1. Digital Electronics & Logic Design
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2. Unit I Number System & Logic Design Minimization Techniques
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3. Digital Electronics & Logic Design
Syllabus
Unit I : (8 Hrs)
Number System& Logic Design Minimization Techniques
Introduction. Binary, Hexadecimal numbers, Octal numbers and number conversion.
Signed Binary number representation. Signed Magnitude, 1’s complement and 2’s complement representation. Binary, Hexadecimal Arithmetic. 2’s complement arithmetic.
Algebra for logic circuits : Logic variables;
Logic function : NOT, AND, NOR, XOR, OR, XNOR, NAND
Codes : BCD, Excess-3, Gray code , Binary Code and their conversion
Boolean algebra. Truth tables and Boolean algebra. Idealized logic gates and symbols. De Morgan's rules Axiomatic definition of Boolean algebra, Basic theorems and properties of Boolean algebra
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Introduction
We all are familiar with the decimal number system consisting of 10 symbols (0 to 9). It is the number system which we use in our day today life.
There are some other systems also, used to represent numbers, like binary, octal & hexadecimal.
These number systems are widely used in digital systems like microprocessors, logic circuits, computers etc. & knowledge of these systems is very essential for understanding, analyzing & designing of digital systems.
Computers & other digital circuits use binary signals but are required to handle data which may be numeric, alphabets or special characters. Therefore it is necessary to convert the information available in these formats into binary format.
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5. Digital Electronics & Logic Design
To achieve this, a process of coding is used & various codes like BCD, gray, excess 3, ASCII etc are used for this purpose.
In this unit we shall study the above mentioned number systems & codes as well the Boolean algebra which is used for logic design minimization.
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Number systems
In any number system there is an ordered set of symbols known as digits with rules defined for performing arithmetic operations. A collection of these digits makes a number which in general has 2 parts-integer & fractional set apart by a radix point (.), that is
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The digits in a number are placed side by side & each position in the number is assigned a weight. The table below gives the details of commonly used number systems.
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Binary number system
The number system with base 2 is known as binary number system. (0 & 1 these 2 symbols are used.)
It is a positional system.
Any number is a collection of the digits 0 & 1 & has 2 parts: integer & fractional. These 2 parts are set apart by the radix point which is also known as binary point.
Table shown next, illustrates counting in binary number system.
The leftmost bit is known as most significant bit (MSB) & the rightmost bit is known as least significant bit (LSB).
Group of 4 bits is known as nibble & group of 8 bits is known as byte.
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Binary number
Decimal number B3 B2 B1 B0 D1 D0 1 2 3 4 5 6 7 8 9
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Binary to decimal conversion:
Any binary number can be converted into its equivalent decimal number using the weights assigned to each bit position.
E.g. Find the decimal equivalent of the binary number
Solution: The equivalent decimal number is
= 1x24 + 1x23 + 1x22 + 1x21 + 1x20 =
= (31)10
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11. Digital Electronics & Logic Design
Decimal to binary conversion:
Any decimal number can be converted into its equivalent binary number. For integers, the conversion is obtained by continuous division by 2 & keeping track of the remainders generated till quotient is 0. While for the fractional parts, the conversion is obtained by continuous multiplication by 2 & keeping track of the integers generated till the fractional part becomes 0 (or up to 4 bits of binary.)
The conversion process is illustrated in the next example.
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E.g. Find the binary equivalent of the following decimal numbers
13 &
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Octal number system
The number system with base 8 is known as octal number system. (0 to 7 these 8 symbols are used.)
It is also a positional system.
Any number is a collection of the digits 0 to 7 & has 2 parts: integer & fractional. These 2 parts are set apart by the radix point which is also known as octal point.
In a number, the leftmost digit is known as most significant digit (MSD) & the rightmost digit is known as least significant digit (LSD).
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Octal to decimal conversion:
Any octal number can be converted into its equivalent decimal number using the weights assigned to each digit position.
