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TRANSCRIPT
CHAPTER
Sydney Smith(1835)
2
2.1
(bit) bit binary digit
( ) 1964 IBM System/360
8
8 (byte)
(words)
(word
size) 16 32
64 ( )
(nibbles nybles)
(positional number sys-tem)
(least-value) (low-
order) (high-order)
2.2 (POSITIONAL NUMBER SYSTEM)
( (base)10)
243
1 10
(radix ) (weighted
numbering system)
0 9 3 ( 3)
0 1 2
9 8
3310 33(
)
( 2.1)
2.1
243.5110 = 2 � 102 + 4 � 101 + 3 � 100 + 5 � 10�1 + 1 � 10�2
2123 = 2 � 32 + 1 � 31 + 2 � 30 = 2310
101102 = 1 � 24 + 0 � 23 + 1 � 22 + 1 � 21 + 0 � 20 = 2210
( 2)
( 16) ( 8) 0 1
0 7 0 9 A B C D E F
10 15 2.1
-
10410 3 34=81
3 104 5
( 0 4) 81 104
23 33 = 27 23
23 32 = 9 23-18 = 5 5
31 = 3 2 2 30 2.2
2.2 10410 3
104�81 = 34 � 1
23
�0 = 33 � 023
�18 = 32 � 25
�3 = 31 � 12
�2 = 30 � 20 10410 = 102123
- -
2.3
2.3 - 10410 3
3 104 104 3 34 23 34 34 3 11 13 11 11 3 3 23 3 3 3 1 0
1 1 3 0 10
10410 = 102123
2.4
2.4 14710
2 147 147 2 73 12 73 73 2 36 12 36 36 2 18 02 18 18 2 9 02 9 9 2 4 12 4 4 2 2 02 2 2 2 1 02 1 1 2 0 1
014710 = 100100112
N 0 2n-1 4
0 15 8 0
255
4
11112(1510) 11112 15 15 30 4
30 (overflow)
2.4
2.3.2
(radix points)
(decimal point)
2/3
0.23(2 3-1 = 2 1/3)
-
2.5 5
2.5 0.430410 5
0.4304� 0.4000 = 5�1 � 2
0.0304� 0.0000 = 5�2 � 0
0.0304� 0.0240 = 5�3 � 3
0.0064� 0.0064 = 5�4 � 4
0.0000
0.430410 = 0.20345
-
2.6
2.6 0.430410 5
.4304� 52.1520 2
.1520� 50.7600 0
.7600� 53.8000 3
.8000� 54.0000
0.430410 = 0.20345
(rounding algorithm)
( )
2.7
2.7 0.3437510 2 4
.34375� 2
0.68750 ( ).68750� 2
1.37500.37500� 2
0.75000.75000� 2
1.50000 ( 4 )
0.3437510 = 0.01012
4 3( 2.8)
10
2
2.8 31214 3
31214 = 3 � 43 + 1 � 42 + 2 � 41 + 1 � 40
= 3 � 64 + 1 � 16 + 2 � 4 + 1 = 21710
3
3 | 217 13 | 72 03 | 24 03 | 8 23 | 2 2
0 31214 = 220013
2.3.3
( )
16=24 4 ( hextet)
8=23 3
( octet)
2.9 1100100111012
110 010 011 1016 2 3 5
1100100111012 = 62358
1100 1001 1101C 9 D
1100100111012 = C9D16
(
)
2.4
1
8
(signed magnitude)
(complements)
2.4.1
(signed integers)
(signed magnitude representation)
(
) ( )
8 1 10000001 +1 00000001
8 7
8 27 1 127(
)
127 N 2(N 1) 1 ~ 2(N 1)
1
(1)
(2)
( )
8
7
2.10 010011112 001000112
1 1 1 1 ⇐ 0 1 0 0 1 1 1 1 (79)0 + 0 1 0 0 0 1 1 + (35)0 1 1 1 0 0 1 0 (114)
010011112 + 001000112 = 011100102
( 2.11 )
(
)
Double-Dabble
double-dabble( double-dib-
ble)
( )
1
100100112
1
1 0 0 1 0 0 1 1
2
1 0 0 1 0 0 1 12
22
3
1 0 0 1 0 0 1 12 4
+ 02
� 2 � 22 4
4 3
1 0 0 1 0 0 1 12 4 8 18 36 72 146
+ 0 + 0 + 1 + 0 + 0 + 1 + 12 4 9 18 36 73 147‹ 100100112 = 14710
