digital logic design number system

number system in digital logic design

1. 1
Digital Logic Design

2. 2
Digital Logic Design
°Digital
-Concerned with the interconnection among digital
components and modules
»Best Digital System example is General Purpose
Computer
°Logic Design
-Deals with the basic concepts and tools used to design
digital hardware consisting of logic circuits
»Circuits to perform arithmetic operations (+, -, x, ÷)

3. 3
DigiDigi
tal tal
SigSig
nalsnals
°Decimal values are difficult to represent in electrical
systems. It is easier to use two voltage values than
ten.
°Digital Signals have two basic states:
1 (logic “high”, or H, or “on”)
0 (logic “low”, or L, or “off”)
°Digital values are in a binary format. Binary means
2 states.
°A good example of binary is a light (only on or off)
on off
Power switches have labels “1” for on and “0” for off.

4. 4
Digital Logic Design
°Bits and Pieces of DLD History
°George Boole
-Mathematical Analysis of Logic (1847)
-An Investigation of Laws of Thoughts; Mathematical
Theories of Logic and Probabilities (1854)
°Claude Shannon
-Rediscovered the Boole
-“ A Symbolic Analysis of Relay and Switching Circuits “
-Boolean Logic and Boolean Algebra were Applied to Digital
Circuitry
---------- Beginning of the Digital Age and/or Computer Age
World War II
Computers as Calculating Machines
Arlington (State Machines) “ Control “

5. 5
Motivation
°Microprocessors/Microelectronics have
revolutionized our world
•Cell phones, internet, rapid advances in medicine, etc.
°The semiconductor industry has grown tremendously

6. 6
Objectives
°Number System, Their Uses, Conversions
°Basic Building Blocks of Digital System
°Minimization
°Combinational And Sequential Logic
°Digital System/Circuit Analysis and Design
°State Minimizations
°Integrated Circuits
°Simulations

7. 7
Text Book
°Primary Text:
“Digital Design” By M. Morris Mano and Michael D.
Ciletti
°Complementary Material
“Logic and Computer Design Fundamentals” By M.
Morris Mano & Charles R Kime.

8. 8
Digital Logic Design
Lecture 1
Number Systems

9. 9
Number Systems
°Decimal is the number system that we use
°Binary is a number system that computers use
°Octal is a number system that represents groups of
binary numbers (binary shorthand). It is used in
digital displays, and in modern times in
conjunction with file permissions under Unix
systems.
°Hexadecimal (Hex) is a number system that
represents groups of binary numbers (binary
shorthand). Hex is primarily used in computing as
the most common form of expressing a human-
readable string representation of a byte (group of 8
bits).

10. 10
Overvie
w
°The design of computers
•It all starts with numbers
•Building circuits
•Building computing machines
°Digital systems
°Understanding decimal numbers
°Binary and octal numbers
•The basis of computers!
°Conversion between different number systems

11. 11
Analog vs. Digital
Consider a faucet
Digital
Water can be flowing or NOT flowing
from the faucet
Two States
•On
•Off
Analog
How much water is flowing from the
faucet?
Advantages of Digital
Replication
•Analog
Try replicating the exact flow
from a faucet
•Digital
Try replicating ON or OFF

12. 12
Advantages of Digital
oError Correction/Detection
•Small errors don’t propagate
oMiniaturization of Circuits
oProgrammability
•Digital computers are programmable
°Two discrete values are used in digital systems.
°How are discrete elements represented?
•Signals are the physical quantities used to represent discrete
elements of information in a digital system.
°Electric signals used:
•Voltage
•Current

13. 13
Advantages of Digital/Representation of Binary Values
V
o
l
t
s
- 1 . 0
0 . 0
1 . 0
2 . 0
3 . 0
4 . 0
5 . 0
6 . 0
H i g h
L o w
°Why are there voltage ranges instead
of exact voltages?
•Variations in circuit behavior & noise
oTwo possible values
•1, 0
•On, Off
•True, False
•High, Low
•Heads, Tails
•Black, White