E.g. Find the decimal equivalent of the octal number
Solution: The equivalent decimal number is
= 6x83 + 3x82 + 2x81 + 7x80 + 4x8-1 =
= (3287.5)10
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Decimal to octal conversion:
Any decimal number can be converted into its equivalent octal number. For integers, the conversion is obtained by continuous division by 8 & keeping track of the remainders till quotient is 0. While for the fractional parts, the conversion is obtained by continuous multiplication by 8 & keeping track of the integers generated till the fractional part becomes 0 (or up to 4 digits of octal.).
The conversion process is illustrated in the next example.
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E.g. Find the octal equivalent of the following decimal numbers
247 &
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Octal to binary conversion:
Octal numbers can be converted into equivalent binary numbers by replacing each octal digit with its 3 bit binary number. Table next, gives octal numbers & their binary equivalents for decimal numbers 0 to 15.
E.g. Find the binary equivalent of the octal number (736.5)8.
Solution: From the table shown next, the binary equivalents of 7, 3, 6 & 5 are 111, 011, 110 & 101 respectively. Therefore, the equivalent binary number is
( )2
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Octal
Decimal
Binary
000 1
001 2
010 3
011 4
100 5
101 6
110 7
111 10 8 11 9 12 13 14 15 16 17
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Binary to octal conversion:
Binary numbers can be converted into equivalent octal numbers by making groups of 3 bits starting from LSB & moving towards MSB for integer part of the number & then replacing each group of 3 bits by its octal representation.
For fractional part, the groupings of 3 bits are made starting from the binary point & moving towards right.
The conversion process is illustrated in the next examples.
From the examples it is clear that in forming the 3 bit groupings 0’s may be required to complete the first (most significant digit) group in the integer part & the last (least significant digit) group in the fractional part.
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E.g. Find the octal equivalent of the following binary numbers
( ) & ( )
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21. Hexadecimal number system
The number system with base 16 is known as hexadecimal number system.
It is also a positional system.
Any number is a collection of the digits 0 to 9 & alphabets A to F. It has 2 parts: integer & fractional. These 2 parts are set apart by the radix point which is also known as hexadecimal point.
In a number, the leftmost digit is known as most significant digit (MSD) & the rightmost digit is known as least significant digit (LSD).
Table next, gives hexadecimal numbers with their binary equivalents for decimal numbers 0 to 15.
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Hexadecimal
Decimal
Binary
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 A 10
1010 B 11
1011 C 12
1100 D 13
1101 E 14
1110 F 15
1111
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Hexadecimal to decimal conversion:
Any hexadecimal number can be converted into its equivalent decimal number using the weights assigned to each bit position.
E.g. Find the decimal equivalent of the hexadecimal number 3A.4
Solution: The equivalent decimal number is
= 3x161 + Ax x16-1 =
= (58.25)10
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Decimal to hexadecimal conversion:
Any decimal number can be converted into its equivalent hexadecimal number. For integers, the conversion is obtained by continuous division by 16 & keeping track of the remainders till quotient is 0. While for the fractional parts, the conversion is obtained by continuous multiplication by 16 & keeping track of the integers generated till the fractional part becomes 0 (or up to 4 digits of hex.).
The conversion process is illustrated in the next example.
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E.g. Find the hexadecimal equivalent of the decimal number
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Hexadecimal to binary conversion:
Hexadecimal numbers can be converted into equivalent binary numbers by replacing each hexadecimal digit with its 4 bit binary number.
E.g. Find the binary equivalent of the hexadecimal number 2F9.A
Solution: From the table, the binary equivalents of 2, F, 9 & A are 0010, 1111, 1001 & 1010 respectively.
Therefore, the equivalent binary number is
( )2
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Binary to hexadecimal conversion:
Binary numbers can be converted into equivalent hexadecimal numbers by making groups of 4 bits starting from LSB & moving towards MSB for integer part of the number & then replacing each group of 4 bits by its hexadecimal representation.
For fractional part, the groupings of 4 bits are made starting from the binary point.
The conversion process is illustrated in the next examples.
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E.g. Find the hexadecimal equivalent of the given binary numbers
( ) & ( )
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Conversion from hex to octal & vice-versa
Hexadecimal numbers can be converted to equivalent octal numbers & octal numbers can be converted to equivalent hexadecimal numbers by converting the hex/octal number to equivalent binary & then to octal/hex respectively. The conversion process is illustrated in the next examples.
