R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
UNIT-1
DATA REPRESENTATION AND DIGITAL CIRCUITS-I
Course Objectives:
• To familiarize with the concepts of different number systems and codes.
• To understand the representation and operations on floating-point numbers
• To understand the representation and Arithmetic operations using complements
• To understand the operation of Boolean algebra and logic gates
• To learn K-map method
• To understand various combinational logic circuits.
Course Outcomes:
Students will be able to
• understand various number systems.
• perform the arithmetic operations using complementary methods.
• understand basic theorems and properties of Boolean algebra.
• understand basic logic operations and gates.
• Determine the minimized Boolean function using K-maps
• Design adders and subtractors.
• Understand other combinational circuits like decoder, multiplexer
Syllabus:
1.1 Binary Numbers
1.2 Fixed Point Representation
1.3 Floating Point Representation
1.4 Number Base Conversions
1.5 Octal and Hexadecimal Numbers
1.6 Complements
1.6.1 Radix and Diminished Radix Complement
1.6.2 Subtraction with Complements
1.7 Signed Binary Numbers
1.7.1 Number Representation
1.7.2 Arithmetic Addition
1.7.3 Arithmetic Subtraction
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 1
,R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
1.8 Binary Codes
1.8.1 Binary Coded Decimal Numbers
1.8.2 BCD Addition
1.8.3 Other Decimal codes
1.8.4 Gray Code
1.9 Basic Logic Functions
1.9.1 Postulates and Theorems
1.10 Logic Gates
1.10.1 Logic Gates
1.10.2 Gates Implementation of Expressions
1.11 Universal Logic Gates
1.11.1 NAND,NOR
1.11.2 Gates Realization using NAND,NOR
1.12 Minimization of Logic Expressions
1.12.1 Example
1.13 K-Map Simplifications
1.13.1 2-Variable K Map
1.13.2 3-Variable K Map
1.13.3 4-Variable K Map
1.13.4 Don’t Care Conditions
1.14 Combinational Circuits
1.14.1 Design Procedure (code conversion)
1.14.2 Half Adder, Half Subtractor
1.14.3 Half Subtractor, Full Subtractor
1.15 Decoders
1.15.1 3x8 Decoder
1.15.2 4x16 with two 3x8 Decoders
1.16 Multiplexers
1.16.1 2x1,4x1 Multiplexers
1.16.2 Boolean Function Implementation
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 2
,R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
1.1 Binary Numbers
• According to digital electronics and mathematics, a binary number is defined as a
number that is expressed in the binary system or base 2 numeral system. It describes
numeric values by two separate symbols; 1 (one) and 0 (zero). The base-2 system is the
positional notation with 2 as a radix.
• The binary system is applied internally by almost all latest computers and computer-
based devices because of its direct implementation in electronic circuits using logic
gates. Every digit is referred to as a bit.
• Binary: allowed digits(0, 1), Radix -2
• The number of numerical values the system uses is called the Base or Radix of the
system
1.2 Fixed Point Representation
• A fixed-point binary number will be stored in a given number of bits called the word
length. Some of the bits in the word would be used for the integer part and the rest of
them for the fractional part. In addition, a bit would also be used to signify the sign of
the number.
• Example:Given that a fixed- point positive binary number is stored in eight bits word
length, where the first six bits are used for the integer part and the next two for the
fractional part, how would the number (13.875)10 is stored ?
• We are given (13.875)10.
• 13 is represented as 1101 and 0.875 =.111
• By fixing 6 bits for integer and 2 bits for fractional the answer will be (001101.11)2.
Two zeros are appended for integer part and one bit is truncated from fractional part.
1.3 Floating Point Representation
• A floating point number in computer registers consists of two parts: a mantissa m and an
exponent e.
• The two parts represent a number obtained from multiplying m times a radix r raised to
the value of e.
• m x re
• The mantissa may be a fraction or an integer. The location of the radix point and the
value of the radix r are assumed and are not included in the registers.
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 3
, R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
• For e.g., the decimal number +6132.789 is represented in floating point with a fraction
and exponent as follows:
Fraction Exponent
+0.6132789 +04
• The above representation is equivalent to scientific notation +0.6132789 X 104
• A floating point binary number is represented in a similar manner to floating point
decimal number except that it uses base 2 for exponent.
• For e.g., the binary number +1001.11 is represented as 8-bit fraction and 6-bit exponent
as follows:
Fraction Exponent
01001110 000100
1.4 Conversion of numbers from one radix to another radix
• To convert among different number systems. We use decimal numbers every day.
Computers understand only binary numbers, which are lengthy and inconvenient to
human beings. Octal and Hexadecimal numbers are introduced to make both happy:
they are easier to be converted to binary numbers and also easier for us to handle.
Fig 1.1 Classification of numbers
Unsigned Numbers
Radices and Characters:
• Binary : 0, 1
• Decimal : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Octal : 0, 1, 2, 3, 4, 5, 6, 7
• Hexadecimal : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 4
UNIT-1
DATA REPRESENTATION AND DIGITAL CIRCUITS-I
Course Objectives:
• To familiarize with the concepts of different number systems and codes.
