Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Chapter 1 Digital Systems and Binary Numbers
Typology: Slides
1 / 15
This page cannot be seen from the preview
Don't miss anything!
Digital System Ch1- 1
Logics Design
Chapter 1 : Digital Systems and Binary
Numbers
Outline of Chapter 1
1.1 Digital Systems
1.2 Binary Numbers
1.3 Number-base Conversions
1.4 Octal and Hexadecimal Numbers
1.5 Complements
1.6 Signed Binary Numbers
1.7 Binary Codes
Digital System Ch1- 3
1.1 Digital Systems and Binary Numbers (p.17)
Digital age and information age
Digital computers
General purposes Many scientific, industrial and commercial applications
Digital systems
Telephone switching exchanges Digital camera Electronic calculators, PDA's Digital TV
Discrete information-processing systems
Manipulate discrete elements of information For example, {1, 2, 3, …} and {A, B, C, …}…
Analog and Digital Signal
Analog system
The physical quantities or signals may vary continuously over a specified range.
Digital system
The physical quantities or signals can assume only discrete values. Greater accuracy
t
X(t)
t
X(t)
Analog signal Digital signal
Digital System Ch1- 7
Binary Numbers (p.20, 21)
Special Powers of 2
2 10 (1024) is Kilo, denoted "K" 2 20 (1,048,576) is Mega, denoted "M" 2 30 (1,073,741,824) is Giga, denoted "G"
Powers of two
Example: Base-2 number (110101) 2 32 16 4 1 (53) 10
Arithmetic operation
Arithmetic operations with numbers in base-r follow the same rules as decimal numbers.
Binary Arithmetic (p.21)
Single Bit Addition with Carry
Multiple Bit Addition
Single Bit Subtraction with Borrow
Multiple Bit Subtraction
Multiplication
BCD Addition
Digital System Ch1- 9
Addition
Multiplication
Subtraction
Augend: 101101 Addend: + Sum: 1010100
Minuend: 101101 Subtrahend: - 100111 Difference: 0000110
Multiplicand 1011 Multiplier × 101 Partial Products 1011 0000 1011 Product 110111
Binary Arithmetic (p.21)
1.3 Number-Base Conversions (p.21)
The six letters (in addition to the 10 integers) in hexadecimal
represent: 10 (A), 11 (B), 12 (C), 13 (D), 14 (E), and 15 (F), respectively.
Name Radix Digits Binary 2 0, Octal 8 0,1,2,3,4,5,6, Decimal 10 0,1,2,3,4,5,6,7,8, Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Digital System Ch1- 13
Number-Base Conversions (p.22, 23)
Example 1.
Convert decimal 153 to octal. The required base r is 8.
Example1.
Convert (0.6875) 10 to binary. The process is continued until the fraction becomes 0 or until the number of digits has sufficient accuracy.
Number-Base Conversions
Example1.3 (cont.)
To convert a decimal fraction to a number expressed in base r, a similar procedure is used. However, multiplication is by r instead of 2, and the coefficients found from the integers may range in value from 0 to r 1 instead of 0 and 1.
Digital System Ch1- 15
Number-Base Conversions (p.23)
Example1. Convert (0.513) 10 to octal.
From Examples 1.1 and 1.3: (41.6875) 10 = (101001.1011) 2 From Examples 1.2 and 1.4: (153.513) 10 = (231.406517) 8
1.4 Octal and Hexadecimal Numbers (p.24)
Numbers with different bases: Table 1.2.
Digital System Ch1- 19
Complements (p.28)
Subtraction with Complements
Example 1.
Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y ; and (b) Y X, by using 2's complement.
There is no end carry. Therefore, the answer is Y – X = (2's complement of 1101111) = 0010001.
1.6 Signed Binary Numbers (p.30)
To represent negative integers, we need a notation for negative
values.
It is customary to represent the sign with a bit placed in the
leftmost position of the number since binary digits.
The convention is to make the sign bit 0 for positive and 1 for
negative.
Example:
Table 1.3 lists all possible four-bit signed binary numbers in the
three representations.
Digital System Ch1- 21
Signed Binary Numbers (p.31)
Signed Binary Numbers (p.32)
Arithmetic addition
The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign of the larger magnitude. The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits. A carry out of the sign-bit position is discarded.
Example:
Digital System Ch1- 25
Binary Code (p.35)
Example:
Consider decimal 185 and its corresponding value in BCD and binary:
BCD addition
Binary Code (p.36)
Example:
Consider the addition of 184 + 576 = 760 in BCD:
Decimal Arithmetic: (+375) + (-240) = +
Digital System Ch1- 27
Binary Codes (p.38)
Gray Code
The advantage is that only bit in the code group changes in going from one number to the next. » Error detection. » Representation of analog data. » Low power design. 000 001
010 100
110 111
101
011
1-1 and onto!!
Binary Codes (p.39, 40)
American Standard Code for Information Interchange (ASCII) Character Code