� 2 � 2 � 2 � 2 � 2 � 2 � 22 4 8 18 36 72 146
double-dabble ( )
2
02CA16
2.11 010011112 011000112
1 ← 1 1 1 1 ⇐
0 1 0 0 1 1 1 1 (79)
0 + 1 1 0 0 0 1 1 + (99)
0 0 1 1 0 0 1 0 (50)
79 + 99 = 50
(minuend)
2.12 011000112 010011112
0 1 1 2 ⇐ 0 1 1 0 0 0 1 1 (99)0 � 1 0 0 1 1 1 1 � (79)0 0 0 1 0 1 0 0 (20)
001000112 010011112 = 000101002
2.13 010011112(79) 011000112(99)
01100011 01000011
2.12 00101002
0 2 C A0000 0010 1100 1010
double-dabbl
1 0 1 1 0 0 1 0 1 02 4 10 22 44 88 178 356 714
+ 0 + 1 + 1 + 0 + 0 + 1 + 0 + 1 + 02 5 11 22 44 89 178 357 714
� 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 22 4 10 22 44 88 178 356 714
02CA16 = 10110010102 = 71410
( ) 010011112
011000112 = 100101002
(
)
(1)
(2)
( )
2.14 100100112( 19) 000011012( 13)
( ) 1 ( )
( )
0 1 2 ⇐1 0 0 1 0 0 1 1 (�19)0 � 0 0 0 1 1 0 1 + (13)1 0 0 0 0 1 1 0 (�6)
100100112 000011012 100001102
2.15 101010112( 43) 100110002( 24)
2 4 2 4
43 43 24
( 43 24)
( 43 24)
0 20 1– 0 1 0 1 1 (43)
� 0 0 1 1 0 0 0 � (24)0 0 1 0 0 1 1 (19)
101010112 000110002
100100112
( ) 10000000
00000000 ( ) ( )
2.4.2
(diminished radix complement) 167 52 = 115 999
52 947 167 52 = 167 947 = 114
167 52 = 115
9(casting out 9s)
700
700 300
001-500 501-999
997
3 997 501-999
001-500 (radix complements)( )
001-500
10 ( 9)
d
r N N (rd 1) N r
= 10 10 1 = 9 2468 9999 2468 =
7531 (2) 1 01012
11112 01012 = 10102
1
0 0 1
10 7 10 ( 7)
(
)
1
0 2.16
2.16 8 2310 910
2310 = + (000101112) = 000101112� 910 = �(000010012) = 111101102
23 9 (9)
(23) 9 23
1 0 ( (end
carry-arround) )
2.17 2310 910
1 ← 1 1 1 1 1 ⇐0 0 0 1 0 1 1 1 (23)
+ 1 1 1 1 0 1 1 0 + (–9)0 0 0 0 1 1 0 1
+ 10 0 0 0 1 1 1 0 1410
2.18 910 2310
0 ← 0 0 0 0 1 0 0 1 (9)+ 1 1 1 0 1 0 0 0 + (–23)
1 1 1 1 0 0 0 1 –1410
111100012 1410
( 1) 111100012
000011102 14
0 0 0 0 0 0 0 0
11111111
(radix complement) r d
N N rd N N 0 N = 0 0
103 2 = 998 ( )
4 00112 24 00112 = 100002
00112 = 11012
( )
2.19
2.19 8 2310 2310 910
2310 = + (000101112) = 000101112�2310 = � (000101112) = 111010002 + 1 = 111010012�910 = � (000010012) = 111101102 + 1 = 111101112
000101112
23
111101112
(
) 000010002 1 000010012
9 11110112
9
(
)
2.20 910 2310
0 0 0 0 1 0 0 1 (9)+ 1 1 1 0 1 0 0 1 + (–23)
1 1 1 1 0 0 1 0 –1410
111100102 1410
2.21 2310 910
1← 1 1 1 1 1 1 ⇐0 0 0 1 0 1 1 1 (23)
+ 1 1 1 1 0 1 1 1 + (–9)0 0 0 0 1 1 1 0 1410
2.21
( )
2.21
( )
( 1)
( )
( )
2.22 ( 1) ( 0)
2.22 12610 810
0← 1 1 1 1 ⇐0 1 1 1 1 1 1 0 (126)
+ 0 0 0 0 1 0 0 0 +(8)1 0 0 0 0 1 1 0 (–122???)