• To understand the representation and operations on floating-point numbers
• To understand the representation and Arithmetic operations using complements
• To understand the operation of Boolean algebra and logic gates
• To learn K-map method
• To understand various combinational logic circuits.
Course Outcomes:
Students will be able to
• understand various number systems.
• perform the arithmetic operations using complementary methods.
• understand basic theorems and properties of Boolean algebra.
• understand basic logic operations and gates.
• Determine the minimized Boolean function using K-maps
• Design adders and subtractors.
• Understand other combinational circuits like decoder, multiplexer
Syllabus:
1.1 Binary Numbers
1.2 Fixed Point Representation
1.3 Floating Point Representation
1.4 Number Base Conversions
1.5 Octal and Hexadecimal Numbers
1.6 Complements
1.6.1 Radix and Diminished Radix Complement
1.6.2 Subtraction with Complements
1.7 Signed Binary Numbers
1.7.1 Number Representation
1.7.2 Arithmetic Addition
1.7.3 Arithmetic Subtraction
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 1
,R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
1.8 Binary Codes
1.8.1 Binary Coded Decimal Numbers
1.8.2 BCD Addition
1.8.3 Other Decimal codes
1.8.4 Gray Code
1.9 Basic Logic Functions
1.9.1 Postulates and Theorems
1.10 Logic Gates
1.10.1 Logic Gates
1.10.2 Gates Implementation of Expressions
1.11 Universal Logic Gates
1.11.1 NAND,NOR
1.11.2 Gates Realization using NAND,NOR
1.12 Minimization of Logic Expressions
1.12.1 Example
1.13 K-Map Simplifications
1.13.1 2-Variable K Map
1.13.2 3-Variable K Map
1.13.3 4-Variable K Map
1.13.4 Don’t Care Conditions
1.14 Combinational Circuits
1.14.1 Design Procedure (code conversion)
1.14.2 Half Adder, Half Subtractor
1.14.3 Half Subtractor, Full Subtractor
1.15 Decoders
1.15.1 3x8 Decoder
1.15.2 4x16 with two 3x8 Decoders
1.16 Multiplexers
1.16.1 2x1,4x1 Multiplexers
1.16.2 Boolean Function Implementation
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 2
,R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
1.1 Binary Numbers
• According to digital electronics and mathematics, a binary number is defined as a
number that is expressed in the binary system or base 2 numeral system. It describes
numeric values by two separate symbols; 1 (one) and 0 (zero). The base-2 system is the
positional notation with 2 as a radix.
• The binary system is applied internally by almost all latest computers and computer-
based devices because of its direct implementation in electronic circuits using logic
gates. Every digit is referred to as a bit.
• Binary: allowed digits(0, 1), Radix -2
• The number of numerical values the system uses is called the Base or Radix of the
system
1.2 Fixed Point Representation
• A fixed-point binary number will be stored in a given number of bits called the word
length. Some of the bits in the word would be used for the integer part and the rest of
them for the fractional part. In addition, a bit would also be used to signify the sign of
the number.
• Example:Given that a fixed- point positive binary number is stored in eight bits word
length, where the first six bits are used for the integer part and the next two for the
fractional part, how would the number (13.875)10 is stored ?
• We are given (13.875)10.
• 13 is represented as 1101 and 0.875 =.111
• By fixing 6 bits for integer and 2 bits for fractional the answer will be (001101.11)2.
Two zeros are appended for integer part and one bit is truncated from fractional part.
1.3 Floating Point Representation
• A floating point number in computer registers consists of two parts: a mantissa m and an
exponent e.
• The two parts represent a number obtained from multiplying m times a radix r raised to
the value of e.
• m x re
• The mantissa may be a fraction or an integer. The location of the radix point and the
value of the radix r are assumed and are not included in the registers.
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 3
, R23 DIGITAL LOGIC & COMPUTER ORGANIZATION II B.TECH II- SEM
• For e.g., the decimal number +6132.789 is represented in floating point with a fraction
and exponent as follows:
Fraction Exponent
+0.6132789 +04
• The above representation is equivalent to scientific notation +0.6132789 X 104
• A floating point binary number is represented in a similar manner to floating point
decimal number except that it uses base 2 for exponent.
• For e.g., the binary number +1001.11 is represented as 8-bit fraction and 6-bit exponent
as follows:
Fraction Exponent
01001110 000100
1.4 Conversion of numbers from one radix to another radix
• To convert among different number systems. We use decimal numbers every day.
Computers understand only binary numbers, which are lengthy and inconvenient to
human beings. Octal and Hexadecimal numbers are introduced to make both happy:
they are easier to be converted to binary numbers and also easier for us to handle.
Fig 1.1 Classification of numbers
Unsigned Numbers
Radices and Characters:
• Binary : 0, 1
• Decimal : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Octal : 0, 1, 2, 3, 4, 5, 6, 7
• Hexadecimal : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Seshadri Rao Gudlavalleru Engineering College Dept. of ECE AS&A Page 4
