1 + p3p2p1g0
P0 = p3p2p1p0

30. Department of Communication Engineering, NCTU 30
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Define a 4-bit carry lookahead generator as
G= g3 + p3g2 + p3p2g1 + p3p2p1g0
P = p3p2p1p0