1. ENG2410 Digital Design “Introduction to Digital Systems”
Fall 2017
S. Areibi
School of Engineering
University of Guelph

2. Resources Chapter #1, Mano Sections 1.1 Digital Computers
1.2 Number Systems
1.3 Arithmetic Operations
1.4 Decimal Codes
1.5 Alphanumeric Codes
School of Engineering

3. Topics Computing Devices and VLSI Design Signals (Digital vs. Analog)
Digital Systems and Computers
Number systems [binary, octal, hex]
Base Conversion
Arithmetic Operations
Decimal Codes [BCD]
Alphanumeric Codes
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4. Digital Systems

5. Computing Devices Everywhere! “Embedded Systems”PC
PDA
Home Networking
Car
Household
Super Computer
Body
Medicine
Game console
Entertainment
Communication
What is the main enabler to such digital systems?
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6. The Transistor Revolution
Intel 4004 processor
Designed in 1971
Almost 3000 transistors
Speed:1 MHz operation
Bipolar logic
1960’s
First transistor
Bell Labs, 1948

7. The VLSI Design Cycle Specification Functional design Logic design
Circuit design
Physical design
Test/Fabrication
Logic design
SYSTEM +
MODULE
GATE
CIRCUIT n+ S G D
DEVICE

8. 1.3 VLSI Design Styles Vdd Vdd IN2 IN2 IN1 OUT OUT IN1 GND GND IN1 OUT
Contact
Metal layer
Vdd
IN2
Poly layer
IN2
IN1
OUT
Diffusion layer
OUT
IN1
p-type transistor
GND
n-type transistor
GND
IN1
OUT
IN2
Power (Vdd)-Rail
Ground (GND)-Rail

9. Signals

10. Signals
An information variable represented by a physical quantity (speech, Temp, humidty, noise, …)

11. Signals Signals can be analog or digital:
Analog signals can have an infinite number of values in a range;
Digital signals can have only a limited number of values.

12. Continuous in value & time
Analog Signals
Time
Analog
Continuous in value & time

13. Digital Signals Time Digital
For digital systems, the variable takes on discrete values (i.e., not continuous)
Time
Digital
Discrete in value

14. Signal Examples Over Time
Digital (Binary) values are represented by:
digits 0 and 1 / False (F) and True (T)
words (symbols) Low (L) and High (H)
words On and Off.
Time
Discrete in value & continuous in time
Asynchronous
Digital
Discrete in value & time
Synchronous

15. Binary Values: Other Physical Quantities
What are other physical quantities represent 0 and 1?
CPU: Voltage
Hard Drive: Magnetic Field Direction
CD: Surface Pits/Light
Dynamic Ram: Electric Charge

16. A Digital Computer Example
Data/Instructions/code
All in
clock
Inputs: Keyboard, mouse, modem, microphone
Outputs: CRT, LCD, modem, speakers
Answer: Part of specification for a PC is in MHz. What does that imply? A clock which defines the discrete times for update of state for a synchronous system. Not all of the computer may be synchronous, however.
School of Engineering

17. Number Representation

18. Number Systems – Representation
A number with radix r is represented by a string of digits: An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m + 1 A- m in which 0 £ Ai < r and “.” is the radix point.
The string of digits represents the power series: ( )
(Number)r = å +
j = - m j i
i = 0 r A
(Integer Portion)
(Fraction Portion)
i = n - 1
j = - 1

19. Decimal Number System Base (also called radix) = 10 Digit Position
10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Digit Position
Integer & fraction
Formal Notation
Digit Weight
Weight = (Base) Position
Magnitude
Sum of “Digit x Weight” 1 -1 2 -2 5 1 2 7 4
(512.74)10 10 1
0.1
100
0.01
500 10 2
0.7
0.04
d2*B2+d1*B1+d0*B0+d-1*B-1+d-2*B-2