( )
( 0)
(self-invert)
7 7 8 7
7
0
1 ( ) 01112
7 11112 11112 1(
1 ) 8
8 10002 1
(0111) 1( 1000 8)
8
2.5
16 32,767
000001102 000010112
0 0 0 0 0 1 1 0 +
+
+
+
=
0 0 0 0 1 1 0 0
0 0 0 1 1 0 0 0
0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 0
0 0 0 1 0 0 1 0
0 0 0 1 0 0 1 0
0 1 0 0 0
1 0 1 1
1 0 1 1
1 0 1 1
1 0 1 1
0 1 0
32,767 32
16
(scientific notation)
(fractional
part) (mantissa)
mantissa
32,767 3.2767 104
2.5.1
(
2 ) (significand
)
( ) ( )
14 5 8
( 2.2) 2.5.2
2.2
17 17
17.0 100 1.7 101 0.17 102 1710 100012 20 1000.12
21 100.012 22 10.0012 23 1.00012 24 0.100012 25
0.100012 25 10001000 00101
00 0 1 0 1 1 0 0 0 1 0 0 0
14 (fixed-point)
( 14 )
65536 0.12 217
10 0 0 0 1 1 0 0 0 0 0 0 0
0.25 2-2 2
(biased exponent)
(bias value)
16 0 31
( 5 25 32 )
16 16
(excess-16) 16
0 1
( ) 0 1
17 1710 0.100012 25
16 5 21
10 0 1 0 1 1 0 0 0 1 0 0 0
0.25 0.1 2-1
00 1 1 1 1 1 0 0 0 0 0 0 0
10 0 1 0 1 1 0 0 0 1 0 0 0 =
10 0 1 1 0 0 1 0 0 0 1 0 0 =
10 0 1 1 1 0 0 1 0 0 0 1 0 =
10 1 0 0 0 0 0 0 1 0 0 0 1
1
(normalization) 1
2.23 -16 0.0312510
0.0312510 0.000012 20 0.0001 2-1 0.001 2-2 0.01 2-3 0.1
2-4 16 4 12
00 1 1 0 0 1 0 0 0 0 0 0 0
2.5.2
1.5 102
3.5 103
1.5 102 3.5 103 0.15 103 3.5 103 3.65
103
2.24 -16 14
10 0 0 1 0 1 1 0 0 1 0 0 0 +
10 0 0 0 0 1 0 0 1 1 0 1 0
11.001000+ 0.1001101011.10111010
10 0 0 1 0 1 1 1 0 1 1 1 0
2-3 24 21
2.25
10 0 0 1 0 1 1 0 0 1 0 0 0 = 0.11001000 � 22
= 0.10011010 � 20� 10 0 0 0 0 1 0 0 1 1 0 1 0
0.11001000 0.10011010 0.0111100001010000
22 20 22 1.1110000101
10 0 0 0 1 1 1 1 1 0 0 0 0
2.5.3
(real number)
(approximation)
(bla tant) (subt le)
(unnoticed) (overflow) (underflow)
.111111112 215 .11111111 215
2-19 2128 128.5
128.5
128.5 10000000.1
(significand)
128.5
128.5 � 128 = 0.00389105 � 0.39%128.5
2.3 14 16.24 0.91
8
100
14
2.5.4 IEEE-754
1980
1985 (Institute of
Electrical and Electronic Engineers IEEE)
IEEE-754(1985)
IEEE-754 8 127
23 32 255
( ) (not a number) ( )
N a N
�
�
�
�
�
�
2.3 14
(error indicator)
64 11 52
(bias) 1023 IEEE 2.4
NaN 2047
FPU 64
IEEE-754
0 1
IEEE-754
IBM
IBM
IEEE-754 1998 IBM 1964
System/360
2.6
2.4 IEEE-754
2.6.1
(Binary-coded decimal BCD) IBM
BCD
4 8
(zone) (digit)
(
)BCD
1111
1100 1101 2.5 BCD
1010 1111
40
0 .3 8
0 .296875 1.05
BCD 0.3 1111 0011( )
BCD
2.5 (BCD)
(packing)
(packed decimal number)
2.26 3 BCD 1265
1256 -
1111 0001 1111 0010 1111 1111 0101
0001 0010 0110 0101
(1101)
1111
0000 0001 0010 0110 0101 1101
2.6.2 EBDIC
IBM IBM System/360 6 BCD
System/360
IBM BCD 6 8
(Extended Binary Coded Decimal Interchange EBCDIC) IBM
EBCDIC 2.6 - EBCDIC
EBCDIC a 1000
0001 3 1111 0011
0000
NUL
DLE
DS
SP
&
–
{
}
\
0
Zone
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0001
SOH
DC1
SOS
/
a
j
~
A
J
1
0010
STX
DC2
FS
SYN
b
k
s
B
K
S
2
0011
ETX
TM
c
l
t
C
L
T
3
0100
PF
RES
BYP
PN
d
m
u
D
M
U
4
0101
HT
NL
LF
RS
e
n
v
E
N
V
5
0110
LC
BS
ETB
UC
f
o
w
F
O
W
6
0111
DEL
IL
ESC
EOT
g
p
x
G
P
X
7
1000
CAN
h
q
y
H
Q
Y
8
1001
RLF
EM
'
i
r
z
I
R
Z
9
1010
SMM
CC
SM
[
]
|
:
1011
VT
CU1
CU2
CU3
.
$
,
#
1100
FF
IFS
DC4
<
*
%
@
1101
CR
IGS
ENQ
NAK
(
)
_
'
1110
SR
IRS
ACK
+
;
>
=
1111
SI
IUS
BEL
SUB
!
