1. Chapter 1 Digital Systems and Binary Numbers
授課教師: 張傳育 博士 (Chuan-Yu Chang Ph.D.)
Tel: (05) ext. 4337
Office: EB212

2. Digital Logic Design Text Book Reference Grade
M. M. Mano and M. D. Ciletti, “Digital Design," 4th Ed., Pearson Prentice Hall, 2007.
Reference
class notes
Grade
Quizzes:
Mid-term:
Final:
Appearance:

3. Chapter 1: Digital Systems and Binary Numbers
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

4. Signal An information variable represented by physical quantity
For digital systems, the variable takes on discrete 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)
and words On and Off.
Binary values are represented by values or ranges of values of physical quantities

5. Signal Example – Physical Quantity: Voltage
Threshold Region
The HIGH range typically corresponds to binary 1 and LOW range to binary 0. The threshold region is a range of voltages
for which the input voltage value cannot be interpreted reliably as either a 0 or a 1.

6. Signal Examples Over Time
Continuous in value & time
Analog
Digital
Discrete in value & continuous in time
Asynchronous
Discrete in value & time
Synchronous

7. A Digital Computer Example
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.
Synchronous or Asynchronous?

8. Binary Numbers and Binary Coding
Information Types
Numeric
Must represent range of data needed
Represent data such that simple, straightforward computation for common arithmetic operations
Tight relation to binary numbers
Non-numeric
Greater flexibility since arithmetic operations not applied.
Not tied to binary numbers

9. Number of Elements Represented
Given n digits in radix r, there are rn distinct elements that can be represented.
But, you can represent m elements, m < rn
Examples:
You can represent 4 elements in radix r = 2 with n = 2 digits: (00, 01, 10, 11).
You can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000).
This second code is called a "one hot" code.

10. 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
Color
Binary Number
000
001
010
011
101
110
111
Red
Orange
Yellow
Green
Blue
Indigo
Violet

11. Binary Numbers Decimal number … a5a4a3a2a1.a1a2a3…
Base or radix
Power
… a5a4a3a2a1.a1a2a3…
Decimal point
Example:
General form of base-r system
Coefficient: aj = 0 to r  1

12. Binary Numbers Example: Base-2 number Example: Base-5 number

13. Converting Binary to Decimal
To convert to decimal, use decimal arithmetic to form S (digit × respective power of 2).
Example:Convert to N10:  
Powers of 2: 43210
=>
1 X = 16
+ 1 X = 8
+ 0 X = 0
+ 1 X 21 = 2
+ 0 X 20 = 0
2610

14. 220 (1,048,576) is Mega, denoted "M" Binary Numbers
Example: Base-2 number
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"
Powers of two
Table 1.1

15. Arithmetic operation
Arithmetic operations with numbers in base r follow the same rules as decimal numbers.

16. Binary Arithmetic Single Bit Addition with Carry Multiple Bit Addition
Single Bit Subtraction with Borrow
Multiple Bit Subtraction
Multiplication
BCD Addition

17. Single Bit Binary Addition with Carry

18. Multiple Bit Binary Addition
Extending this to two multiple bit examples:
Carries
Augend
Addend
Sum
Note: The 0 is the default Carry-In to the least significant bit.
Carries
Augend
Addend
Sum

19. Binary Arithmetic Subtraction Addition Multiplication Minuend: 101101
Subtrahend: 100111
Difference:
Augend:
Addend:
Sum:
Multiplication

20. Number-Base Conversions
Name
Radix
Digits
Binary 2
0,1
Octal 8
0,1,2,3,4,5,6,7
Decimal 10
0,1,2,3,4,5,6,7,8,9
Hexadecimal 16
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
The six letters (in addition to the 10 integers) in hexadecimal represent: 10, 11, 12, 13, 14, and 15, respectively.

21. Number-Base Conversions
Example1.1
Convert decimal 41 to binary. The process is continued until the integer quotient becomes 0.

22. Number-Base Conversions
 The arithmetic process can be manipulated more conveniently as follows:

23. Number-Base Conversions
Example 1.2
Convert decimal 153 to octal. The required base r is 8.
Example1.3
Convert (0.6875)10 to binary.
The process is continued until the fraction becomes 0 or until the number of digits has sufficient accuracy.

