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binary addition, multiplexers, bcd,and many much
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II B. Tech I Semester (Autonomous-R17)
CHADALAWADA RAMANAMMA ENGINEERING COLLEGE
Department of Computer Science and Engineering
II B.Tech I Semester: CSE Course Code Category Hours / Week Credits Maximum Marks
17CA0430 6 Core
L T P C CI AE SEEE Total
2 2 - 3 30 70 100 Contact Classes: 34 Tutorial Classes: 34 Practical Classes: Nil Total Classes: 6 8 Objectives: The course should enable the students to: Analyze and explore the uses of Logic Functions for Building Digital Logic Circuits Explore the Combinational Logic Circuits. Examine the Operation of Sequential (Synchronous and Asynchronous) Circuits. Unit^ ^ Know the Concepts of Basic Memory System. - I NUMBERS SYSTEMS AND CODES Classes: Review of Number Systems, Number Base Conversion. Binary Arithmetic: Binary Weighted and Non-Weighted Codes. Complements: Signed Binary Numbers, Error detection and correcting codes, Binary Storage and Registers, Binary logic. Unit-II^ Boolean Algebra and Gate Level Minimization^ Classes: Postulates and Theorems, Representation of Switching Functions, SOP and POS Forms, Canonical Forms, Digital Logic Gate. Karnaugh Maps: Minimization using Three variable, Four variable, Five variable K- Maps; Don‟t Care Conditions, NAND and NOR implementation, Other Two-level Implementation, Exclusive – OR function. Unit-III Design of Combinational Circuits Classes: Combinational circuits: Analysis and Design Procedure, Binary Adder and Subtractors, Carry Look-a-head Adder, Binary Multiplier, Magnitude comparator, BCD Adder, Decoders, Encoders, Multiplexers, Demultiplexer. Unit-IV Design of Sequential Circuits Classes: 13 Combinational versus Sequential circuits , Latches, Flip Flops: RS Flip Flop, JK Flip Flop, T Flip Flop, D Flip Flop, Master-Slave Flip Flop, Flip Flops Excitation Functions, Conversion of one Flip Flop to another Flip Flop, Shift Registers, Design of Asynchronous and Synchronous Circuits, State Table, State Diagram, State Reduction and State Assignment for Mealy and Moore Machines. Unit-V Memory Classes: 14 Random Access Memory, Types of ROM, Memory Decoding, Address and Data Bus, Sequential Memory, Cache Memory, Programmable Logic Arrays, Memory Hierarchy in terms of Capacity and Access time.
Text Book:
Analog Vs Digital Digital Systems Binary numbers Number base conversions Compliments Octal and Hexadecimal Numbers Signed Binary Numbers
ANALOG Vs DIGITAL:
To learn and understand about the digital logic design, the initial knowledge we require is to differentiate between analog and digital. The following are fews that differentiate between analog and digital.
Compliments are used in digital computers to simplify the subtraction operation and for logical manipulation. Simplifying operations leads to simpler, less expensive circuits to implement the operations.
There are 2 types of complements for each base r system.
(1) The radix complement (2) Diminished radix compliment
Radix compliment: Also referred to as the r‟s compliment.
Diminished radix compliment: Also referred to as (r-1)‟s compliment.
OCTAL NUMBERS
The Octal Number System is another type of computer and digital base number system. The Octal Numbering System is very similar in principle to the previous hexadecimal numbering system except that in Octal, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 7 ). Octal numbers therefore have a range of just “8” digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base- numbering system and therefore, q is equal to “8”.
HEXADECIMAL NUMBERING SYSTEM: The one main disadvantage of binary numbers is that the binary string equivalent of a large decimal base-10 number can be quite long. When working with large digital systems, such as computers, it is common to find binary numbers consisting of 8, 16 and even 32 digits which makes it difficult to both read and write without producing errors especially when working with lots of 16 or 32-bit binary numbers. One common way of overcoming this problem is to arrange the binary numbers into groups or sets of four bits (4-bits). These groups of 4-bits uses another type of numbering system also commonly used in computer and digital systems called Hexadecimal Numbers
The “Hexadecimal” or simply “Hex” numbering system uses the Base of 16 system and are a popular choice for representing long binary values because their format is quite compact and much easier to understand compared to the long binary strings of 1‟s and 0‟s.
