Digital Systems
and
Logic Design
1
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
2
Chapter Objectives



Convert the number between decimal and binary.
Apply arithmetic operation to binary numbers.
Understand Binary Coded Decimal (BCD), Gray Code,
ASCII
3
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:
4
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.
5
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
6
Integers and Fractional Parts

Binary numbers can contain fractional parts
as well as integer parts
# quantization error.
7
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
5010
= 32 + 18
= 32 + 16 + 2
= 1x25 + 1x24 + 1x21
= 1100102
322 = ?
8
Conversion: Decimal to Binary
(Method 2)

Repeated division
50/2 =
25/2 =
12/2 =
6/2 =
3/2 =
1/2 =
Quotient
25
12
6
3
1
0
stop
Remainder
0
LSB
1
0
0
1
1
MSB
50 = 110010
9
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:
110010 = 1x25 + 1x24 + 1x21 = 50
10
Example Convert the binary whole
number 100011012 to decimal
100011012 = 27 + 23 + 22 + 20
= 128 + 8 + 4 + 1 = 14110
11
Example Convert the fractional binary
number 0.10112 to decimal
0.1011012 = 2-1 + 2-3 + 2-4 + 2-6
= 0.5 + 0.125 + 0.0625
= 0.687510
12
Example Convert the binary whole
number 101.01012 to decimal
101.01012 = 22 + 20 + 2-2 + 2-4
= 4 + 1 +0.25 + 0.125 + 0.0625
= 5.437510
13
Conversion: Fractional to Binary

ิ เป็ น
เลขทศนิยมฐานสบ
เลขฐานสอง


0.782 = 0.110012
ผลบวกของค่าน้ าหนัก
การคูณซา้ ด ้วยสอง
ผลบวกของค่านา้ หนัก :
การคูณซา้ ด้วยสอง :
0.782 = 0.5 + 0.25 + 0.03125
= (0.78125)
0.782 x 2 = 1.564
= 2-1 + 2-2 + 2-5
0.128 x 2 = 0.256
= 0.110012
0.256 x 2 = 0.512
0.564 x 2 = 1.128
0.512 x 2 = 1.024
14
Binary Number Systems


Unsigned Binary Code
Signed Binary Codes




2’s Complement System
BCD Code
Excess Codes
Floating-Point System
15
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)
16
Unsigned Binary Code

Example 1: Represent (26)10 in Unsigned
Binary Code
2610 =
11010

Example 2: Represent (26)10 in Unsigned
Binary Code using 8 bits.
2610 = 00011010

Example 3: Represent (26)10 in Unsigned
Binary Code using 4 bits.
Can’t do. Not enough bits.
17
Unsigned Binary Code ( 4 bits)
Unsigned
Decimal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
18
Unsigned Binary Code

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)
19
Unsigned Binary Code: Arithmetic
& Logic Operations

Arithmetic Operations:





Addition
Subtraction
Multiplication
Division
Logic Operations




AND
OR
NOT
XOR
CONJUNCTION
DISJUNCTION
NEGATION
EXCLUSIVE OR
20
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
21
Sign & Magnitude Code
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
01101
11101
00101
10101
Decimal
+13
-13
+5
-5
22
Sign &Magnitude (4 bits)
Unsigned
Decimal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
-0
-1
-2
-3
-4
-5
-6
-7
What is the range of values that can be represented in S&M
code with n bits?
23
Sign&Magnitude

Example 1: Represent (26)10 in Sign & Magnitude
Code.
2610 =
011010

Example 2: Represent (26)10 in Sign & Magnitude
Code using 8 bits
2610 =
0001 1010

Example 3: Represent (26)10 in Sign & Magnitude
Code using 5 bits.
 Need at least 6 bits.
24
Sign&Magnitude

Example 1: Represent (-26)10 in Sign & Magnitude
Code.
 26
=
11010
 -2610 =
111010

Example 2: Represent (-26)10 in Sign & Magnitude
Code using 8 bits
 26
=
00011010


- 2610 =
10011010
Example 3: Represent (-26)10 in Sign & Magnitude
Code using 5 bits.
 Need at least 6 bits.
25
Binary Arithmetic
Addition
Subtraction
Multiplication
Division
26
Binary Addition

Equation
A0  B0   0 COUT
Recall decimal addition
Binary addition
27
Example Add the following binary number:
(a) 10010011 + 01001011
(b) 00011111 + 00000111
1 1
(a)
(b)
10010011
+01001011
1 1 0 1 1 1 10
1 1 1 1 1
0001 1111
+00000111
0 0 1 0 0 1 10
147
+ 75
= 222
31
+7
= 3810
28
Binary Subtraction
0-0=0 
0-1=1
1-0=1
1-1=0
ผลลบ 0 ตัวยืม 0
 ผลลบ 1 ตัวยืม 1
 ผลลบ 1 ตัวยืม 0
 ผลลบ 0 ตัวยืม 0
สมการทัว่ ไป :
A0  B0  R0  BOUT
R0 : ผลต่าง Bout : ตัวยืม
ตารางความจริงการลบ
A0 B0
0 0
0 1
1 0
1 1
R0 B0ut
0 0
1 1
1 0
0 0
A0 < B 0
ต้องการ
ตัวยืม
29
30
Binary Multiplication
A Basic Roles for Multiplication
31
32
Binary Division
33
1’s Complement Code

Positive 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
34
1’s Complement Code

Example: represent 2610 in 1’s complement
code



2610 =
11010
=
011010
Pad a 0:
Example: Represent (-26)10 in 1’s complement
code.
1.
2. Pad a 0:
3. Complement:
26
=
11010
011010
100101
35
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 = 0001 1010
2. Complement every bit
11100101
-2610 = (1110 0101) 1’s comp
36
1’s Complement Code (4 bits)
Unsigned
Decimal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
-0
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]
37
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
38
2’s Complement Code



Example 1: Represent 26 in 2’s complement
code.
26 =
011010
Example 2: Represent 26 in 2’s complement
code using 8 bits
26 = 00011010
Example 3: Represent 26 in 2’s complement
code using 5 bits

Need at least 6 bits.
39
2’s Complement Code

Example 3: Represent - 26 in 2’s comp.
Code
1.
2. Pad with a 0:
3. Complement:
4. Add 1:
+26 =
11010
011010
100101
+
1
--------------100110
40
2’s Complement Code

Example 4: Represent - 26 in 2’s comp.
Code using 8 bits
1.
2. Pad 0s:
3. Complement:
4. Add 1:
+26 =
11010
00011010
11100101
+
1
--------------11100110
41
2’s Complement Code

More example: represent 65 in 2’s comp.
Code.


65 =
(0100 0001)2’s comp
Represent - 65 in 2’s comp


65 = 0100 0001
-65 = 1011 1111
42
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
43
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.
44
2’s Complement Code (4 bits)
Unsigned
Decimal
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
-8
-7
-6
-5
-4
-3
-2
-1
Range of values with n bits:
[ -2
(n-1)
, 2
(n-1)
-1]
45
1’s and 2’s Complement of Binary
Numbers
46
47
Alternative Method to find 2’s
Complement
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
Hexadecimal Notation


Hexadecimal system: base 16
There are 16 digits:

0 123456789ABCDEF

Each Hex digit represents a group of 4 bits
(i.e. half of a byte) 0000 thru 1111

Generally used as shorthand notation for
binary numbers => easier to read



Binary:
Decimal:
Hex:
0101 1010
5
10
5
A
1001
9
9
1110
14
E
65
Hexadecimal number
66
Hexadecimal number conversions
67
68
69
70
71
Addition in Hex.
72
Subtration in Hex.
73
74
Octal number
75
76
77
78
79
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.
80
Binary Coded Decimal (BCD)
Decimal digit
BCD
Decimal digit
BCD
0
0000
5
0101
1
0001
6
0110
2
0010
7
0111
3
0011
8
1000
4
0100
9
1001
 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)
81
Binary Coded Decimal (BCD)
Decimal digit
BCD
Decimal digit
BCD
0
0000
5
0101
1
0001
6
0110
2
0010
7
0111
3
0011
8
1000
4
0100
9
1001
 Examples:
(234)10 = (0010 0011 0100)BCD
(7093)10 = (0111 0000 1001 0011)BCD
(1000 0110)BCD = (86)10
(1001 0100 0111 0010)BCD = (9472)10
Notes: BCD is not equivalent to binary.
Example: (234)10 = (11101010)2
CS1104-2
Binary Coded Decimal (BCD)
82
BCD : Binary Coded Decimal
83
84
85
86
The Gray Code
 Unweighted (not an arithmetic code).
 Only a single bit change from one code
number to the next.
 Good for error detection.
Decimal
0
1
2
3
4
5
6
7
Binary
0000
0001
0010
0011
0100
0101
0110
0111
Gray Code
0000
0001
0011
0010
0110
0111
0101
0100
Decimal
8
9
10
11
12
13
14
15
Binary
1000
1001
1010
1011
1100
1101
1110
1111
Gray code
1100
1101
1111
1110
1010
1011
1001
1000
Q. How to generate 5-bit standard Gray code?
Q. How to generate n-bit standard Gray code?
CS1104-2
The Gray Code
87
The Gray Code
0000
0001
0011
0001
0010
0000
0010
0110
0111
0011
0101
0001
0100
0000
1100
0100
0101
1101
0111
1111
0110
1110
0010
1010
0011
1011
0001
1001
0000
1000
Generating 4-bit standard Gray code.
CS1104-2
The Gray Code
88
The Gray Code
sensors
mis-aligned
sensors
Binary coded: 111  110  000
CS1104-2
The Gray Code
mis-aligned
sensors
Gray coded: 111  101
89
Binary-to-Gray Code Conversion
 Retain most significant bit.
 From left to right, add each adjacent pair of binary
1

1
code bits to get the next Gray code bit, discarding
carries.
 Example: Convert binary number 10110 to Gray
1 0 + 1 1 0 Binary
1 + 0 1 1 0 Binary
0 1 1 0 Binary
code.


1
Gray
1
1
0
1
1
1
+
1

0
0
Binary
Gray
1
1
Gray
1
1
0
1
1
1
1
0
+
1
1
0
Binary

1
Gray
Gray
(10110)2 = (11101)Gray
CS1104-2
Binary-to-Gray Code Conversion
90
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 11011 to binary.
1
1
0
1
1
1
Gray
1
+

1
1
Binary
1
1
0
1
+
1
0
0

1
1
0
1
1
1
Binary
1
1
0
1
+
Binary
1
0
+

0
Gray
1
Gray
1
0
0
1
0
1
Gray

0
Binary

0
1
1
Gray
Binary
(11011)Gray = (10010)2
CS1104-2
Gray-to-Binary Conversion
91
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
92
Gray code
93
Binary to Gray Conversion
94
95
Gray to Binary
96
97
Other Decimal Codes
Decimal Digit
0
1
2
3
4
5
6
7
8
9
BCD
8421
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Excess-3
84-2-1
2*421
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
0000
0111
0110
0101
0100
1011
1010
1001
1000
1111
0000
0001
0010
0011
0100
1011
1100
1101
1110
1111
Biquinary
5043210
0100001
0100010
0100100
0101000
0110000
1000001
1000010
1000100
1001000
1010000
 Self-complementing codes: excess-3, 84-2-1, 2*421 codes.
 Error-detecting code: biquinary code (bi=two, quinary=five).
CS1104-2
Other Decimal Codes
98
Self-Complementing Codes
 Examples: excess-3, 84-2-1, 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
CS1104-2
Self-Complementing Codes
241: 0101 0111 0100
758: 1010 1000 1011
99
Alphanumeric Codes
 Apart from numbers, computers also handle textual data.
 Character set frequently used includes:
alphabets:
digits:
special symbols:
non-printable:
‘A’ .. ‘Z’, and ‘a’ .. ‘z’
‘0’ .. ‘9’
‘$’, ‘.’, ‘,’, ‘@’, ‘*’, …
SOH, NULL, BELL, …
 Usually, these characters can be represented using 7 or 8 bits.
CS1104-2
Alphanumeric Codes
100
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
Character
0
1
...
9
:
A
B
...
Z
[
\
ASCII Code
0110000
0110001
...
0111001
0111010
1000001
1000010
...
1011010
1011011
1011100
101
Alphanumeric Codes
 ASCII table:
LSBs
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
CS1104-2
000
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
HT
LF
VT
FF
CR
O
SI
001
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
010
SP
!
“
#
$
%
&
‘
(
)
*
+
,
.
/
MSBs
011 100
0
@
1
A
2
B
3
C
4
D
5
E
6
F
7
G
8
H
9
I
:
J
;
K
<
L
=
M
>
N
?
O
Alphanumeric Codes
101
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
110
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
111
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
DEL
102
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
103
Error Detection Codes
 Parity bit.
 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.
Character
ASCII Code
 Example: Odd parity.
CS1104-2
0
1
...
9
:
A
B
...
Z
[
\
Error Detection Codes
0110000 1
0110001 0
...
0111001 1
0111010 1
1000001 1
1000010 1
...
1011010 1
1011011 0
1011100 1
Parity bits
104
Error Detection Codes
 Parity bit can detect odd number of errors but not even number
of errors.
Example: For odd parity numbers,
10011  10001 (detected)
10011  10101 (non detected)
 Parity bits can also be
applied to a block of data:
0110 1
0001 0
1011 0
1111 1
1001 1
0101 0
Column-wise parity
Row-wise parity
CS1104-2
Error Detection Codes
105
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
106
Alphanumeric Codes
107
ASCII
108
109
Parity Method for error checking
110
111
Put it together
112