Digital System Design
Lecture 1 - Number Systems and Code
Instructors:
Dr. Nguyen Tuan Khanh, postdoc, khanh.nt@vgu.edu.vn
M.Sc. Nguyen Vo That Thuyet, lab engineer, thuyet.nvt@vgu.edu.vn
Electrical and Computer Engineering (ECE)
Faculty of Engineering, Vietnamese-German University, Vietnam
Sep. 2024
1
Outline
•Binary-to-Decimal Conversions
•Putting It All Together
•Decimal-to-Binary Conversions
•The Byte, Nibble, and Word
•Hexadecimal Number System
•Alphanumeric Codes
•BCD Code
•The Gray Code
•Parity Method for Error
Detection
•Applications
Binary-to-Decimal Conversions
LSB
MSB
Decimal-to-Binary Conversions
Decimal-to-Binary Conversions
•Repeated Division
Decimal-to-Binary Conversions
•Repeated Division
Start
Is Q = 0
Divide by 2
Record quotient (Q)
and remainder (R)
Collect R’s into desired
binary number with first
as LSB and last R as MSB
Start
Decimal-to-Binary Conversions
•Counting Range
Using N bits, we can represent decimal numbers ranging from 0 to 2N -1,
a total of 2N different numbers
Hexadecimal Number System
Hexadecimal Number System
•Hex-to-Decimal Conversion
Hexadecimal Number System
•Decimal-to-Hex Conversion
Hexadecimal Number System
•Hex-to-Binary Conversion
Hexadecimal Number System
•Binary-to-Hex Conversion
•Counting in Hexadecimal
•Usefulness of Hex
Conversion Summary
•Why is conversion is necessary?
•Summary
•Binary/Hex 🡪 Decimal
•Decimal 🡪 Binary/Hex
•Hex 🡪 Binary
•Binary 🡪 Hex
BCD Code
•Decimal ↔ Straight binary coding
•Binary-coded-decimal (hereafter abbreviated BCD)
•Comparison of BCD and Binary
The Gray Code
•Only one bit ever changes between two successive
numbers in the sequence
The Gray Code
•Converting (a) binary to Gray and (b) Gray to binary
The Gray Code
•An eight-position, three-bit shaft encoder.
Putting All Together
The Byte, Nibble, and Word
•Byte
•Nibble
•Word
Alphanumeric Codes
•All of the various characters and functions that are
found on a computer keyboard
•A, B, C, …, a, b, c, …, 1, 2, 3, …, ., ,, ?, …, +, /, #, …
•American Standard Code for Information Interchange
(ASCII)
•27 = 128 possible code groups
•Transfer of alphanumeric info.: a computer ↔ external
devices
•Store the information that an operator types in at the
computer’s keyboard.
Alphanumeric Codes
•Example
padding with 0s
Parity Method for Error Detection
•Movement of binary data and codes from one
location
•Error can occur
•Electrical noise
•Detection code
•Correction code
Parity Method for Error Detection
•Parity bit
•Even-parity method
•Odd-parity method
•Detection code, not correction code
•Single-bit error detection
•Checking the parity, request for retransmission,
retransmission
Applications
• Transmitting an ASCII character
(Application 2-5)
• Example: transmit a Delete