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Department of Communication Engineering, NCTU 1
Unit 4 Application of Boolean
Algebra
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
Department of Communication Engineering, NCTU 3
4.1 Conversion of English
Sentences to Boolean Equations
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'
Department of Communication Engineering, NCTU 5
4.2 Combinational Logic Design
Using a Truth Table
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
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
Department of Communication Engineering, NCTU 8
4.3 Minterm and Maxterm
Expansions
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
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)
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      
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)
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)
Department of Communication Engineering, NCTU 14
4.5 Incompletely Specified
Functions
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
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
Department of Communication Engineering, NCTU 17
4.5 Examples of Truth Table
Construction
Department of Communication Engineering, NCTU 18
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
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
Department of Communication Engineering, NCTU 20
Logic Design Unit 4 Application of Boolean Algebra Sau-Hsuan Wu
 Switching Expression

Department of Communication Engineering, NCTU 21
4.5 Design of Binary Adders
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
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
     
       
  
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    
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
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
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
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
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
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

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Unit 04

  • 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