1. Digital Systems and Logic Design

2. General Overview
This chapter is Number Systems, Operations, and Codes.
Decimal numbers
Binary numbers
Decimal-to-binary numbers
Binary arithmetic
1’s and 2’s Complements of Binary numbers
Signed number
Arithmetic operations with signed numbers
Hexadecimal Numbers
Octal Numbers
Binary Coded Decimal (BCD)
Digital Codes and Parity
Digital System Application

3. Chapter Objectives Convert the number between decimal and binary.
Apply arithmetic operation to binary numbers.
Understand Binary Coded Decimal (BCD), Gray Code, ASCII

4. Decimal Numbers Decimal numbers has ten digits. (0, 1, 2, 3, …, 9)
The decimal numbering system has a base of 10 with each position weighted by a factor of 10:

5. Binary Numbers Binary numbers has two digits, 1 and 0.
The binary numbering system has a base of 2 with each position weight by a factor of 2.

6. Binary numbers: Example
10010 in binary is
1x24 + 0x23 + 0x22 + 1x21 + 0x20 = 18
Binary is the base 2 number system
Most common in digital electronics
Weight: Most Significant bit (MSB)
Least Significant bit (LSB)
10010
MSB
LSB

7. Integers and Fractional Parts
Binary numbers can contain fractional parts as well as integer parts
# quantization error.

8. Conversion: Decimal to Binary (Method 1)
The decimal numbers is simply expressed as a sum of power of 2, and then 1s and 0s are written in the appropriate bit positions.
50 = =
= 1x25 + 1x24 + 1x21
5010 =
322 = ?

9. Conversion: Decimal to Binary (Method 2)
Repeated division
Quotient Remainder
50/2 = LSB
25/2 =
12/2 =
6/2 =
3/2 =
1/2 = MSB
stop
50 =

10. Conversion: Binary to Decimal
The simplest way is to represent the binary number as
anx2n a2x22 + a1x21 + a0x20
The conversion can be done by substituting the a’s with the given bits then multiplying and adding:
= 1x25 + 1x24 + 1x21 = 50

11. Example Convert the binary whole number 100011012 to decimal=
27 +
23 +
22 + 20
= = 14110

12. Example Convert the fractional binary number 0.10112 to decimal=
2-1 +
2-3 +
2-4 +
2-6 = =

13. Example Convert the binary whole number 101.01012 to decimal= = =

14. Conversion: Fractional to Binary
เลขทศนิยมฐานสิบเป็นเลขฐานสอง
ผลบวกของค่าน้ำหนัก
การคูณซ้ำด้วยสอง
0.782 =
ผลบวกของค่าน้ำหนัก :
การคูณซ้ำด้วยสอง :
0.782 =
= ( )
0.782
x 2 = 1.564
0.564 x 2 = 1.128 =
0.128 x 2 = 0.256 =
0.256 x 2 = 0.512
0.512 x 2 = 1.024

15. Binary Number Systems Unsigned Binary Code Signed Binary Codes
2’s Complement System
BCD Code
Excess Codes
Floating-Point System

16. Unsigned Binary Code
Given a number N in Unsigned Binary code, find the value of N in the decimal system
Use series substitution method
Given a number N in the decimal system, find the value of N in the Unsigned Binary Code.
Use successive division method (for integer part)
Use successive multiplication method (for fraction part)

17. Unsigned Binary Code Can’t do. Not enough bits. 2610 = 00011010
Example 1: Represent (26)10 in Unsigned Binary Code
2610 =
Example 2: Represent (26)10 in Unsigned Binary Code using 8 bits.
2610 =
Example 3: Represent (26)10 in Unsigned Binary Code using 4 bits.
Can’t do. Not enough bits.

18. Unsigned Binary Code ( 4 bits)

19. Unsigned Binary Code [0, 2n-1]
The Unsigned Binary Code is used to represent positive integer numbers.
What is the range of values that can be represented with n bits in the Unsigned Binary Code?
[0, 2n-1]
How many bits are required to represent a given number N?
number of bits = smallest integer greater than or equal to log(N)

20. Unsigned Binary Code: Arithmetic & Logic Operations
Arithmetic Operations:
Addition
Subtraction
Multiplication
Division
Logic Operations
AND CONJUNCTION
OR DISJUNCTION
NOT NEGATION
XOR EXCLUSIVE OR

21. Signed Binary Codes
These are codes used to represent positive and negative numbers.
Sign and Magnitude Code
1’s Complement Code
2’s Complement Code

22. Sign & Magnitude Code Sign & Mag. Code Decimal 01101 +13 11101 -13
The leftmost bit is the sign bit
0 for positive numbers
1 for negative numbers
The remaining bits represent the magnitude of the number
Example:
Sign & Mag. Code Decimal

23. Sign &Magnitude (4 bits)
What is the range of values that can be represented in S&M code with n bits?

24. Sign&Magnitude Example 1: Represent (26)10 in Sign & Magnitude Code.
2610 =
Example 2: Represent (26)10 in Sign & Magnitude Code using 8 bits
2610 =
Example 3: Represent (26)10 in Sign & Magnitude Code using 5 bits.
Need at least 6 bits.