24. Number-Base Conversions
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.

25. Number-Base Conversions
Example1.4
Convert (0.513)10 to octal.
 From Examples 1.1 and 1.3:
( )10 = ( )2
 From Examples 1.2 and 1.4:
( )10 = ( )8

26. Octal and Hexadecimal Numbers
 Numbers with different bases: Table 1.2.

27. Octal and Hexadecimal Numbers
 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 is similar, except that the binary number is divided into groups of four digits:
 Conversion from octal or hexadecimal to binary is done by reversing the preceding procedure.

28. Complements
 There are two types of complements for each base-r system: the radix complement and diminished radix complement.
the r's complement and the second as the (r  1)'s complement.
■ Diminished Radix Complement
Example:
 For binary numbers, r = 2 and r – 1 = 1, so the 1's complement of N is (2n  1) – N.
Example:

29. Complements (cont.) ■ Radix Complement
The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r  1) 's complement, we note that the r's complement is obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn  1) – N] + 1.
Example: Base-10
The 10's complement of is
The 10's complement of is
Example: Base-10
The 2's complement of is
The 2's complement of is

30. Complements (cont.) ■ Subtraction with Complements
The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:

31. Complements (cont.) Example 1.5
Using 10's complement, subtract – 3250.
Example 1.6
Using 10's complement, subtract 3250 – 72532
There is no end carry.
Therefore, the answer is – (10's complement of 30718) = 

32. Complements (cont.) Example 1.7
Given the two binary numbers X = and Y = , 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 ) = 

33. Complements (cont.)
 Subtraction of unsigned numbers can also be done by means of the (r  1)'s complement. Remember that the (r  1) 's complement is one less then the r's complement.
Example 1.8
Repeat Example 1.7, but this time using 1's complement.
There is no end carry, Therefore, the answer is Y – X =  (1's complement of ) = 

34. Signed Binary Numbers
 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.
The convention is to make the sign bit 0 for positive and 1 for negative.
Example:
 Table 3 lists all possible four-bit signed binary numbers in the three representations.

35. Signed Binary Numbers (cont.)

36. Signed Binary Numbers (cont.)
■ 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 if 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:

37. Signed Binary Numbers (cont.)
■ Arithmetic Subtraction
 In 2’s-complement form:
Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit).
A carry out of sign-bit position is discarded.
Example:
( 6)  ( 13)
(  )
( )
(+ 7)

38. Binary Coded Decimal (BCD)
The BCD code is the 8,4,2,1 code.
This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9.
Example: (9) = 1000 (8) (1)
How many “invalid” code words are there?
What are the “invalid” code words? 6
Answer 1: 6
Answer 2: 1010, 1011, 1100, 1101, 1110, 1111
1010, 1011, 1100, 1101, 1110, 1111

39. Binary Codes
■ BCD Code
A number with k decimal digits will require 4k bits in BCD. Decimal 396 is represented in BCD with 12bits as , with each group of 4 bits representing one decimal digit. A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9. A BCD number greater than 10 looks different from its equivalent binary number, even though both contain 1's and 0's. Moreover, the binary combinations 1010 through 1111 are not used and have no meaning in BCD.

40. Warning: 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. 
1310 = (This is conversion) 
13  0001|0011 (This is coding)

41. BCD Arithmetic
Given a BCD code, we use binary arithmetic to add the digits: 8
1000
Eight +5
+0101
Plus 5 13
1101
is 13 (> 9)
Note that the result is MORE THAN 9, so must be represented by two digits!
To correct the digit, subtract 10 by adding 6 modulo 16. 8
1000
Eight +5
+0101
Plus 5 13
1101
is 13 (> 9)
+0110
so add 6
carry = 1
0011
leaving 3 + cy
0001 | 0011
Final answer (two digits)
If the digit sum is > 9, add one to the next significant digit

42. Binary Codes ■ BCD Addition Example:
Consider decimal 185 and its corresponding value in BCD and binary:
■ BCD Addition

43. Binary Codes ■ Decimal Arithmetic Example:
Consider the addition of = 760 in BCD:
■ Decimal Arithmetic

44. BCD Addition Example
Add 2905BCD to 1897BCD showing carries and digit corrections.
+0110
+0001
+0110
+0001
10111
1010
10010
0100
+0001
+0110 4 8 2
11000
10000

