UNIT V.pptx

Introduction to digital system
• A Digital system is an interconnection of digital modules and it is a
system that manipulates discrete elements of information that is
represented internally in the binary form.
• Now a day’s digital systems are used in wide variety of industrial and
consumer products such as automated industrial machinery, pocket
calculators, microprocessors, digital computers, digital watches, TV
games and signal processing and so on.

Analog systems vs Digital systems
• Analog system process information that varies continuously i.e; they
process time varying signals that can take on any values across a
continuous range of voltage, current or any physical parameter.
• Digital systems use digital circuits that can process digital signals
which can take either 0 or 1 for binary system

Advantages of Digital system over Analog system
1. Ease of programmability
2. Reduction in cost of hardware
3. gh speed
4. High Reliability
5. Design is easy
6. Result can be reproduced easily

Disadvantages of Digital Systems
 Use more energy than analog circuits to accomplish the same tasks, thus producing
more heat as well.
 Digital circuits are often fragile, in that if a single piece of digital data is lost or
misinterpreted the meaning of large blocks of related data can completely change.
 Digital computer manipulates discrete elements of information by means of a
binary code.
 Quantization error during analog signal sampling

NUMBER SYSTEM
• Number system is a basis for counting varies items.
• Modern computers communicate and operate with binary numbers which use only the
digits 0 &1. Basic number system used by humans is Decimal number system.
• For Ex: Let us consider decimal number 18. This number is represented in binary as
10010.
• We observe that binary number system take more digits to represent the decimal
number. For large numbers we have to deal with very large binary strings. So this fact
gave rise to three new number systems.
• i) Octal number systems
• ii) Hexa Decimal number system
• iii) Binary Coded Decimal number (BCD) system
• To define any number system we have to specify
•  Base of the number system such as 2,8,10 or 16.
•  The base decides the total number of digits available in that number system.
•  First digit in the number system is always zero and last digit in the number system is
always base-1.

Minimization Of Boolean Expressions-
• There are following two methods of minimizing or reducing the
boolean expressions-
1.By using laws of Boolean Algebra
2.By using Karnaugh Maps also called as K Maps

Karnaugh Map-
• The Karnaugh Map also called as K Map is a graphical
representation that provides a systematic method for simplifying
the boolean expressions.
• For a boolean expression consisting of n-variables, number of cells
required in K Map = 2n cells.

Two Variable K Map-
• Two variable K Map is drawn for a boolean expression consisting of
two variables.
• The number of cells present in two variable K Map = 22 = 4 cells.
• So, for a boolean function consisting of two variables, we draw a 2 x 2
K Map.
• Two variable K Map may be represented as-
Here, A and B are the two variables of the given
boolean function.

Three Variable K Map-
• Three variable K Map is drawn for a
boolean expression consisting of three
variables.
• The number of cells present in three
variable K Map = 23 = 8 cells.
• So, for a boolean function consisting of
three variables, we draw a 2 x 4 K Map.
• Three variable K Map may be
represented as-
• Here, A, B and C are the three variables
of the given boolean function.

Four Variable K Map-
• Four variable K Map is drawn for a
boolean expression consisting of four
variables.
• The number of cells present in four
variable K Map = 24 = 16 cells.
• So, for a boolean function consisting of
four variables, we draw a 4 x 4 K Map.
• Four variable K Map may be
represented as
• Here, A, B, C and D are the four
variables of the given boolean function.

Karnaugh Map Simplification Rules-
To minimize the given boolean function,
• We draw a K Map according to the number of variables it contains.
• We fill the K Map with 0’s and 1’s according to its function.
• Then, we minimize the function in accordance with the following rules.
Rule-01:
• We can either group 0’s with 0’s or 1’s with 1’s but we can not group 0’s and 1’s
together.
• X representing don’t care can be grouped with 0’s as well as 1’s.
NOTE
• There is no need of separately grouping X’s i.e. they can be ignored if all 0’s and
1’s are already grouped.

Rule-02:
• Groups may overlap each other.
Rule-03:
• We can only create a group whose number of cells can be represented in
the power of 2.
• In other words, a group can only contain 2n i.e. 1, 2, 4, 8, 16 and so on
number of cells.

Rule-04:
• Groups can be only either horizontal or vertical.
• We can not create groups of diagonal or any other shape.