25. Sign&Magnitude Example 1: Represent (-26)10 in Sign & Magnitude Code.= =
Example 2: Represent (-26)10 in Sign & Magnitude Code using 8 bits = =
Example 3: Represent (-26)10 in Sign & Magnitude Code using 5 bits.
Need at least 6 bits.

26. Binary Arithmetic
Addition
Subtraction
Multiplication
Division

27. Binary Addition
Equation
Recall decimal addition
Binary addition

28. Example Add the following binary number:. (a) 10010011 + 01001011
Example Add the following binary number: (a) (b) 1 1
(a)
147
+ 75 1 1 1 1 1 1
= 222 1 1 1 1 1 31
(b)
+ 7
= 3810 1 1 1

29. Binary Subtraction สมการทั่วไป : ตารางความจริงการลบ A0 < B0 ต้องการ
0 - 0 = 0  ผลลบ 0 ตัวยืม 0
0 - 1 = 1  ผลลบ 1 ตัวยืม 1
1 - 0 = 1  ผลลบ 1 ตัวยืม 0
1 - 1 = 0  ผลลบ 0 ตัวยืม 0
ตารางความจริงการลบ
สมการทั่วไป :
A0 B0 R0 B0ut
A0 < B0
ต้องการ
ตัวยืม
R0 : ผลต่าง Bout : ตัวยืม

31. Binary Multiplication
A Basic Roles for Multiplication

33. Binary Division

34. 1’s Complement Code Positive numbers: Negative numbers:
same as in unsigned binary code
pad a 0 at the leftmost bit position
Negative numbers:
1. Represent the magnitude of the number in unsigned binary system
2. pad a 0 at the leftmost bit position
3. complement every bit

35. 1’s Complement Code Example: represent 2610 in 1’s complement code
2610 =
Pad a 0: =
Example: Represent (-26)10 in 1’s complement code. =
2. Pad a 0:
3. Complement: (-26)10 = (100101)1’s comp

36. 1’s Complement Code
Example: Represent (-26)10 in 1’s comp. code using 8 bits
1. Represent magnitude in unsigned binary using 8 bits
26 =
2. Complement every bit
-2610 = ( ) 1’s comp

37. 1’s Complement Code (4 bits)
What is the range of values that can be represented in S&M code with n bits?
[ -(2 (n-1) -1) , 2(n-1) -1]

38. 2’s Complement Code
This is the code commonly used to represent integer numbers.
Positive Numbers:
same as in unsigned binary code
pad with a 0 leftmost bit position
Negative Numbers
1. represent magnitude in unsigned binary code
2. pad leftmost positions with 0s
3. complement every bit
4. add 1

39. 2’s Complement Code Example 1: Represent 26 in 2’s complement code.
26 =
Example 2: Represent 26 in 2’s complement code using 8 bits
26 =
Example 3: Represent 26 in 2’s complement code using 5 bits
Need at least 6 bits.

40. 2’s Complement Code Example 3: Represent - 26 in 2’s comp. Code=
2. Pad with a 0:
3. Complement:
4. Add 1:
100110

41. 2’s Complement Code
Example 4: Represent in 2’s comp. Code using 8 bits =
2. Pad 0s:
3. Complement:
4. Add 1:

42. 2’s Complement Code More example: represent 65 in 2’s comp. Code.
65 =
-65 =

43. Conversion from 2’s comp code to decimal code
How to convert a number in 2’s Comp. Code into the decimal code.
There 2 cases:
Case 1: If leftmost bit of the number is 0
=> number is positive
=> conversion is the same as in unsigned binary code

44. Conversion from 2’s comp code to decimal code
Case 2: If leftmost bit is 1
=> the number is negative
step1: complement every bit
step2: add 1
step3: convert result to decimal code using same method as in unsigned binary code.
Answer = the negative of the result of step 3.

45. 2’s Complement Code (4 bits)
Range of values with n bits:
[ -2 (n-1), 2(n-1) -1]

46. 1’s and 2’s Complement of Binary Numbers

48. Alternative Method to find 2’s Complement

65. Hexadecimal Notation Hexadecimal system: base 16 There are 16 digits:
A B C D E F
Each Hex digit represents a group of 4 bits (i.e. half of a byte) thru 1111
Generally used as shorthand notation for binary numbers => easier to read
Binary:
Decimal:
Hex: 5 A E

66. Hexadecimal number

67. Hexadecimal number conversions

72. Addition in Hex.

73. Subtration in Hex.

75. Octal number

80. Binary Coded Decimal (BCD)
Decimal numbers are more natural to humans. Binary numbers are natural to computers. Quite expensive to convert between the two.
If little calculation is involved, we can use some coding schemes for decimal numbers.
One such scheme is BCD, also known as the 8421 code.
Represent each decimal digit as a 4-bit binary code.