20. Octal Number System Base = 8 Weights Formal Notation Magnitude 5 1 2 7
8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }
Weights
Weight = (Base) Position
Formal Notation
Magnitude
Sum of “Digit x Weight” 1 -1 2 -2 8
1/8 64
1/64 5 1 2 7 4
(512.74)8
5 *8 2+1 *8 1+2 *8 0+7 * *8 -2
=( )10

21. Octal Number System: Example
For Example,
(27)8 can be expressed as: ( )10
(17.1)8 can be expressed as: ( )10

22. Hexadecimal Number System
Base = 16
16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }
Weights
Weight = (Base) Position
Formal Notation
Magnitude
Sum of “Digit x Weight” 1 -1 2 -2 16
1/16
256
1/256 1 E 5 7 A
(1E5.7A)16
1 * *161+5 *160+7 * *16-2
=( )10

23. Hex to Decimal (2ac)16 2 • 256 + 10 • 16 + 12 • 1 = (684)101 2 3 4 5 6 7 8 9 10 a 11 b 12 c 13 d 14 e 15 f
Just multiply each hex digit by decimal value, and add the results.
(2ac)16
2 • 256
+ 10 • 16
+ 12 • 1
= (684)10

163
162
161
160
4096
256 16 1

24. Binary Numbers

25. Binary Number System Base = 2 Weights Magnitude Formal Notation
2 digits { 0, 1 }, called binary digits or “bits”
Weights
Weight = (Base) Position
Magnitude
Sum of “Bit x Weight”
Formal Notation
Groups of bits bits = Nibble
8 bits = Byte 1 -1 2 -2
1/2 4
1/4 1 1 1
1 *22+0 *21+1 *20+0 *2-1+1 *2-2
=(5.25)10
(101.01)2

26. Binary  Decimal: Example7 6 5 4 3 2 1 27 26 25 24 23 22 21 20
128 64 32 16 8
position
value
What is in decimal? 7 6 5 4 3 2 1
position 1
Binary #
= (156)10

27. Binary Numbers Strings of binary digits (“bits”) Examples:
(00)2  (0)10
(01)2  (1)10
( )2  (1)10
(10)2  (2)10
(010)2  (2)10
(11)2  (3)10
(100)2  (4)10
( )2
Strings of binary digits (“bits”)
One bit can store a number from 0 to 1
n bits can store numbers from 0 to 2n-1
School of Engineering

28. Binary Fractions 2i 2i-1 4 2 1 1/2 1/4 1/8 2-j Integer Values
bi bi b2 b1 b b-1 b-2 b-3 … b-j
1/2
Fractional Values
1/4
1/8
2-j
decimal number =

29. Example 1
1 x x x x x2-2
6 and 3/4

30. Example 2
0 x x x x x x x2-6
63/64
Note: (1) Numbers of the form …2 are just below 1.0
(2) Short form notation for such numbers is ε

31. Why Binary?
This is easier to implement in hardware than a unit that can take on 10 different values.
For instance, it can be represented by a transistor being off (0) or on (1).
Alternatively, it can be a magnetic stripe that is magnetized with North in one direction (0) or the opposite (1).
Binary also has a convenient and natural association with logical values of:
False (0) and
True (1).

32. The Power of 2 n 2n
20=1 1
21=2 2
22=4 3
23=8 4
24=16 5
25=32 6
26=64 7
27=128 n 2n 8
28=256 9
29=512 10
210=1024 11
211=2048 12
212=4096 20
220=1M 30
230=1G 40
240=1T
Kilo
Mega
Giga
Tera

33. Base Conversion

34. Number Base Conversions
Evaluate Magnitude
Octal
(Base 8)
Evaluate Magnitude
Decimal
(Base 10)
Binary
(Base 2)
Hexadecimal
(Base 16)
Evaluate Magnitude

35. Conversion Between Bases
To convert from one base to another:
1) Convert the Integer Part
2) Convert the Fraction Part
3) Join the two results with a radix point