Rule-05:
• Each group should be as large as possible.
• Example-

Rule-06:
• Opposite grouping and corner grouping are allowed.
• The example of opposite grouping is shown illustrated in Rule-05.
• The example of corner grouping is shown below.
Rule-07:
There should be as few groups as possible.

PROBLEMS BASED ON KARNAUGH MAP-
Problem-01:
Minimize the following boolean function-
F(A, B, C, D) = Σm(0, 1, 2, 5, 7, 8, 9, 10, 13, 15)
Solution-
• Since the given boolean expression has 4
variables, so we draw a 4 x 4 K Map.
• We fill the cells of K Map in accordance
with the given boolean function.
F(A, B, C, D)
= (A’B + AB)(C’D + CD) + (A’B’ + A’B + AB + AB’)C’D
+ (A’B’ + AB’)(C’D’ + CD’)
= BD + C’D + B’D’
Thus, minimized boolean expression is-
F(A, B, C, D) = BD + C’D + B’D’

Problem-02:
• Minimize the following boolean function-
• F(A, B, C, D) = Σm(0, 1, 3, 5, 7, 8, 9, 11, 13, 15)
Solution-
• We fill the cells of K Map in accordance with
the given boolean function.
• Then, we form the groups in accordance with
the above rules.
Now,
F(A, B, C, D)
= (A’B’ + A’B + AB + AB’)(C’D + CD) + (A’B’ +
AB’)(C’D’ + C’D)
= D + B’C’
F(A, B, C, D) = B’C’ + D

Problem-03:
• F(A, B, C, D) = Σm(1, 3, 4, 6, 8, 9, 11, 13, 15) +
Σd(0, 2, 14)
• Solution-
• We fill the cells of K Map in accordance with
the given boolean function.
• Then, we form the groups in accordance with
the above rules.
F(A, B, C, D)
= (AB + AB’)(C’D + CD) + (A’B’ + AB’)(C’D + CD) +
(A’B’ + AB’)(C’D’ + C’D) + (A’B’ + A’B)(C’D’ + CD’)
= AD + B’D + B’C’ + A’D’
F(A, B, C, D) = AD + B’D + B’C’ + A’D’

Problem-04:
• F(A, B, C) = Σm(0, 1, 6, 7) + Σd(3, 5)
Solution-
• We fill the cells of K Map in accordance
with the given boolean function.
• Then, we form the groups in accordance
with the above rules.
Now,
F(A, B, C)
= A'(B’C’ + B’C) + A(BC + BC’)
= A’B’ + AB
F(A, B, C) = AB + A’B’
NOTE-
It may be noted that there is no need of considering the
quad group.
This is because even if we consider that group, we will
have to consider the other two duets.
So, there is no use of considering that quad grou

Boolean algebra
• The logical symbol 0 and 1 are used for representing the digital input or output.
• The symbols "1" and "0" can also be used for a permanently open and closed digital circuit.
• The digital circuit can be made up of several logic gates.
• To perform the logical operation with minimum logic gates, a set of rules were invented,
known as the Laws of Boolean Algebra.
• These rules are used to reduce the number of logic gates for performing logic operations.
• The Boolean algebra is mainly used for simplifying and analyzing the complex Boolean
expression.
• It is also known as Binary algebra because we only use binary numbers in this.
• George Boole developed the binary algebra in 1854.

Rules in Boolean algebra
Following are the important rules used in Boolean algebra.
• Variable used can have only two values. Binary 1 for HIGH and Binary 0 for
LOW.
• Complement of a variable is represented by an overbar (-). Thus,
complement of variable B is represented as . Thus if B = 0 then = 1 and B
= 1 then = 0.
• ORing of the variables is represented by a plus (+) sign between them. For
example ORing of A, B, C is represented as A + B + C.
• Logical ANDing of the two or more variable is represented by writing a dot
between them such as A.B.C. Sometime the dot may be omitted like ABC.

Boolean Laws
Commutative law
• A. B = B. A
• A + B = B + A
Associative Law
• ( A. B ). C = A . ( B . C )
• ( A + B ) + C = A + ( B + C)
Distributive Law
• A. ( B + C) = (A. B) + (A. C)
• A + (B. C) = (A + B) . ( A + C)
AND Law
A .0 = 0
A . 1 = A
A. A = A
A.À = 0
OR Law
A + 0 = A
A + 1 = 1
A + A = A
A+À = 1
Inversion Law

UNIT V.pptx

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UNIT V.pptx