14. 14

15. 15
Digital Computer Systems
°Digital systems consider discrete amounts of data.
°Examples
•26 letters in the alphabet
•10 decimal digits
°Larger quantities can be built from discrete values:
•Words made of letters
•Numbers made of decimal digits (e.g. 239875.32)
°Computers operate on binary values (0 and 1)
°Easy to represent binary values electrically
•Voltages and currents.
•Can be implemented using circuits
•Create the building blocks of modern computers

16. 16
Understanding Decimal Numbers
°Decimal numbers are made of decimal digits:
(0,1,2,3,4,5,6,7,8,9)
°But how many items does a decimal number
represent?
•8653 = 8x10
3
+ 6x10
2 +
5x10
1 +
3x10
0

°What about fractions?
•97654.35 = 9x10
4
+ 7x10
3 +
6x10
2 +
5x10
1 +
4x10
0
+ 3x10
-1 +
5x10
-2
•In formal notation -> (97654.35)
10
°Why do we use 10 digits, anyway?

17. 17
Understanding Octal
Numbers
°Octal numbers are made of octal digits:
(0,1,2,3,4,5,6,7)
°How many items does an octal number represent?
•(4536)
8
= 4x8
3
+ 5x8
2 +
3x8
1 +
6x8
0
= (1362)
10

°What about fractions?
•(465.27)
8
= 4x8
2 +
6x8
1 +
5x8
0
+ 2x8
-1 +
7x8
-2
°Octal numbers don’t use digits 8 or 9
°Who would use octal number, anyway?

18. 18
Understanding Binary Numbers
°Binary numbers are made of binary digits (bits):
•0 and 1
°How many items does an binary number represent?
•(1011)
2
= 1x2
3
+ 0x2
2 +
1x2
1 +
1x2
0
= (11)
10

°What about fractions?
•(110.10)
2
= 1x2
2 +
1x2
1 +
0x2
0
+ 1x2
-1 +
0x2
-2
°Groups of eight bits are called a byte
•(11001001)
2
°Groups of four bits are called a nibble.
• (1101)
2

19. 19
Why Use Binary Numbers?
°Easy to represent 0 and 1 using
electrical values.
°Possible to tolerate noise.
°Easy to transmit data
°Easy to build binary circuits.
AND Gate
1
0
0

20. 20
BinaryBinary
Base 2 = Base 10
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
In Binary, there are only 0’s and 1’s. These numbers are called “Base-2” ( Example:
010
2
)
B
i
n
a
r
y

t
o

D
e
c
i
m
a
l
We count in “Base-10”
(0 to 9)
°Binary number has base 2
°Each digit is one of two
numbers: 0 and 1
°Each digit is called a bit
°Eight binary bits make a byte
°All 256 possible values of a
byte can be represented using
2 digits in hexadecimal
notation.

21. 21
Binary as a VoltageBinary as a Voltage
°Voltages are used to represent logic values:
°A voltage present (called Vcc or Vdd) = 1
°Zero Volts or ground (called gnd or Vss) = 0
A simple switch can provide a logic high or a logic low.

22. 22
A Simple SwitchA Simple Switch
°Here is a simple switch used to provide a logic value:
Vcc
Gnd, or 0
Vcc
Vcc, or 1
There are other ways to connect a switch.

23. 23
Binary digits
Bit: single binary digit
Byte: 8 binary digits
10010111
2
Bit
Byte
Radix

24. 24
Conversion Between Number Bases
Decimal(base 10)
Octal(base 8)
Binary(base 2)
Hexadecimal
(base16)
°Learn to convert between bases.
°Already demonstrated how to convert
from binary to decimal.
°Hexadecimal described in next
lecture.