36. Decimal to Binary Conversion

37. Decimal (Integer) to Binary Conversion
Divide the number by the ‘Base’ (=2)
Take the remainder (either 0 or 1) as a coefficient
Take the quotient and repeat the division
Example: (13)10
Quotient
Remainder
Coefficient 13
/ 2 =
a0 = 1 6
/ 2 =
a1 = 0 3
/ 2 =
a2 = 1 1
/ 2 =
a3 = 1
Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2
MSB LSB

38. Decimal (Fraction) to Binary Conversion
Multiply the number by the ‘Base’ (=2)
Take the integer (either 0 or 1) as a coefficient
Take the resultant fraction and repeat multiplication
Example: (0.625)10
Integer
Fraction
Coefficient
0.625
* 2 =
a-1 = 1
0.25
* 2 = a-2 = 0
0.5
* 2 = a-3 = 1
Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2
MSB LSB

39. Decimal to Octal Conversion
Example: (175)10
Quotient
Remainder
Coefficient
175
/ 8 =
a0 = 7 21
/ 8 =
a1 = 5 2
/ 8 =
a2 = 2
Answer: (175)10 = (a2 a1 a0)8 = (257)8
Example: (0.3125)10
Integer
Fraction
Coefficient
0.3125
* 8 =
a-1 = 2
0.5
* 8 = a-2 = 4
Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8

40. Decimal to Hex c ac 2ac (684)10 684/16 = 42 rem 12=c1 2 3 4 5 6 7 8 9 10 a 11 b 12 c 13 d 14 e 15 f
(684)10 c
684/16 = 42 rem 12=c ac
42/16 = 2 rem 10=a
2/16 = rem 2
2ac

163
162
161
160
4096
256 16 1

41. Hexadecimal (Base 16) Strings of 0’s and 1’s too hard to write
Use base-16 or hexadecimal – 4 bits
Dec
Bin
Hex
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111
Dec
Bin
Hex 8
1000 9
1001 10
1010 a 11
1011 b 12
1100 c 13
1101 d 14
1110 e 15
1111 f
Why use
base 16?
Power of 2
Size of byte

42. Hex to Binary 0x2ac (2ac)16 No magic – remember hex digit = 4 bits
0000
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 a
1011 b
1100 c
1101 d
1110 e
1111 f
Convention – write 0x (prefix)
before number
Hex to Binary – just convert digits
0x2ac
(2ac)16
0010
1010
1100
0x2ac = ( )2
No magic – remember hex digit = 4 bits

43. Binary − Hexadecimal Conversion
16 = 24
Each group of 4 bits represents a hexadecimal digit
Hex
Binary 1 2 3 4 5 6 7 8 9 A B C D E F
Example:
Pad with Zeros
( )2
( )16
Works both ways (Binary to Hex & Hex to Binary)

44. Binary to Hex
Bin
Hex
0000
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
1000 8
1001 9
1010 a
1011 b
1100 c
1101 d
1110 e
1111 f
Just convert groups of 4 bits
( )2
0101 
0011 
0111 
1011 5 3 7 b
( )2 = 0x537b = (537b)16

45. Octal − Hexadecimal Conversion
Convert to Binary as an intermediate step
Example:
( )8
Assume Zeros
Assume Zeros
( )2
( )16
Works both ways (Octal to Hex & Hex to Octal)

46. Binary Arithmetic

47. Princess Sumaya University
Addition
Digital Logic Design
Decimal Addition 1 1
Carry 5 5 + 5 5 1 1
= Ten ≥ Base
 Subtract a Base
Dr. Bassam Kahhaleh

48. Binary Addition Adding bits: Adding integers: carry carry 1 1
0 + 0 =
0 + 1 =
1 + 0 =
1 + 1 = (1) 0
= (1) 1
Adding integers:
carry
carry 1 1
(1)2 = (7)10
(0)2 = (6)10
(0) (1)2
= = (13)10
(1)1
(1)0