E.g. Convert (247.36)8 to equivalent hex number.
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E.g. Convert the following hex numbers to octal numbers.
A72E & 0.BF85
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31. Signed binary number representation
In the decimal number system, a plus (+) sign is used to denote a positive number & a minus (-) sign is used to denote a negative number. This representation of numbers is known as signed number.
As the digital circuits can understand only 2 symbols 0 & 1; same is used to indicate the sign of the number also.
The 3 types of signed binary number representation are-
i) Sign magnitude representation
ii) One’s complement representation &
iii) Two’s complement representation.
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i) Sign magnitude representation
In this representation, an additional bit is used as the sign bit & is placed as the most significant bit. A 0 is used to represent a positive number & a 1 to represent a negative number. The remaining bits of the number give the magnitude of it. It is illustrated in the following examples.
E.g. Find the decimal equivalent of the following binary numbers assuming sign magnitude representation.
&
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ii) One’s complement representation
In a binary number if each 1 is replaced by 0 & each 0 by 1, the resulting number is known as the one’s complement of the first number.
Both the numbers are complement of each other. If one of these numbers is positive, then the other number will be negative with the same magnitude. In this representation also, MSB is 0 for positive numbers & 1 for negative numbers.
E.g. Find the one’s complement of the following numbers
&
Solution: = =
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iii) Two’s complement representation
If 1 is added to one’s complement of a binary number, the resulting number is known as the two’s complement of the binary number.
Both the numbers are complement of each other. If one of these numbers is positive, then the other number will be negative with the same magnitude.
In this representation also, MSB is 0 for positive numbers & 1 for negative numbers.
The two’s complement of the two’s complement of a number is the number itself.
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E.g. Find the two’s complement of the following numbers
&
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36. Digital Electronics & Logic Design
Binary arithmetic
Just like decimal numbers, arithmetic operations such as addition, subtraction, multiplication & division can be performed on binary numbers also.
The rules of binary addition are given in the following table. A B
Sum
Carry
Result 1 10
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In case of multibit addition, if the carry is generated as shown in last row of above table, it is added to the next higher binary position. It is as shown in the following example.
E.g. Add the binary numbers: 0101 & 1111.
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The rules of binary subtraction are given in the following table.
In case of multibit subtraction, if the borrow is generated as shown in second row of the table, it is transferred to the next higher binary position.
This is as shown in the next example. A B
Difference
Borrow 1
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In binary multiplication, each partial product is either 0 (multiplication by 0) or exactly same as the multiplicand (multiplication by 1). An example is given below.
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Binary division is obtained using the same procedure as decimal division. An example is given below.
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42. Hexadecimal arithmetic
The arithmetic operations in hexadecimal number system are actually performed in binary number system as each hex operand is first converted into binary & then the specified operation is carried out using the rules of binary arithmetic.
At the end, the binary result is converted back to the hex equivalent. It is shown in the next examples.
E.g. Add (7F)16 & (BA)16
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43. Two’s complement arithmetic
Digital circuits are used for performing binary arithmetic operations.
With the two’s complement arithmetic, a problem of subtraction can be converted into a problem of addition.
This eliminates the need of additional circuits for subtraction as the circuits of addition can be used for performing both addition and subtraction.
This makes design of arithmetic circuits very convenient and cheaper.
Rules of two’s complement arithmetic are different for the operands with opposite sign and the operands with same sign.
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I) Rules for operands with opposite sign
First, we find the two’s complement representation of both the operands & then carry out their addition.
It should be noted that the two’s complement representation of positive numbers is same as that of their sign magnitude representation. Also the two’s complement representation for negative numbers is obtained from the corresponding positive number.
After performing the addition, if a final carry is generated, discard the carry and the answer is positive & the magnitude is given by the remaining bits.
After performing the addition, if a final carry is not generated, the answer is negative & is in two’s complement form.
This is illustrated in the next example.
