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Week-2 (Introduction to Digital Logic Design)

CE102 Digital Logic Design

Çiğdem Sazak Turgut 2022

Week-2 (Introduction to Digital Logic Design)

Spring Semester, 2021-2022

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PART 1: BINARY SYSTEMS

Binary Systems

  • Analog Vs Digital
  • Digital Systems Binarynumbers
  • Number base conversions Compliments Binary Systems
  • Octal and Hexadecimal Numbers
  • Signed Binary Numbers

Analog and Digital

  • Analog information is made up of a continuum of values within a given range.

  • At its most basic, digital information can assume only one of two possible values:

  • one/zero,
  • on/off,
  • high/low,

  • true/false, etc.

  • Digital Information is less susceptible to noise than analog information

  • Exact voltage values are not important, only their class (1 or 0)

  • The complexity of operations is reduced, thus it is easier to implement them with high accuracy in digital form.

Digital Systems

  • Digital;
  • generates stores
  • processes data \(\downarrow\)
    • two states:
    • positive (\(1\)) and
    • non-postitive (\(0\))

Digital Systems

  • A "digital system" is a data technology that uses discrete (discontinuous) values represented by high and low states known as bits.
  • non-digital (or analog) systems use a continuous range of values to represent information

Binary Number System

  • Binary;

  • describes a numbering scheme in which there are only two possible values for each digit: 0 and 1

  • Binary Number System

  • a numbering system
  • represents numeric values using 0 and 1
  • known as the base-2 number system

BINARY NUMBER EXAMPLE

  • 10
  • 111
  • 10101
  • 11110

COMPLIMENTS

  • used in digital computers to simplify the subtraction operation and for logical manipulation
  • 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

  • a binary number is divided up into groups of only 3 bits

  • 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-8 numbering system and therefore, q is equal to “8”

HEXADECIMAL NUMBERING SYSTEM

  • main disadvantage of binary numbers
  • the binary string equivalent of a large decimal base-10 number can be quite long
  • Working with large digital systems, such as computers, it is common to find binary numbers consisting of 8, 16 and even 32 digits
  • Overcome the above problem:
  • 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
  • uses the Base of 16 system
  • Hexdecimal system format is quite compact and much easier to understand

HEXADECIMAL NUMBERING SYSTEM

Decimal   Binary    Octal   Hexadecimal
----------------------------------------
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

SIGNED BINARY NUMBERS

  • In mathematics,
  • positive numbers (including zero) are represented as unsigned numbers we do not put the (\(+\)) ve sign in front of them to show that they are positive numbers
  • When dealing with negative numbers we do use a (\(-\)) sign in front of the number to show that the number is negative in value and different from a positive unsigned value and the same is true with signed binary numbers
  • However in digital circuits
  • there is no provision made to put a plus or even a minus sign to a number
  • digital systems operate with binary numbers that are represented in terms of "\(0\)"s" and "\(1\)"s"
  • to represent a positive (N) and a negative (-N) binary number we can use the binary numbers with sign
  • For signed binary numbers the most significant bit (MSB) is used as the sign
  • If the sign bit is "\(0\)":
  • the number is positive
  • If the sign bit is "\(1\)":

  • the number is negative

  • The remaining bits are used to represent the magnitude of the binary number in the usual unsigned binary number format.

Positive Signed Binary Number

  • 8-bit word
\[ \begin{bmatrix} & | & \overbrace{0}^{+sign} & | & 0 & | & 1 &| &\overbrace{1}^{magnitude} & | & 0 & | & 1 & | & 0 & | & 1 & | & \end{bmatrix} = 53 \]

Negative Signed Binary Number

  • 8-bit word
\[ \begin{bmatrix} & | & \overbrace{1}^{-sign} & | & 0 & | & 1 &| &\overbrace{1}^{magnitude} & | & 0 & | & 1 & | & 0 & | & 1 & | & \end{bmatrix} = -53 \]

BINARY CODES

  • In the coding,
  • when numbers, letters or words are represented by a specific group of symbols, it is said that the number, letter or word is being encoded
  • The group of symbols is called as a code
  • digital data is represented, stored and transmitted as group of binary bits
  • called BINARYCODE

Advantages of Binary Code

  • 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

  • Weighted Codes
  • Non-Weighted Codes
  • Binary Coded Decimal Code
  • Alphanumeric Codes
  • Error Detecting Codes
  • Error Correcting Codes

Weighted Codes

  • 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
\[ \begin{align*} & = 24 \\ & = \overbrace{2}^{0010} \overbrace{4}^{0100} \\ & \downarrow \\ & \overbrace{0}^{8}\overbrace{0}^{4}\overbrace{1}^{2}\overbrace{0}^{1} = 2 \\ & \overbrace{0}^{8}\overbrace{1}^{4}\overbrace{0}^{2}\overbrace{0}^{1} = 4 \end{align*} \]