49. Princess Sumaya University
Binary Addition
Digital Logic Design
Column Addition 1 1 1 1 1 1 1 1 1 1 1
= (61)10
= (23)10 + 1 1 1 1
= (84)10 1 1 1
≥ (2)10
Dr. Bassam Kahhaleh

50. Binary Codes

51. Binary Numbers and Binary Coding
Flexibility of representation
Within constraints below, can assign any binary combination (called a code word) to any data as long as data is uniquely encoded.
Information Types
Numeric
Must represent range of data needed
Very desirable to represent data such that simple, straightforward computation for common arithmetic operations permitted
Tight relation to binary numbers
Non-numeric
Greater flexibility since arithmetic operations not applied.
Not tied to binary numbers

52. Non-numeric Binary Codes
Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers.
Example: A binary code for the seven colors of the rainbow
Code 100 is not used
Binary Number
000
001
010
011
101
110
111
Color
Red
Orange
Yellow
Green
Blue
Indigo
Violet

53. Number of Bits Required
Given M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships:
2n > M > 2(n – 1)
n = log2 M where x , is called the ceiling function, i.e the integer greater than or equal to x.
M = 10
Therefore n = 4 since:
24 =16 is  10 > 23 = 8
and the ceiling function for log2 10 is 4.
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54. Number of Bits Required
Given M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships:
Example: How many bits are required to represent decimal digits with a binary code?
M = 10, hence n = ceiling (log2 10) = ceiling (3.3219) = 4
Checking:
M = 10
Therefore n = 4 since:
24 =16 is  10 > 23 = 8
and the ceiling function for log2 10 is 4.

55. Binary Codes Group of n bits Binary Coded Decimal (BCD)
Up to 2n combinations
Each combination represents an element of information
Binary Coded Decimal (BCD)
Each Decimal Digit is represented by 4 bits
(0 – 9)  Valid combinations
(10 – 15)  Invalid combinations
Decimal
BCD 1 2 3 4 5 6 7 8 9

56. Gray Code One bit changes from one code to the next code
Different than Binary
Decimal
Gray 00
0000 01
0001 02
0011 03
0010 04
0110 05
0111 06
0101 07
0100 08
1100 09
1101 10
1111 11
1110 12
1010 13
1011 14
1001 15
1000
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

57. Binary Representations
A bit is the most basic unit of information in a computer.
It is a state of “on” or “off” in a digital circuit.
Sometimes these states are “high” or “low” voltage instead of “on” or “off..”
A group of four bits is called a nibble (or nybble).
Bytes, therefore, consist of two nibbles: a “high-order nibble,” and a “low-order” nibble.
A byte is a group of eight bits.
A byte is the smallest possible addressable (can be found via its location) unit of computer storage.
A word is a contiguous group of bytes.
Words can be any number of bits (16, 32, 64 bits are common).

58. Advantages/Disadvantages?
Conversion or Coding?
Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE. 
(13)10 = (1101)2 (This is conversion) 
(13)BCD  (0001|0011)BCD (This is coding)
Advantages/Disadvantages?

59. BCD: Advantages/Disadvantages
It is obvious that a BCD number needs more bits than its equivalent binary value
(26)10 = (11010)2
(26)10 = ( )BCD
Advantages:
Computer input/output data are handled by people who use the decimal system. So it is easier to convert back/forth to BCD.

60. ASCII Codes

61. Character Codes From numbers to letters ASCII Unicode
Stands for American Standard Code for Information Interchange
Only 7 bits defined
Unicode

62. ASCII Code American Standard Code for Information Interchange Info
7-bit Code A B . Z a b z @ ? +

63. ASCII table

64. Reading Read Chapter 1 Check the lecture notes from the Web site.
Make sure you’re comfortable with material
Check the lecture notes from the
Web site.
Solve the assignment.

65. Homework See Assignment #1 On Web
I expect you to know number systems well and be able to do conversions and arithmetic
Decimal – Binary
Binary – Decimal
Decimal – Hex
Hex – Decimal
Will be on test!

66. End Slides