Digital Systems
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Transcript Digital Systems
ENEL 111 Digital Electronics
Richard Nelson
G.1.29
[email protected]
Second Half of ENEL 111
Digital Electronics
Number Systems and Logic
Electronic Gates
Combinational Logic
Sequential Circuits
ADC – DAC circuits
Memory and Microprocessors
Hardware Description Languages
Weekly Structure
Lectures Monday, Tuesday, Wednesday
Slides in ppt and pdf format on support website:
http://wand.cs.waikato.ac.nz/~111/2005/
(follow link from course website)
Friday Tutorials - Sample Questions on website.
The lecture today
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)
1
0
1
0
For example, Binary Alphabet: 0, 1.
1
Slide example
Consider a child’s slide in a playground:
a set of discrete steps
continuous movement
levels
Relationship between Analogue and
Digital systems
5 Volt
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
Input
Range
for 1
Output
Range
for 1
2.8
2.4
Input
Range
for 0
0.8
0.4
0 Volt
Output
Range
for 0
Exercise
Explain whether the following are analog or
digital:
A photograph or painting
A scanned image
Sound from a computer’s loud speaker
Sound file stored on disc
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
00
01
Each additional input doubles the number of
combinations we can represent
i.e. with n inputs it is possible to represent 2n
combinations
10
1 1
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
100 (102)
10 (101)
1 (100)
4
0
3
8 (23)
4 (22)
2 (21)
1 (20)
1
0
0
1
decimal
binary
400 + 0 + 3
8+0+0+1
Binary to decimal
Multiply each 1 bit by the appropriate power of 2 and add
them together.
?
?
1
128
64
32
16
8
4
2
1
1
0
0
0
0
0
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
77
38
1
38
19
0
19
9
1
9
4
1
4
2
0
2
1
0
1
0
1
Least Significant Bit
Most Significant bit
15510 = 100110112
Decimal to Binary
An alternative way is to use the “placement” method
128
64
32
16
8
4
2
128 goes into 155 once leaving 27 to be placed
1
So 64 and 32 are too big (make them zero)
16 goes in once leaving 11
1
and so on…
0
0
1
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
0
1
2
3
4
5
6
7
8
9
BCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
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
hexadecimal
512 (83)
64 (82)
8 (81)
1 (80)
1
2
0
7
256 (162)
16 (161)
1 (160)
1
A
F
512 + 128 + 7
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
Thus 100111010100112 = 235238
3
Similarly with Hexadecimal
Each hex digit corresponds to 4 binary bits
binary
hex
binary
hex
0000
0
1000
8
0001
1
1001
9
0010
2
1010
A
0011
3
1011
B
0100
4
1100
C
0101
5
1101
D
0110
6
1110
E
0111
7
1111
F
To convert a binary string:
10011101010011
Split into groups of 4:
0010 0111 0101 0011
Thus 100111010100112 = ……………16 ?
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) :
00
nul
10
dle
20
sp
30
0
40
@
50
P
60
`
70
p
01
soh
11
dc1
21
!
31
1
41
A
51
Q
61
a
71
q
02
sot
12
dc2
22
"
32
2
42
B
52
R
62
b
72
r
03
etx
13
dc3
23
#
33
3
43
C
53
S
63
c
73
s
04
eot
14
dc4
24
$
34
4
44
D
54
T
64
d
74
t
05
enq
15
nak
25
%
35
5
45
E
55
U
65
e
75
u
06
ack
16
syn
26
&
36
6
46
F
56
V
66
f
76
v
07
bel
17
etb
27
'
37
7
47
G
57
W
67
g
77
w
08
bs
18
can
28
(
38
8
48
H
58
X
68
h
78
x
09
ht
19
em
29
)
39
9
49
I
59
Y
69
i
79
y
0a
nl
1a
sub
2a
*
3a
:
4a
J
5a
Z
6a
j
7a
z
0b
vt
1b
esc
2b
+
3b
;
4b
K
5b
[
6b
k
7b
{
0c
np
1c
fs
2c
,
3c
<
4c
L
5c
\
6c
l
7c
0d
cr
1d
gs
2d
3d
=
4d
M
5d
]
6d
m
7d
}
0e
so
1e
rs
2e
.
3e
>
4e
N
5e
^
6e
n
7e
~
0f
si
1f
us
2f
/
3f
?
4f
O
5f
_
6f
o
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.
Dec
0
1
2
3
E.g. changing from 3 to 4, normal binary has 4
all three bits changing 011 -> 100.
5
Depending on the order in which the bits
change any intermediate value may be
6
created.
7
Gray
000
001
011
010
110
111
101
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.