Chapter 1. Digital Systems and Binary Numbers - Tong In Oh · Chapter 1. Digital Systems and ......

Digital Design, Kyung Hee Univ.

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Chapter 1. Digital Systems and Binary Numbers

Tong In Oh


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1.1 Digital Systems• Digital age• Characteristic of digital system

• Generality and flexibility• Represent and manipulate discrete elements of information

• Binary (bit)• ‘0’ and ‘1’• Binary codes

• General purpose digital computer• Memory / central processing / input-output

• HDL• Digital system = manipulate discrete quantities of information that are

represented in binary form


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1.2 Binary Numbers


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1.3 Number-Base Conversions


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1.4 Octal and Hexadecimal Numbers


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1.5 Complements of Numbers


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Subtraction with Complements


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1.6 Signed Binary Numbers• Plus / minus sign• A bit at leftmost position (0 for positive, 1 for negative)• Ex. 01001

• 9 (unsigned binary)• +9 (signed binary)

• Ex. 11001• 25 (unsigned binary)• -9 (signed binary)

• Type of representation• Signed-magnitude system

• 10001001• Signed-complement system

• 1’s complement: 11110110• 2’s complement: 11110111


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Arithmetic Addition• Addition of two signed binary numbers with negative numbers

represented in signed- 2’s-complement form • Addition of the two numbers, including their sign bits• Discard a carry out of the sign-bit position

• Overflow is a problem in computers• Number of bits that hold a number is finite• A result that exceeds the finite value by 1 cannot be accommodated


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Arithmetic Subtraction• Take the 2’s complement of the subtrahend (including the sign bit)• Add it to the minuend (including the sign bit)• Discard a carry out of the sign-bit position

• (-6)-(-13)=+7• 1111 1010 – 1111 0011 = 1111 1010 + 0000 1101 = 1 0000 0111


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1.7 Binary Codes• Digital system

• Signal of two distinct values• Circuit elements for two stable states• Binary digits

• Any discrete element of information• Represent with a binary code• Patterns of 0’s and 1’s• N-bit binary code: a group of n bits

that assumes up to 2n distinct combinations of 1’s and 0’s

• Each combination representing one element of the set being coded

• Ex. (185)10 = (0001 1000 0101)BCD = (1011 1001) 2


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BCD Addition• Maximum 9+9+1 = 19 (0001 1001)BCD


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Decimal Arithmetic• Represent sign of a decimal number with four bits to conform to the

four-bit code of the decimal digits• To designate a plus with four 0’s and a minus with the BCD equivalent

of 9, which is 1001• 9’s or 10’s complement• Addition by summing all digits, including the sign digit• Discarding the end carry• Assume that all negative numbers are in 10’s-complement form


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Other Decimal Codes• BCD and 2421 code:

weighted code• Assign each bit position to a

weighting factor• Each digit can be evaluated

by adding the weights of all the 1’s in the coded combination

• 2421 and excess-3 code: self-complementing codes

• Obtain 9’s complement of a decimal number

• Directly changing 1’s to 0’s and 0’s to 1’s

• Excess-3: unweighted code • Corresponding binary value

plus 3


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Gray Code• Physical quantities are continuous• Only one bit in the code group changes

in going from one number to the next• Normal sequence of binary numbers

generated by the hardware may produce an error or ambiguity during the transition from one number to the next

• 0111 → 1000

• Angular position of a shaft


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ASCII Character Code


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• Control characters• Format effectors / information separators / communication-control


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Error-Detecting Code• To detect errors in data communication and processing• Add an eighth bit to the ASCII character to indicate its parity• A parity bit is an extra bit included with a message to make the total

number of 1’s either even or odd• Leftmost position of the code to produce an even number of 1’s in the

character for even parity or an odd number of 1’s in the character for odd parity

• Detects one, three, or any odd combination of errors in each character


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1.8 Binary Storage and Registers• Binary cell: a device that possesses two stable states and is capable of

storing one bit (0 or 1) of information• Register: a group of binary cells

• N-tuple of 1’s and 0’s• Store discrete elements of information and that the same bit configuration may

be interpreted differently for different types of data depending on the application

• Digital logic circuits• Process the binary information stored in the registers


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Register Transfer• The basic elements for storing and holding the binary information


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1.9 Binary logic


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Logic Gates• Electronic circuits that operate on one or

more input signals to produce an output signal

• In a digital system, interpret voltages as two recognizable values (0 or 1)

• Voltage operated logic circuits: two separate voltage levels

• Logic 0 = 0V• Logic 1 = 3V• Acceptable range


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• Gate: block of hardware that produces the equivalent of logic 1 or logic 0 output signals if input logic requirements are satisfied

• Transition between logic values occur quickly, but not instantaneously

Logic Gate Circuit

Chapter 1. Digital Systems and Binary Numbers - Tong In Oh · Chapter 1. Digital Systems and ......

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