Being a Base-16 system, the hexadecimal numbering system therefore uses 16 (sixteen) different digits with a combination of numbers from 0 through to 15. In other words, there are 16 possible digit symbols. Decima Binar Octal Hexadeci l y mal 0 0000 0 0 1 0001 1 1 2 0010 2 2 3 0011 3 3 4 0100 4 4 5 0101 5 5 6 0110 6 6 7 0111 7 7 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
In mathematics, positive numbers (including zero) are represented as unsigned numbers. That is we do not put the +ve sign in front of them to show that they are positive numbers. However, when dealing with negative numbers we do use a -ve sign in front of the number to show that the number is negative in value and different from a positive unsigned value and the
is also called as binary code. The binary code is represented by the number as well as alphanumeric letter.
Advantages of Binary Code
Following is the list of advantages that binary code offers.
Binary codes are suitable for the computer applications. Binary codes are suitable for the digital communications. Binary codes make the analysis and designing of digital circuits if we use the binary codes. Since only 0 & 1 are being used, implementation becomes easy. Classification of binary codes
The codes are broadly categorized into following four categories.
Weighted Codes Non-Weighted Codes Binary Coded Decimal Code Alphanumeric Codes Error Detecting Codes Error Correcting Codes
Weighted Codes
Weighted binary codes are those binary codes which obey the positional weight principle. Each position of the number represents a specific weight. Several systems of the codes are used to express the decimal digits 0 through 9. In these codes each decimal digit is represented by a group of four bits.
Non-Weighted Codes
In this type of binary codes, the positional weights are not assigned. The examples of non- weighted codes are Excess-3 code and Gray code.
Excess-3 code
The Excess-3 code is also called as XS-3 code. It is non-weighted code used to express decimal numbers. The Excess-3 code words are derived from the 8421 BCD code words adding (0011) or (3)10 to each code word in 8421. The excess-3 codes are obtained as follows.
Gray Code
It is the non-weighted code and it is not arithmetic codes. That means there are no specific weights assigned to the bit position. It has a very special feature that, only one bit will change each time the decimal number is incremented as shown in fig. As only one bit changes at a time, the gray code is called as a unit distance code. The gray code is a cyclic code. Gray code cannot be used for arithmetic operation. Application of Gray code
Gray code is popularly used in the shaft position encoders. A shaft position encoder produces a code word which represents the angular position of the shaft. Binary Coded Decimal (BCD) code
In this code each decimal digit is represented by a 4-bit binary number. BCD is a way to express each of the decimal digits with a binary code. In the BCD, with four bits we can
Extended Binary Coded Decimal Interchange Code (EBCDIC). Five bit Baudot Code.
ASCII code is a 7-bit code whereas EBCDIC is an 8-bit code. ASCII code is more commonly used worldwide while EBCDIC is used primarily in large IBM computers.
Error Codes
There are binary code techniques available to detect and correct data during data transmission.
NUMBER BASE CONVERSIONS There are many methods or techniques which can be used to convert code from one format to another. We'll demonstrate here the following Binary to BCD Conversion BCD to Binary Conversion BCD to Excess-3 Excess-3 to BCD
Binary to BCD Conversion
Steps
Step 1 -- Convert the binary number to decimal. Step 2 -- Convert decimal number to BCD.
Example − convert (11101) 2 to BCD.
