Transcript Digital Systems

ENEL 111 Digital Electronics
Richard Nelson
G.1.29
[email protected]
Second Half of ENEL 111
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Digital Electronics
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Number Systems and Logic
Electronic Gates
Combinational Logic
Sequential Circuits
ADC – DAC circuits
Memory and Microprocessors
Hardware Description Languages
Weekly Structure
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Lectures Monday, Tuesday, Wednesday
Slides in ppt and pdf format on support website:
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http://wand.cs.waikato.ac.nz/~111/2005/
(follow link from course website)
Friday Tutorials - Sample Questions on website.
The lecture today
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Digital vs Analog data
Binary inputs and outputs
Binary, octal, decimal and hexadecimal number
systems
Other uses of binary coding.
Analog/Analogue Systems
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Analogue Systems
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V(t) can have any value between its minimum and
maximum value
V(t)
Digital Systems
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Digital Systems
 V(t) must take a value
selected from a set of
values called an
alphabet
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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
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Consider a child’s slide in a playground:
a set of discrete steps
continuous movement
levels
Relationship between Analogue and
Digital systems
5 Volt
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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
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Explain whether the following are analog or
digital:
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A photograph or painting
A scanned image
Sound from a computer’s loud speaker
Sound file stored on disc
Binary Inputs and Outputs
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Coding:
 A single binary input can only have two
values: True or False (Yes or No) (1 or
0)
Binary
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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
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Example 1:
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How many combinations are possible with 10 binary
inputs?
Example 2:
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What is the minimum number of bits needed to
represent the digits ‘0’ to ‘9’ as a binary code?”
Decimal systems
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Number Representation
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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
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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
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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
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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
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There are different ways of representing decimal
numbers in a binary coding
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BCD or Binary Coded Decimal is one example.
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Each decimal digit is replaced by 4 binary digits
Binary Inputs and Outputs
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6 of the possible 16 values unused
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example 45310 = 0100 0101 0011BCD
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Note that BCD code is longer than a direct
representation in natural binary code:
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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
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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
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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
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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
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Colour codes
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You often see hex used in graphic design programs for
the red, blue and green components of a colour:
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FF0000 represents red, for example.
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How many bits are used to represent each colour?
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How many different colours can be represented?
Binary Inputs and Outputs
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Characters
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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
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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.
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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
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Support website
Analogue and Digital
Binary Number Systems
Coding schemes considered were:
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Natural Binary
BCD
Octal representation
Hexadecimal representation
ASCII
Exercises
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You should practice conversions between binary,
octal, decimal and hexadecimal.
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You should be able to code decimal to BCD (and
BCD to decimal).
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You should be able to explain and give examples
of digital and analogue data.