45. Binary Codes
■ Other Decimal Codes

46. Binary Codes ■ Gray Code 0000 0001 0011 0010 0110 0111 0101 0100 1100
1101
1111
1110
1010
1011
1001
1000

47. Gray Code
What special property does the Gray code have in relation to adjacent decimal digits?
Decimal
8,4,2,1
Gray
0000
0000 1
0001
0001 2
0010
0011 3
0011
0010 4
0100
0110 5
0101
0111 6
0110
0101 7
0111
0100 8
1000
1100
Answer: As we “counts” up or down in decimal, the code word for the Gray code changes in only one bit position as we go from decimal digit to digit including from 9 to 0. 9
1001
1101

48. Does this special Gray code property have any value?
Gray Code (Continued)
Does this special Gray code property have any value?
An Example: Optical Shaft Encoder B
111
110
000
001
010
011
100
101 1 2
(a) Binary Code for Positions 0 through 7 G
(b) Gray Code for Positions 0 through 7
Yes, as illustrated by the example that followed.

49. Binary Codes
■ ASCII Character Code

50. Binary Codes
■ ASCII Character Code

51. American Standard Code for Information Interchange
ASCII Character Codes
American Standard Code for Information Interchange
A popular code used to represent information sent as character-based data.
It uses 7-bits to represent:
94 Graphic printing characters.
34 Non-printing characters
Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return)
Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).
(Refer to Table 1.7)

52. ASCII Properties ASCII has some interesting properties:
Digits 0 to 9 span Hexadecimal values 3016
to 3916 .
Upper case A -
Z span 4116
to 5A16 .
Lower case a -
z span 6116
to 7A16 .
Lower to upper case translation (and vice versa)
occurs by
flipping bit 6.
Delete (DEL) is all bits set,
a carryover from when
punched paper tape was used to store messages.
Punching all holes in a row erased a mistake!

53. Binary Codes ■ Error-Detecting Code
To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity.
A parity bit is an extra bit included with a message to make the total number of 1's either even or odd.
Example:
Consider the following two characters and their even and odd parity:

54. Binary Codes
■ Error-Detecting Code
Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.
A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.
A code word has even parity if the number of 1’s in the code word is even.
A code word has odd parity if the number of 1’s in the code word is odd.

55. 4-Bit Parity Code Example
Fill in the even and odd parity bits:
The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data.
Even Parity
Odd Parity
Message
Message
Parity -
Parity - 1
000
000 - -
001
001 - 1 -
010
010 - 1 -
011
011 - - 1
100
100 - - 1
101
101 - - 1
110
110 - 1
Even Parity Bits: 0, 1, 1, 0, 1, 0, 0, 1
Odd Parity Bits: 1, 0, 0, 1, 0, 1, 1, 0 -
111
111 - 1 -

56. UNICODE extends ASCII to 65,536 universal characters codes
For encoding characters in world languages
Available in many modern applications
2 byte (16-bit) code words
See Reading Supplement – Unicode on the Companion Website

57. Binary Storage and Registers
 A binary cell is a device that possesses two stable states and is capable of storing one of the two states.
 A register is a group of binary cells. A register with n cells can store any discrete quantity of information that contains n bits.
n cells
2n possible states
A binary cell
two stable state
store one bit of information
examples: flip-flop circuits, ferrite cores, capacitor
A register
a group of binary cells
AX in x86 CPU
Register Transfer
a transfer of the information stored in one register to another
one of the major operations in digital system
an example

58. Transfer of information

59. The other major component of a digital system
circuit elements to manipulate individual bits of information

60. Binary Logic ■ Definition of Binary Logic
 Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0, There are three basic logical operations: AND, OR, and NOT.

61. Binary Logic
■ The truth tables for AND, OR, and NOT are given in Table 1.8.

62. Binary Logic
■ Logic gates
 Example of binary signals

63. Binary Logic ■ Logic gates
 Graphic Symbols and Input-Output Signals for Logic gates:
Fig. 1.4 Symbols for digital logic circuits
Fig. 1.5
Input-Output signals for gates

64. Binary Logic ■ Logic gates
 Graphic Symbols and Input-Output Signals for Logic gates:
Fig Gates with multiple inputs