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Digital Electronics Number Systems and Logic Electronic Gates Combinational Logic Sequential Circuits ADC – DAC circuits Memory and Microprocessors Hardware Description Languages
Materi Hari ini
Digital vs Analog data Binary inputs and outputs Binary, octal, decimal and hexadecimal number
systems Other uses of binary coding.
Analog/Analogue Systems
Analogue Systems V(t) can have any value between its minimum and
maximum value
V(t)
Digital Systems
Digital Systems V(t) must take a value
selected from a set of values called an alphabet
Binary digital systems form the basis of almost all hardware systems currently
V(t)
For example, Binary Alphabet: 0, 1.
1 0 1 0 1
Slide example
Consider a child’s slide in a playground:
continuous movement
a set of discrete steps
levels
Relationship between Analogue and Digital systems
Advantages of Digital Systems Analogue systems: slight error in
input yields large error in output Digital systems more accurate
and reliable Computers use digital circuits
internally Interface circuits (for instance,
sensors and actuators) are often analogue
5 Volt
0 Volt
0.80.4
2.42.8
InputRangefor 1
InputRangefor 0
OutputRangefor 0
OutputRangefor 1
Binary Inputs and Outputs
Coding: A single binary input can only have two
values: True or False (Yes or No) (1 or 0)
Binary
More bits = more combinations
0 0 0 1 1 0 1 1
Each additional input doubles the number of combinations we can representi.e. with n inputs it is possible to represent 2n combinations
Combinations
Example 1: How many combinations are possible with 10 binary
inputs?
Example 2: What is the minimum number of bits needed to
represent the digits ‘0’ to ‘9’ as a binary code?”
Decimal systems
Number Representation Difficult to represent Decimal numbers directly in a
digital system Easier to convert them to binary There is a weighting system:
eg 403 = 4 x 100 + 0 x 10 + 3 x 1
or in, powers of 10:
40310= 4x102 + 0x101 + 3x100 = 400 + 0 + 3
Binary Inputs and Outputs
Both Decimal and Binary numbers use a positional weighting system, eg:
10102 = 1x23+0x22+1x21+0x20 = 1x8 + 0x4 + 1x2 + 0x1 = 1010
decimal 100 (102) 10 (101) 1 (100)
4 0 3 400 + 0 + 3
binary 8 (23) 4 (22) 2 (21) 1 (20)
1 0 0 1 8 + 0 + 0 + 1
Binary to decimal
Multiply each 1 bit by the appropriate power of 2 and add them together.
? ? 128 64 32 16 8 4 2 1
1 0 0 0 0 0 1 1
1 0 1 0 0 1 1 0 0
100000112 = ……………….10 ?
1010011002 = ……………………10 ?
Binary Inputs and Outputs Number Representation - Binary to decimal A decimal number can be converted to binary by repeated division by 2
number /2 remainder
155 77 1 Least Significant Bit
77 38 1
38 19 0
19 9 1
9 4 1
4 2 0
2 1 0
1 0 1 Most Significant bit
15510 = 100110112
Decimal to Binary
An alternative way is to use the “placement” method
128 goes into 155 once leaving 27 to be placed
So 64 and 32 are too big (make them zero)16 goes in once leaving 11
and so on…
128 64 32 16 8 4 2 1
1
1 0 0 1
Representations
There are different ways of representing decimal numbers in a binary coding
BCD or Binary Coded Decimal is one example.
Each decimal digit is replaced by 4 binary digits
Binary Inputs and Outputs
6 of the possible 16 values unused
example 45310 = 0100 0101 0011BCD
Note that BCD code is longer than a direct representation in natural binary code:
453 = 111000101
Decimal BCD0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001
Binary Inputs and Outputs
Hexadecimal and Octal Writing binary numbers as strings of 1s and 0s can be very
tedious Octal (base 8) and Hexadecimal (base 16) notations can be
used to reduce a long string of binary digits.
octal 512 (83) 64 (82) 8 (81) 1 (80)
1 2 0 7 512 + 128 + 7
hexadecimal 256 (162) 16 (161) 1 (160)
1 A F 256 + 160 + 15
Notice that hexadecimal requires 15 symbols (each number system needs 0 – base-1 symbols) and therefore A – F are used after 9.
Octal as shorthand for Binary
Each octal digit corresponds to 3 binary bits
binary octal
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
To convert a binary string: 10011101010011
Split into groups of 3:
010 011 101 010 011
2 3 5 2 3
Thus 100111010100112 = 235238
Similarly with Hexadecimal
Each hex digit corresponds to 4 binary bits
binary hex
0000 0
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
To convert a binary string: 10011101010011
Split into groups of 4:
0010 0111 0101 0011
Thus 100111010100112 = ……………16 ?
binary hex
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
Binary inputs and outputs
Colour codes
You often see hex used in graphic design programs for the red, blue and green components of a colour:
FF0000 represents red, for example.
How many bits are used to represent each colour?
How many different colours can be represented?
Binary Inputs and Outputs Characters
Three main coding schemes used: ASCII (widespread use), EBCDIC (not used often) and UNICODE (new)
ASCII table (in hex) : 00nul
01soh
02sot
03 etx
04 eot
05enq
06 ack
07 bel
08 bs
09 ht
0a nl
0b vt
0c np
0d cr
0e so
0f si
10 dle
11 dc1
12 dc2
13 dc3
14 dc4
15 nak
16 syn
17 etb
18 can
19 em
1a sub
1b esc
1c fs
1d gs
1e rs
1f us
20 sp
21 !
22 "
23 #
24 $
25 %
26 &
27 '
28 (
29 )
2a *
2b +
2c ,
2d -
2e .
2f /
30 0
31 1
32 2
33 3
34 4
35 5
36 6
37 7
38 8
39 9
3a :
3b ;
3c <
3d =
3e >
3f ?
40 @
41 A
42 B
43 C
44 D
45 E
46 F
47 G
48 H
49 I
4a J
4b K
4c L
4d M
4e N
4f O
50 P
51 Q
52 R
53 S
54 T
55 U
56 V
57 W
58 X
59 Y
5a Z
5b [
5c \
5d ]
5e ̂
5f _
60 ̀
61 a
62 b
63 c
64 d
65 e
66 f
67 g
68 h
69 i
6a j
6b k
6c l
6d m
6e n
6f o
70 p
71 q
72 r
73 s
74 t
75 u
76 v
77 w
78 x
79 y
7a z
7b {
7c 7d }
7e ~
7f del
Gray Codes
Other codes exist for specific purposes Gray codes provide a sequence where
only one bit changes for each increment Allows increments without ambiguity due
to bits changing at different times. E.g. changing from 3 to 4, normal binary has
all three bits changing 011 -> 100. Depending on the order in which the bits change any intermediate value may be created.
Dec Gray
0 000
1 001
2 011
3 010
4 110
5 111
6 101
7 100
Summary
Support website Analogue and Digital Binary Number Systems Coding schemes considered were:
Natural Binary BCD Octal representation Hexadecimal representation ASCII
Exercises
You should practice conversions between binary, octal, decimal and hexadecimal.
You should be able to code decimal to BCD (and BCD to decimal).
You should be able to explain and give examples of digital and analogue data.
