digital systems and binary numbers - national chiao tung ... · pdf file digital systems and...
Post on 07-Mar-2020
14 views
Embed Size (px)
TRANSCRIPT
Digital Circuit Design
Digital Systems and Binary Numbers
Lan-Da Van (范倫達), Ph. D.
Department of Computer Science
National Chiao Tung University Taiwan, R.O.C.
Spring, 2017
ldvan@cs.nctu.edu.tw
http://www.cs.nctu.edu.tw/~ldvan/
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-2
Digital Systems
Binary Numbers
Number-Base Conversion
Octal and Hexadecimal Number
Signed Binary Numbers
Binary Codes
Binary Storage and Registers
Binary Logic
Outline
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-3
Digital age and information age
Digital computers
– general purposes
– many scientific, industrial and commercial applications
Digital systems
– smartphone
– digital camera
– electronic calculators, PDA's
– digital TV
Discrete information-processing systems
– manipulate discrete elements of information
Digital System
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-4
A Digital Computer Example
Synchronous or
Asynchronous?
Inputs: keyboard,
mouse, modem,
microphone,
posture
Outputs: LCD,
modem, speakers
Memory
Control unit
Datapath
Input/Output
CPU
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-5
Signal
An information variable represented by physical quantity
For digital systems, the variable takes on digital values
Two level, or binary values are the most prevalent values
Binary values are represented abstractly by:
digits 0 and 1
words (symbols) False (F) and True (T)
words (symbols) Low (L) and High (H)
words On and Off.
Binary values are represented by values or ranges of
values of physical quantities
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-6
Signal
Analog signal
t->y: y=f(t), y:C, n:C
Discrete-time signal
n->y: y=f(nT), y:C, n:Z
Digital signal
n->y: y=D{f(nT)}, y:Z,n:Z
)3(
)1(
)2(
)1(
2)1110(
2)1000( 2)1011( 2)1000(
t n n
y y y
Analog Signal Discrete-Time Signal Digital Signal
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-7
Binary Numbers
Decimal number
5 4 3 2 1 0 1 2 3
5 4 3 2 1 0 1 2 310 10 10 10 10 10 10 10 10a a a a a a a a a
… a5a4a3a2a1a0.a1a2a3…
Decimal point
3 2 1 07,329 7 10 3 10 2 10 9 10
Example:
ja
Base or radix
Power
General form of base-r system 1 2 1 1 2
1 2 1 0 1 2
n n m
n n ma r a r a r a r a a r a r a r
Coefficient: aj = 0 to r 1
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-8
Example: Base-2 number
Binary Numbers
2 10
4 3 2 1 0 1 2
(11010.11) (26.75)
1 2 1 2 0 2 1 2 0 2 1 2 1 2
Example: Base-5 number
5
3 2 1 0 1
10
(4021.2)
4 5 0 5 2 5 1 5 2 5 (511.5)
Example: Base-8 number
Example: Base-16 number
3 2 1 0
16 10(B65F) 11 16 6 16 5 16 15 16 (46,687)
4
10 1012
8
)5.87(84878281
)4.127(
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-9
Binary Numbers
Example: Base-2 number
2 10(110101) 32 16 4 1 (53)
Special Powers of 2
210 (1024) is Kilo, denoted "K"
220 (1,048,576) is Mega, denoted "M"
230 (1,073, 741,824) is Giga, denoted "G"
240 (1.0995116e+12) is Tera, denoted “T"
250 (1.1258999e+15) is Peta, denoted “P"
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-10
Arithmetic operations with numbers in base r follow the same rules as decimal numbers.
Binary Numbers
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-11
Binary Arithmetic
Addition
Augend: 101101
Addend: +100111
Sum: 1010100
Subtraction
Minuend: 101101
Subtrahend: 100111
Difference: 000110
The binary multiplication table is simple:
0 0 = 0 | 1 0 = 0 | 0 1 = 0 | 1 1 = 1
Extending multiplication to multiple digits:
Multiplicand 1011
Multiplier 101
Partial Products 1011
0000 -
1011 - -
Product 110111
Multiplication
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-12
Octal and Hexadecimal Numbers
Numbers with different bases: Table 1.2.
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-13
Example1.1
Convert decimal 41 to binary. The process is continued until the integer quotient
becomes 0.
Number-Base Conversions
10/2
5/2
2/2
1/2
5
2
1
0
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-14
The arithmetic process can be manipulated more conveniently as follows:
Number-Base Conversions
Answer=(101001)2
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-15
Example 1.2
Convert decimal 153 to octal. The required base r is 8.
Number-Base Conversions
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-16
Example1.3
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.
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
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-17
Example1.4 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
Number-Base Conversions
Lecture 1
Digital Circuit Design
Lan-Da Van DCD-01-18
Conversion from binary to octal can be done by positioning the binary number into
groups of three digits each, starting from the binary point and proceeding to the left
and to the right.
Conversion from binary to hexadecimal i