Non-Weighted Codes

  • In this type of binary codes,
  • The positional weights are not assigned
  • The examples of nonweighted codes are Excess-3 code and Gray code

Excess-3 Code

  • also called 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)2 or (3)10 to each code word in 8421

The excess-3 codes are obtained as follows Example : Decimal \(\Longrightarrow\) \(8421_{BCD}\) \(\Longrightarrow\) Excess-3

Decimal   BCD   Excess-3
          8421  BCD+0011
-------------------------
0         0000  0011
1         0001  0100
2         0010  0101
3         0011  0110
4         0100  0111
5         0101  1000
6         0110  1001
7         0111  1010

Gray Code

  • It is the non-weighted code and it is not arithmetic codes
  • 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 represent sixteen numbers (0000 to 1111)

Decimal    0    1    2    3    4    5    6    7    8   9
------------------------------------------------------------
BCD     0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Alphanumeric Codes

  • Abinary digit or bit can represent only two symbols as it has only two states '0' or '1'
  • But this is not enough for communication between two computers because there we need many more symbols for communication.
  • These symbols are required to represent 26 alphabets with capital and small letters, numbers from 0 to 9, punctuation marks and other symbols
  • The alphanumeric codes are the codes that represent numbers and alphabetic characters
  • Mostly such codes also represent other characters such as symbol and various instructions necessary for conveying information
  • The following three alphanumeric codes are very commonly used for the data representation.
  • American Standard Code for Information Interchange (ASCII)
  • Extended Binary Coded Decimal Interchange Code (EBCDIC)
  • Five bit Baudot Code

Number Base Conversions

  • Binary to BCD Conversion
  • BCD to Binary Conversion
  • BCD to Excess-3
  • Excess-3 to BCD

Binary to BCD Conversion

  • Step-1: Convert the binary number to decimal
  • Step-2: Convert decimal number to BCD

Step-1 : Binary to Decimal Conversion

Convert to Decimal Equivalent

Example : Convert \((11101)_2\) to BCD

\[ \begin{align*} & = (11101)_2 \\ & = ((1\times 2^4) + (1 \times 2^3)+ (1 \times 2^2)+(0 \times 2^1)+(1 \times 2^0))_{10} \\ & = (16+8+4+0+1)_{10} \\ & = 29_{10} \\ & \downarrow \\ & (11101)_2 = 29_{10} \end{align*} \]

Step-2: Decimal to BCD Conversion

Convert to BCD Equivalent

Example : Convert \((11101)_2\) to BCD

Convert each digit into groups of four binary digits equivalent

\[ \begin{align*} & = (11101)_2 = 29_{10}\\ & = 29_{10} \\ & = 0010_21001_2 \\ & = (00101001)_{BCD} \\ & \downarrow \\ & (11101)_2 = (00101001)_{BCD} \end{align*} \]

BCD to Decimal Conversion

  • Calculating Decimal Equivalent
  • Convert each four digit into a group and get decimal equivalent for each group
\[ \begin{align*} & = (00101001)_{BCD}\\ & = 0010_21001_2 \\ & = 2_{10}9_{10} \\ & = 29_{10} \\ & \downarrow \\ & (00101001)_{BCD} = 29_{10} \end{align*} \]
  • Calculating Binary Equivalent of \(29_{10}\)
  • Used long division method for decimal to binary conversion
\[ \begin{align*} \text{Step-1} & = 29/2 \Longrightarrow result:14 \ remainder:1\\ \text{Step-2} & = 14/2 \Longrightarrow result:7 \ remainder:0\\ \text{Step-3} & = 7/2 \Longrightarrow result:3 \ remainder:1\\ \text{Step-4} & = 3/2 \Longrightarrow result:1 \ remainder:1\\ \text{Step-5} & = 1/2 \Longrightarrow result:0 \ remainder:1\\ & \downarrow \\ & 29_{10} = (11101)_{2} = (00101001)_{BCD} \end{align*} \]

BCD to Excess-3 Conversion

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

BCD to Excess-3 Conversion

Example − convert \((1001)_{BCD}\) to Excess-3

\[ \begin{align*} & = \text{Step-1:Convert to Decimal} \rightarrow (1001)_{BCD} = 9_{10}\\ & = \text{Step-2:Add 3 to decimal} \rightarrow 9_{10} + 3_{10} = 12_{10} \\ & = \text{Step-3:Convert to Excess-3} \rightarrow 12_{10} = (1100)_2 \\ & \downarrow \\ & (1001)_{BCD} = (1100)_{XS-3} \end{align*} \]

Excess-3 to BCD Conversion

  • Subtract \((0011)_2\) from each \(4\) bit of excess-3 digit to obtain the corresponding BCD code

Excess-3 to BCD Conversion

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

\[ \begin{align*} (10011010)_{XS-3} = (01100111)_{BCD} \end{align*} \]

PART 2: BINARY ARITHMETIC SYSTEMS

References

\(End-Of-Week-2-Module\)

Last update: March 4, 2022
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