Upload
atush-jain
View
49
Download
3
Embed Size (px)
Citation preview
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
2
Lesson 4
BOOLEAN EXPRESSION, TRUTH TABLE and
SUM OF THE PRODUCTS (SOPs) [MINITERMS]
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
3
Outline
• SOP two variables cases• SOP for three variable case • SOP for four variable case• Conversion of Boolean expression
into SOPs {Finding Miniterms]
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
4
Two variable Miniterms
0 0 mn0= A.B 0 0 0 1
Inputs Miniterms Outputs A B XOR AND OR NAND
1 0 mn1= A.B 1 0 1 10 1 mn2= A.B 1 0 1 11 1 mn3= A.B 0 1 1 0
A B S0 S1 S2 S3
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
5
XOR,AND, OR and NAND Outputs Two XOR,AND, OR and NAND Outputs Two Variable Cases Sum of Product Terms (SOPs)Variable Cases Sum of Product Terms (SOPs)
XOR: S0 = A.B + A.B =
ΣΣΣΣmn(1, 2)
AND: S1 = A.B =
mn(3)
OR: S2 = A.B + A.B + A.B =
ΣΣΣΣmn(1, 2, 3)
NAND: S3 = A.B + A.B + A.B =
ΣΣΣΣmn(0, 1, 2)
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
6
Advantage of using SOP form is that functions of any two input logic gate functions can be represented by maximum four ANDs at the inputs and four ORs at an output.
SOP form advantage
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
7
Outline
• SOP two variable cases•• SOP for three variable caseSOP for three variable case• SOP for four variable case• Conversion of Boolean expression
into SOPs {Finding Miniterms]
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
8
Three variable Miniterms 0 to 3
0 0 0 mn0= A.B.C 0 0 0 1
Inputs Miniterms Outputs A B C F1 F2 F3 F4
1 0 0 mn1= A.B. C 1 0 1 10 1 0 mn2= A.B .C 1 0 1 11 1 0 mn3= A.B .C 0 1 1 0
A B C S S’ S’’ S’’’
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
9
Three variable Miniterms 4 to 7
0 0 1 mn4= A.B.C 0 0 0 1
Inputs Miniterms Outputs A B C F1 F2 F3 F4
1 0 1 mn5= A.B. C 1 0 1 10 1 1 mn6= A.B .C 1 0 1 11 1 1 mn7= A.B .C 0 1 1 0
A B C S S’ S’’ S’’’
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
10
F1, F2, F3 and F4 outputs Three Variable F1, F2, F3 and F4 outputs Three Variable Cases Sum of Product Terms (SOPs)Cases Sum of Product Terms (SOPs)
F1 = S = A.B.C + A.B.C +A.B.C + A.B.C =
ΣΣΣΣmn(1, 2, 5, 6)F2 = S’= A.B.C + A.B.C =
ΣΣΣΣmn(3, 7)
F3 = S’’ = A.B .C + A.B .C + A.B .C + A.B .C + A.B .C + A.B .C =
ΣΣΣΣmn(1, 2, 3, 5, 6, 7)F4 = S’’’ = A.B .C + A.B .C + A.B .C + A.B .C + A.B .C + A.B .C =
ΣΣΣΣmn(0, 1, 2, 4, 5, 6)
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
11
Advantage of using SOP form is that functions of any three input logic gate functions can be represented by maximum eight ANDs at the inputs and eight ORs at an output.
SOP form advantage
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
12
Outline
• SOP two variable cases• SOP for three variable cases • SOP for four variable case• Conversion of Boolean expression
into SOPs {Finding Miniterms]
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
13
Four variable Miniterms 0 to 3
0 0 0 0 mn0= A.B.C .D 1
Inputs Miniterms Output A B C D F5
0 1 0 0 mn1= A.B. C .D 01 0 0 0 mn2 = A.B .C .D 11 1 0 0 mn3= A.B .C .D 0
A B C D S
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
14
Four variable Miniterms 0 to 3
0 0 1 0 mn4= A.B.C .D 0
Inputs Miniterms Output A B C D F5
0 1 1 0 mn5= A.B. C .D 01 0 1 0 mn6 = A.B .C .D 01 1 1 0 mn7= A.B .C .D 1
A B C D S
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
15
Four variable Miniterms 0 to 3
0 0 0 1 mn8= A.B.C .D 0
Inputs Miniterms Output A B C D F5
0 1 0 1 mn9= A.B. C .D 11 0 0 1 mn10 = A.B.C .D 01 1 0 1 mn11= A.B .C .D 0
A B C D S
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
16
Four variable Miniterms 0 to 3
0 0 1 1 mn12= A.B.C .D 1
Inputs Miniterms Output A B C D F5
0 1 1 1 mn13= A.B. C .D 01 0 1 1 mn14= A.B .C .D 01 1 1 1 mn15= A.B .C .D 0
A B C D S
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
17
F5 output Four Variable Case Sum of Product Terms (SOPs)
F5 = S = A.B.C.D + A.B.C .D +A.B.C .D + A.B.C .D + A.B.C .D= =
ΣΣΣΣmn(0, 2, 7, 9, 12)
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
18
Advantage of using SOP form is that functions of any four input logic gate functions can be represented by maximum sixteen ANDs at the inputs and sixteen ORs at an output.
SOP form advantage
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
19
Outline
• SOP two variable case• SOP for three variable case • SOP for four variable case• Conversion of Boolean expression
into SOPs [Finding Miniterms]
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
20
• Suppose in a four variable SOP, there is a term with two variables, only C.D. We perform AND operation with (A + A).(B + B). [Using OR rule Equation (4) that OR of a complement of a variable or term with itself is always 1.]
Example
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
21
C.D = (A + A).(B+B).C.D = (A + A)(B.C.D + B.C.D)= A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D= Σ mn (15, 11, 7, 3)
= Σ mn (3, 7, 11, 15) and obtain the SOP standard form.
Conversion
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
23
We learnt:• A Boolean expression output can be
written as an SOP expression• SOP expression has the miniterms• Each miniterm represent that row of
truth table in which output = 1• Each miniterm is implemented by
AND gate(s)• Miniterms after ORing gives the
output
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
24
We learnt:• Using OR rules, a Boolean
expression with lesser number of variables can be expanded into SOP form to get all the miniterms and obtain SOP standard form
Ch04L4--"Digital Principles and Design", Raj Kamal, Pearson Education, 2006
25
End of Lesson 4
BOOLEAN EXPRESSION, TRUTH TABLE and
SUM OF THE PRODUCT