Step 1 − Convert to Decimal
Binary Number − 11101 2
Calculating Decimal Equivalent −
Step Binary Number Decimal Number
Step 1 111012 ((1 × 2^4 ) + (1 × 2^3 ) + (1 × 2^2 ) + (0 × 2^1 ) + (1 × 2^0 )) 10
Step 2 111012 (16 + 8 + 4 + 0 + 1) 10
Step 3 (^111012 )
Binary Number − 11101 2 = Decimal Number − 29 10
Step 2 − Convert to BCD
Decimal Number − 29 10
Calculating BCD Equivalent. Convert each digit into groups of four binary digits equivalent
Step Decimal Number Conversion
Step 1 (^2910 00102 )
Step 2 (^2910 00101001) BCD
Result
(11101) 2 = (00101001)BCD
BCD to Binary Conversion
Used long division method for decimal to binary conversion.
Decimal Number − 29 10
Calculating Binary Equivalent
Step Operation Result Remainder
Step 1 29 / 2 14 1
Step 2 14 / 2 7 0
Step 3 7 / 2 3 1
Step 4 3 / 2 1 1
Step 5 1 / 2 0 1
As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the least significant digit (LSD) and the last remainder becomes the most significant digit (MSD).
Decimal Number − 29 10 = Binary Number − 11101 2
Result (00101001)BCD = (11101) 2
BCD to Excess-
Steps
Step 1 -- Convert BCD to decimal.
Step 2 -- Add (3) 10 to this decimal number. Step 3 -- Convert into binary to get excess-3 code.
Example − convert (1001)BCD to Excess-3.
Step 1 − Convert to decimal
(1001)BCD = 9 10
Step 2 − Add 3 to decimal
(9) 10 + (3) 10 = (12) 10
Step 3 − Convert to Excess-
(12) 10 = (1100) 2
Result (1001)BCD = (1100)XS-
Excess-3 to BCD Conversion
Steps
Step 1 -- Subtract (0011) 2 from each 4 bit of excess-3 digit to obtain the
corresponding BCD code.
Example − convert (10011010)XS-3 to BCD.
Given XS-3 number = 1 0 0 1 1 0 1 0 Subtract (0011) 2 = 0 0 1 1 0 0 1 1
BCD = 0 1 1 0 0 1 1 1
Result
(10011010)XS-3 = (01100111)BCD
^ ^ Basic Definitions Axiomatic Definition of Boolean Algebra
^ ^ Basic Theorems and properties of Boolean Algebra ^ ^ Boolean Functions ^ ^ Canonical and Standard Forms, Other Logic Operations ^ ^ Digital Logic Gates Integrated Circuits Boolean Algebra: Boolean algebra, like any other deductive mathematical system, may be defined with aset of elements, a set of operators, and a number of unproved axioms or postulates. A set of elements is anycollection of objects having a common property. If S is a set and x and y are certain objects, then x Î S denotes that x is a member of the set S , and y Ï S denotes that y is not an element of S. A set with adenumerable number of elements is specified by braces: A = {1,2,3,4}, i.e. the elements of set A are thenumbers 1, 2, 3, and 4. A binary operator defined on a set S of elements is a rule that assigns to each pair ofelements from S a unique element from S._ Example: In a * b=c , we say that * is a binary operator if it specifies a rule for finding c from the pair ( a , b )and also if a , b , c Î S.
CLOSURE: The Boolean system is closed with respect to a binary operator if for every pair of Boolean values,it produces a Boolean result. For example, logical AND is closed in the Boolean system because it accepts only Boolean operands and produces only Boolean results.
_ A set S is closed with respect to a binary operator if, for every pair of elements of S , the binary operator specifies a rule for obtaining a unique element of S.
For example, the set of natural numbers N = {1, 2, 3, 4, … 9} is closed with respect to the binary operator plus (+) by the rule of arithmetic addition, since for any a , b Î N we obtain a unique c Î N by the operation a + b = c. ASSOCIATIVE LAW: A binary operator * on a set S is said to be associative whenever ( x * y ) * z = x * ( y * z ) for all x , y , z Î S, forall Boolean values x, y and z.
COMMUTATIVE LAW:
A binary operator * on a set S is said to be commutative whenever x * y = y * x for all x , y , z є IDENTITY ELEMENT:
