1. Chapter 1
Introduction

2. 1.1 Logic Design Digital System Simple Example of Digital System
Chapter 1 Introduction
1.1 Logic Design
Digital
System A B
n inputs W X
m outputs
Digital System
Inputs & Outputs
Clock
Simple Example of Digital System
A system with three inputs, A,B, and C, and one output Z, such that Z = 1
iff two of the inputs are 1. A B C Z 1 P

3. 1.1 Logic Design Digital vs. Analog Two types of digital systems
Chapter 1 Introduction
1.1 Logic Design
Digital vs. Analog
Values: discrete vs. continuous
Time: discrete vs. continuous
Two types of digital systems
Combinational (조합)
Sequential (순차) P

4. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Number systems
n : number of digits
r : radix or base
ai : coefficients
Decimal number
Binary number

5. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Power of 2
First 32 binary integer
Decimal
Binary
4-bit
0000 16
10000 1
0001 17
10001 2 10
0010 18
10010 3 11
0011 19
10011 4
100
0100 20
10100 5
101
0101 21
10101 6
110
0110 22
10110 7
111
0111 23
10111 8
1000 24
11000 9
1001 25
11001
1010 26
11010
1011 27
11011 12
1100 28
11100 13
1101 29
11101 14
1110 30
11110 15
1111 31
11111 n 2ⁿ 1 2 11
2,048 4 12
4,096 3 8 13
8,192 16 14
16,384 5 32 15
32,768 6 64
65,536 7
128 17
131,072
256 18
262,144 9
512 19
524,288 10
1,024 20
1,048,576
P. 4

6. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Decimal to Binary
 2 = 373 remainder 0
373  2 = 186 remainder 1 10
186  2 = 93 remainder 0
010
93  2 = 46 remainder 1
1010
46  2 = 23 remainder 0
01010
23  2 = 11 remainder 1
101010
11  2 = 5 remainder 1
5  2 = 2 remainder 1
2  2 = 1 remainder 0
1  2 = 0 remainder 1 =

7. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Hexadecimal
16 digits(0 ~ 9, A ~ F)
4-bit string ~ 1 hexadecimal digit
= = 2EA 16
2EA 16

8. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Binary Addition 13 + + 5 18
One-bit adder a b
cin
cout s 1
P. 9

9. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Binary Addition
4-bit adder a4 b4 a3 b3 a2 b2 a1 b1 c1
Full
Adder
Full
Adder
Full
Adder
Full
Adder c4 s4 s3 s2 c1 s1
P. 9

10. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Signed Numbers
The sign is represented by a single additional bit in binary numbers
sign bit 0 : positive
01012 = + 510
11012 = – 510
sign bit 1 : negative
Two’s Complement
Range of representable number – (2n – 1) ~ + (2n – 1 – 1)
Positive number : stored in normal binary
Negative number : 2n – a in n-bit system
Radix-complement
n = number of digits
D = rn – D
D = complement of n-digit number D

11. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Signed Numbers
Converting steps for negative numbers in two’s complement
Find the binary equivalent of the magnitude
Complement each bit (that is, change 0’s to 1’s and 1’s to 0’s)
Add 1 -5 -1 -0
0101
1010 1
1011
0001
1110 1
1111
0000
1111 1

12. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Signed and unsigned 4-bit numbers
Binary
Positive
Signed
(two’s complement)
0000
0001 1 +1
0010 2 +2
0011 3 +3
0100 4 +4
0101 5 +5
0110 6 +6
0111 7 +7
1000 8 -8
1001 9 -7
1010 10 -6
1011 11 -5
1100 12 -4
1101 13 -3
1110 14 -2
1111 15 -1
P. 12

13. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Binary Subtraction
a – b = = a + (–b)
7 - 5 1
0111
1010
0010 7
0111
1011
0010 -5 +
(1’s complement) 2
Overflow
7 - (- 5)
5 - 7 1
0101
1000
1110 1
0111
0100
1100
-2 (true)
12 (overflow)

14. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Fractions, Mixed Numbers and Floating Point Representation
Fraction
Conversion from decimal to binary
.625 · 2 = 1.25 .1
.25 · 2 = 0.50
.10
.50 · 2 = 1.00
.101

15. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Fractions, Mixed Numbers and Floating Point Representation
Conversion from decimal to binary
.3 · 2 = 0.6 .0
.6 · 2 = 1.2
.01
.2 · 2 = 0.4
.010
.4 · 2 = 0.8
.0100
.8 · 2 = 1.6
.01001
.6 · 2 = 1.2
 Use BCD(binary coded decimal) to represent fractions exactly

16. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Fractions, Mixed Numbers and Floating Point Representation
Mixed numbers
Mixed numbers are converted separately
24.375
24 =
.375 = =

17. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Fractions, Mixed Numbers and Floating Point Representation
Floating-point representation
mantissa x (radix)exponent
Floating-point arithmetic
Lined up radix point
Increase exponent of the smaller number and shift mantissa

18. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Fractions, Mixed Numbers and Floating Point Representation
Floating-point arithmetic
If mantissa is greater than 1
shift mantissa right and increase exponent
If addition results in leading 0’s
shift mantissa left to remove 0’s and decrease exponent

19. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Binary Coded Decimal(BCD)
Decimal digit
8421 code
5421 code
2421 code
Excess 3 code
2 of 5 code
0000
0011
11000 1
0001
0100
10100 2
0010
0101
10010 3
0110
10001 4
0111
01100 5
1000
1011
01010 6
1001
1100
01001 7
1010
1101
00110 8
1110
00101 9
1111
00011
unused
any of
the 22
patterns
with 0, 1,
3, 4, or 5
1’s
P. 18

20. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
ASCII Code
a3a2a1a0
a6a5a4
010
011
100
101
110
111
0000
space @ P ` p
0001 ! 1 A Q a q
0010 “ 2 B R b r
0011 # 3 C S c s
0100 $ 4 D T d t
0101 % 5 E U e u
0110
& 6 F V f v
0111 ‘ 7 G W g w
1000 ( 8 H X h x
1001 ) 9 I Y i y
1010 * : J Z j z
1011 + ; K [ k {
1100 ,
< L \ l |
1101 = M ] m }
1110 .
> N ^ n ~
1111 / ? O _ o
delete
P. 19

21. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
Gray Code
Number
Gray code
0000 8
1100 1
0001 9
1101 2
0011 10
1111 3
0010 11
1110 4
0110 12
1010 5
0111 13
1011 6
0101 14
1001 7
0100 15
1000
P. 20

22. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
Hamming Code
Single-error-correcting code
4 information bit and 3 check bit a1 a2 a3 a4 a5 a6 a7
Bit 1 X
Bit 2
Bit 3
P. 20

23. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
Hamming Code
16 coded hamming words
Date/Bit a1 a2 a3 a4 a5 a6 a7
0000
0001 1
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
P. 21

24. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
Hamming Code
Checking and correcting
Received :
e1 = 0 e2 = 1 e4 = 0
bit 2(check bit) error => a2 = 1
Correct word : , Data : 1011
Received :
e1 = 1 e2 = 0 e4 = 1
bit 5 error => a5 = 0
Correct word : , Data : 0001

25. 1.2 A Brief Review Of Number Systems
Chapter 1 Introduction
1.2 A Brief Review Of Number Systems
Other Codes
Hamming Code
n check( n ≥ 2 ) bits : 2n – n – 1 information bits
Check bits
Data bits 2 1 3 4 11 5 26
P. 22