81. Binary Coded Decimal (BCD)
Some codes are unused, eg: (1010)BCD, (1011) BCD, …, (1111) BCD. These codes are considered as errors.
Easy to convert, but arithmetic operations are more complicated.
Suitable for interfaces such as keypad inputs and digital readouts.
CS1104-2
Binary Coded Decimal (BCD)

82. Binary Coded Decimal (BCD)
Examples:
(234)10 = ( )BCD
(7093)10 = ( )BCD
( )BCD = (86)10
( )BCD = (9472)10
Notes: BCD is not equivalent to binary.
Example: (234)10 = ( )2
CS1104-2
Binary Coded Decimal (BCD)

83. BCD : Binary Coded Decimal

87. The Gray Code Unweighted (not an arithmetic code).
Only a single bit change from one code number to the next.
Good for error detection.
Q. How to generate 5-bit standard Gray code?
Q. How to generate n-bit standard Gray code?
CS1104-2
The Gray Code

88. The Gray Code
0000
0001
0100
0101
0111
0110
0010
0011
0001
0000
1100
1101
1111
1110
1010
1011
1001
1000
0001
0000
0011
0010
0010
0011
0001
0000
0110
0111
0101
0100
Generating 4-bit standard Gray code.
CS1104-2
The Gray Code

89. The Gray Code sensors mis-aligned sensors Gray coded: 111  101
sensors
mis-aligned sensors
Gray coded: 111  101
Binary coded: 111  110  000
mis-aligned sensors
CS1104-2
The Gray Code

90. Binary-to-Gray Code Conversion
Retain most significant bit.
From left to right, add each adjacent pair of binary code bits to get the next Gray code bit, discarding carries.
Example: Convert binary number to Gray code.
(10110)2 = (11101)Gray
CS1104-2
Binary-to-Gray Code Conversion

91. Gray-to-Binary Conversion
Retain most significant bit.
From left to right, add each binary code bit generated to the Gray code bit in the next position, discarding carries.
Example: Convert Gray code to binary.
(11011)Gray = (10010)2
CS1104-2
Gray-to-Binary Conversion

92. Error Detection Codes
Sometimes, it is not enough to do error detection. We may want to do error correction.
Error correction is expensive. In practice, we may use only single-bit error correction.
Popular technique: Hamming Code (not covered).
CS1104-2
Error Detection Codes

93. Gray code

94. Binary to Gray Conversion

96. Gray to Binary

98. Other Decimal Codes
Self-complementing codes: excess-3, , 2*421 codes.
Error-detecting code: biquinary code (bi=two, quinary=five).
CS1104-2
Other Decimal Codes

99. Self-Complementing Codes
Examples: excess-3, , 2*421 codes.
The codes that represent the pair of complementary digits are complementary of each other.
Excess-3 code
0: 0011
1: 0100
2: 0101
3: 0110
4: 0111
5: 1000
6: 1001
7: 1010
8: 1011
9: 1100
241:
758:
CS1104-2
Self-Complementing Codes

100. Alphanumeric Codes
Apart from numbers, computers also handle textual data.
Character set frequently used includes:
alphabets: ‘A’ .. ‘Z’, and ‘a’ .. ‘z’
digits: ‘0’ .. ‘9’
special symbols: ‘$’, ‘.’, ‘,’, ‘*’, …
non-printable: SOH, NULL, BELL, …
Usually, these characters can be represented using 7 or 8 bits.
CS1104-2
Alphanumeric Codes

101. Alphanumeric Codes Two widely used standards:
ASCII (American Standard Code for Information Interchange)
EBCDIC (Extended BCD Interchange Code)
ASCII: 7-bit, plus a parity bit for error detection (odd/even parity).
EBCDIC: 8-bit code.
CS1104-2
Alphanumeric Codes

102. Alphanumeric Codes
ASCII table:
CS1104-2
Alphanumeric Codes

103. Error Detection Codes
Errors can occur data transmission. They should be detected, so that re-transmission can be requested.
With binary numbers, usually single-bit errors occur.
Example: 0010 erroneously transmitted as 0011, or 0000, or 0110, or 1010.
Biquinary code uses 3 additional bits for error-detection. For single-error detection, one additional bit is needed.
CS1104-2
Error Detection Codes

104. Error Detection Codes Parity bit. Example: Odd parity.
Even parity: additional bit supplied to make total number of ‘1’s even.
Odd parity: additional bit supplied to make total number of ‘1’s odd.
Example: Odd parity.
Parity bits
CS1104-2
Error Detection Codes

105. Error Detection Codes
Parity bit can detect odd number of errors but not even number of errors.
Example: For odd parity numbers,
10011  (detected)
10011  (non detected)
Parity bits can also be
applied to a block of data:
Column-wise parity
Row-wise parity
CS1104-2
Error Detection Codes

106. Error Detection Codes
Sometimes, it is not enough to do error detection. We may want to do error correction.
Error correction is expensive. In practice, we may use only single-bit error correction.
Popular technique: Hamming Code (not covered).
CS1104-2
Error Detection Codes

107. Alphanumeric Codes

108. ASCII

110. Parity Method for error checking

112. Put it together