25. 25
Number Systems
SystemBaseSymbols
Used by
humans?
Used in
computers?
Decimal100, 1, … 9Yes No
Binary 20, 1 No Yes
Octal 80, 1, … 7 No No
Hexa-
decimal
160, 1, … 9,
A, B, … F
No No

26. 26
Con
ver
sio
n
Am
ong
Bas
es
The possibilities:
Hexadecimal
Decimal Octal
Binary

27. 27
Convert an Integer from Decimal to Another
Base
1.Divide decimal number by the base (e.g. 2)
2.The remainder is the lowest-order digit
3.Repeat first two steps until no divisor remains.
For each digit position:
Example for (13)
10:
Integer
Quotient
13/2 = 6 + ½ a
0
= 1
6/2 = 3 + 0 a
1
= 0
3/2 = 1 + ½ a
2
= 1
1/2 = 0 + ½ a
3
= 1

RemainderCoefficient
Answer (13)
10
= (a
3
a
2
a
1
a
0
)
2
= (1101)
2

28. 28
Convert an Fraction from Decimal to Another
Base
1.Multiply decimal number by the base (e.g. 2)
2.The integer is the highest-order digit
3.Repeat first two steps until fraction becomes
zero.
For each digit position:
Example for (0.625)
10:
Integer
0.625 x 2 = 1 + 0.25 a
-1
= 1
0.250 x 2 = 0 + 0.50 a
-2
= 0
0.500 x 2 = 1 + 0 a
-3
= 1

Fraction Coefficient
Answer (0.625)
10
= (0.a
-1
a
-2
a
-3
)
2
= (0.101)
2

29. 29
The Growth of Binary
Numbers
n 2
n
0 2
0
=1
1 2
1
=2
2 2
2
=4
3 2
3
=8
4 2
4
=16
5 2
5
=32
6 2
6
=64
7 2
7
=128
n 2
n
8 2
8
=256
9 2
9
=512
10 2
10
=1024
11 2
11
=2048
12 2
12
=4096
20 2
20
=1M
30 2
30
=1G
40 2
40
=1T
Mega
Giga
Tera

30. 30
Binary
Addition
°Binary addition is very simple.
°This is best shown in an example of adding two
binary numbers…
1 1 1 1 0 1
+ 1 0 1 1 1
---------------------
0
1
0
1
1
1111
1 100
carries

31. 31
Binary Subtraction
°We can also perform subtraction (with borrows in place of
carries).
°Let’s subtract (10111)
2
from (1001101)
2
…
1 10
0 10 10 0 0 10
1 0 0 1 1 0 1
- 1 0 1 1 1
------------------------
1 1 0 1 1 0
borrows

32. 32
Binary
Multiplication
°Binary multiplication is much the same as decimal
multiplication, except that the multiplication
operations are much simpler…
1 0 1 1 1
X 1 0 1 0
-----------------------
0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
-----------------------
1 1 1 0 0 1 1 0

33. 33
Convert an Integer from Decimal to
Octal
1.Divide decimal number by the base (8)
2.The remainder is the lowest-order digit
3.Repeat first two steps until no divisor remains.
For each digit position:
Example for (175)
10:
Integer
Quotient
175/8 = 21 + 7/8 a
0
= 7
21/8 = 2 + 5/8 a
1
= 5
2/8 = 0 + 2/8 a
2
= 2

RemainderCoefficient
Answer (175)
10
= (a
2
a
1
a
0
)
2
= (257)
8

34. 34
Convert an Fraction from Decimal to
Octal
1.Multiply decimal number by the base (e.g. 8)
2.The integer is the highest-order digit
3.Repeat first two steps until fraction becomes
zero.
For each digit position:
Example for (0.3125)
10:
Integer
0.3125 x 8 = 2 + 5 a
-1
= 2
0.5000 x 8 = 4 + 0 a
-2
= 4

Fraction Coefficient
Answer (0.3125)
10
= (0.24)
8

35. 35
Summary
°Binary numbers are made of binary digits (bits)
°Binary and octal number systems
°Conversion between number systems
°Addition, subtraction, and multiplication in binary