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II) Rules for operands with same sign
In case of the two operands with the same sign, after performing the addition, the sign bit of the result (MSB) is required to be compared with the sign bit of the operands after performing the addition.
In case the sign bits are same, the result is correct & is in two’s complement form.
If the sign bits are not same, there is a problem of overflow i.e., the result is too large to fit in the destination.
This is illustrated in the next example.
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E. g. Perform the following. Use 8 bit representation:
i) 48 -(-23) ii)
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Logic variables
Logic variables can be represented by a letter symbol such as A, B, X, Y,…..
The variable can have only one of the two values 0 or 1.
These variables are used for denoting the various inputs & outputs in a digital circuit.
George Boole developed rules for manipulations of binary variables which is known as Boolean algebra. This is the basis of all digital systems.
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Logic functions
In a digital system there are only a few basic operations performed, irrespective of the complexities of the system. These operations are also known as logic functions
These are required to be performed a number of times in a large digital system.
The basic operations are AND, OR, NOT & FLIP FLOP.
The AND, OR, NOT operations are discussed here & the FLIP FLOP will be covered n unit IV.
Along with the basic logic functions, we shall study the other logic functions such as NAND, NOR, EXOR & EXNOR.
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The NOT operation
Figure below shows the logic diagram, logic equation & the truth table of a NOT gate, which is also known as an inverter.
It has one input (A) and one output (Y).
The presence of a small circle, known as the bubble, always denotes inversion in digital circuits.
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The AND operation
Figure below shows the logic diagram, logic equation & the truth table of a AND gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 1, if and only if both the inputs (A) & (B) are 1.
Thus, in general for a N input AND gate, the output (Y) is 1, when all its inputs are high. If any input is low, the output (Y) is zero.
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The OR operation
Figure below shows the logic diagram, logic equation & the truth table of a OR gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 1, if any one of the inputs (A) or (B) is 1.
Thus in general for a N input OR gate, the output (Y) is 1, when any one of the inputs is high. If all inputs are low, then only the output (Y) is zero.
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NAND & NOR operations
Any logic expression can be realized by using he AND, OR & NOT gates discussed earlier.
From these 3 operations 2 more operations are derived; the NAND operation & the NOR operation. NAND is NOT AND operation & NOR is NOT OR operation.
These operations are widely used because only one type of gates, NAND or NOR are sufficient for the realization of any logical expression.
Because of this, these gates are known as universal gates.
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The NAND operation:
Figure below shows the logic diagram, logic equation & the truth table of a NAND gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 0, if and only if both the inputs (A) & (B) are 1.
Thus, in general for a N input NAND gate, the output (Y) is 0, when all its inputs are high. If any input is low, the output (Y) is one.
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The 3 basic logic operations, AND, OR & NOT can be performed by using only NAND gates. These are as shown in the figure below.
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The NOR operation:
Figure below shows the logic diagram, logic equation & the truth table of a NOR gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 1, if and only if both the inputs (A) & (B) are 0.
Thus, in general for a N input NOR gate, the output (Y) is 0, when any of its inputs is high. If all inputs are low, the output (Y) is one.
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The 3 basic logic operations, AND, OR & NOT can be performed by using only NOR gates. These are as shown in the figure below.
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The EX-OR operation
Figure below shows the logic diagram, logic equation & the truth table of a EX-OR gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 0, if the inputs (A) & (B) are same (either 0 or 1) whereas when the inputs are not same (one of them is 0 & the other one is 1) the output is 1.
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The EX-NOR operation
Figure below shows the logic diagram, logic equation & the truth table of a EX-NOR gate with two inputs.
The two inputs are (A) and (B) and single output (Y).
From the truth table it is clear that, the output (Y) is 0, if the inputs (A) & (B) are not same (one of them is 0 & the other one is 1) whereas when the inputs are same (either 0 or 1) ) the output is 1.
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Codes
Computers & other digital circuits use binary signals but are required to handle data which may be numeric, alphabets or special characters. Therefore it is necessary to convert the information available in these formats into binary format.
To achieve this, a process of coding is used & various codes like BCD, gray, excess 3, ASCII etc are used for this purpose.
As every code uses the binary representation, the interpretation of it is possible only if the name of code is known.
E.g. binary number represents 65 decimal in straight binary, 41 decimal in BCD, & an alphabet A in ASCII code.
The above mentioned codes are discussed here.
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Straight binary code
This is used to represent numbers using natural binary form as discussed earlier.
Various arithmetic operations can be performed in this form.
It is a weighted code as weight is assigned to every position.
Binary codes for decimal numbers 0 to 15 are shown in the next table.
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Binary code
Decimal number B3 B2 B1 B0 D1 D0 1 2 3 4 5 6 7 8 9
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Natural BCD code
In this code, decimal digits 0 to 9 are represented by their natural binary equivalents using 4 bits as shown in the next table..
Each decimal digit of a decimal number is represented by this 4 bit code individually.
E.g. (23)10 = in BCD
For the same representation, only 5 bits are required in case of binary. Still, BCD is popular as it is very convenient & useful code for input & output operations in digital systems.
This is a weighted code & is also known as a code which represent the weights of the 4 bits.
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BCD
Decimal number D C B A 1 2 3 4 5 6 7 8 9
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Excess 3 code
This is another form of BCD code, in which each decimal digit is coded into a 4 bit binary code as shown in the next table..
The code for each decimal digit is obtained by adding 3 to the natural BCD code of the digit
E.g. decimal 2 is coded as = 0101 in excess 3 code.
This code is self complementing code, which means 1’s complement of the coded number yields 9’s complement of the number itself.
E.g. excess 3 code of decimal 2 is 0101, its 1’s complement is 1010 which is excess 3 code for 7, which is 9’s complement of 2. this property is useful in performing subtraction operation in digital systems.
This is a non weighted code.
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Excess 3 code
Decimal number E3 E2 E1 E0 1 2 3 4 5 6 7 8 9
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Gray code
This is a very useful code in which a decimal number is represented in binary form in such a way that each gray code number differs from the preceding & succeeding number by a single bit.
Gray codes for decimal numbers 0 to 15 are shown in the next table.
Due to this property this code is extensively used for shaft encoders.
It is a non weighted code.
The gray code is a reflected code & can be constructed using this property as explained next:
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Gray code
Decimal number G3 G2 G1 G0 D1 D0 1 2 3 4 5 6 7 8 9
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A 1 bit code has 2 code words 0 & 1 representing decimal numbers 0 & 1 respectively.
An n bit (n >= 2) gray code will have first 2n-1 gray codes of n-1 bits written in order with a leading 0 appended.
The last 2n-1 gray codes will be equal to the gray code words of an n-1 bit gray code, written in reverse order (assuming a mirror placed between first 2n-1 & last 2n-1 gray codes) with a leading 1 appended.
It is illustrated in the next example.
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Boolean Algebra
Binary variables can be represented by a letter symbol such as A, B, X, Y,….. The variable can have only one of 2 values 0 or 1.
George Boole developed rules for manipulations of binary variables which is known as Boolean algebra. This is the basis of all digital systems.
The Bolean algebraic theorems are given in the next tables.
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Theorem No Theorems
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Theorem No Theorems
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From these theorems we observe that the even numbered theorems can be obtained from their preceding odd numbered theorems by (i) interchanging + & . Signs, and (ii) interchanging 0 & 1.
The theorems which are related in this way are called duals.
Theorems 1.1 to 1.8 involve a single variable & can be proved by considering every possible value of the variable.
E. g. in theorem 1.1,
If A = 0 then = 0 = A
and if A = 1 then = 1 = A
and hence the theorem is proved.
Theorems 1.9 to 1.20 involve more than 1 variable & can be proved by making a truth table.
E. g. theorem 1.10 can be proved with the truth table shown next.
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From the table, it is clear that there are 8 possible combinations of the 3 variables A, B & C.
For each combination, the value of A + BC is the same as that of (A + B) . (A + C), which proves the theorem.
Theorems 1.21 & 1.22 are known as De Morgan’s theorems. These theorems can be proved by first considering the 2 variable case & then extending this result.
The proof is as shown in the next table.
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