ˆ
?
"
Digit
NULSOHSTXETXPFHTLCDELRLFSMMVTFFCRSOSIDLEDC1DC2
NullStart of headingStart of textEnd of textPunch offHorizontal tabLowercaseDeleteReverse linefeedStart manual messageVertical tabForm FeedCarriage returnShift outShift inData link escapeDevice control 1Device control 2
TMRESNLBSILCANEMCCCU1IFSIGSIRSIUSDSSOSFSBYPLF
Tape markRestoreNew lineBackspaceIdleCancelEnd of mediumCursor ControlCustomer use 1Interchange file separatorInterchange group separatorInterchange record separatorInterchange unit separatorDigit selectStart of significanceField separatorBypassLine feed
ETBESCSMCU2ENQACKBELSYNPNRSUCEOTCU3DC4NAKSUBSP
End of transmission blockEscapeSet modeCustomer use 2EnquiryAcknowledgeRing the bell (beep)Synchronous idlePunch onRecord separatorUppercaseEnd of transmissionCustomer use 3Device control 4Negative acknowledgementSubstituteSpace
Abbreviations:
2.6 EBCDIC ( - )
2.6.3 ASCII
IBM System/360
(Ameri-
can Standard Code for Information Interchange ASCII)
ASCII 5
(Murray) 1880 Baudot 1960 5
(International Organization
for Standardization ISO) 7
5(International Alphabet Number 5) 1976
ASCII
2.7 ASCII 32 10 52
( ) 32 ( $ #) ( 8
)
(Parity)
( ) (1)
(0) 1
ASCII A 7
100 0001 1 0
0100 0001 ASCII C 100 0011
1 1100 0011 2.8
ASCII
1980
12810 25510
n ASCII
2.7 ASCII ( )
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
TAB
LF
VT
FF
CR
SO
SI
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
!
"
#
$
%
&
'
(
)
*
+
,
-
.
/
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
ˆ
_
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
DEL
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
HT
LF
VT
FF
CR
SO
SI
Null
Start of heading
Start of text
End of text
End of transmission
Enquiry
Acknowledge
Bell (beep)
Backspace
Horizontal tab
Line feed, new line
Vertical tab
Form feed, new page
Carriage return
Shift out
Shift in
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
DEL
Data link escape
Device control 1
Device control 2
Device control 3
Device control 4
Negative acknowledge
Synchronous idle
End of transmission block
Cancel
End of medium
Substitute
Escape
File separator
Group separator
Record separator
Unit separator
Delete/Idle
Abbreviations:
2.6.4 (Unicode)
EBCDIC ASCII
1991
(Unicode)
(Unicode Consortium)
16 ASCII Latin 1
ISO/IEC 10646-1
16
2.8
´ A Á
Java
ASCII EBCDIC
2.7
ASCII EBCDIC (Unicode)
( )
( )
( ASCII)
(data encoding) (encoded data)
2.7.1 (Non-Return-to-Zero Code)
(NRZ Non-Return-to-Zero)
high low
3 5
3 5 ( )
OK ASCII 11001111 01001011
2.9 NRZ
(bit cells)
2.9 ASCIIO 1 OK
OKAY 0 1 11001111 01001011
01000001 01011001
(out-of-phase) OKAY 10011 0100101
010001 0101001 ASCII <ETX>( )
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1
Zero
High
Low
a.
b.
2.9 OK NRZ
a.
b. ( )
( <ETX> ASCII End-of-Text 26 )
NRZ
2.7.2 (Non-Return-to-Zero-Invert Code)
(NRZI Non-Return-to-Zero-Invert)
NRZI 1 high-to-low low-to-high
0 2.10 OK( ) NRZI
NRZI 1
2.7.3 (Phase Modulation Manchester Coding)
(phase modulatoin
PM) PM PM
1 up( )
0 down( )
2.11 OK PM
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1
2.10 OK NRZI
PM PM NRZ
( 2.11b )
NRZ
2.7.4
(frequency modulation FM)
1
2.12 OK FM
FM PM
FM (modified frequency modu-
lation MFM) FM
PM FM
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1a.
b.
2.11 OK (Manchester )
a.
b.
MFM
MFM PM NRZ
MFM MFM
2.7.5 (Run-Length-Limited Code)
(Run-length-limited, RLL) ASCII EBCDIC
RLL(d, k) d k 0 1
RLL
RLL NRZI RLL
flat NRZI RLL
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1
2.12 OK
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1
2.13 OK MFM
RLL RLL(2, 7)
8 ASCII EBCDIC 16
(flux reversals) MFM
50% ( )
RLL (
) Huffman (bit patterns)
( )
1
10
0.25 (P(bi=1)=1/2 P(bj=0)=1/2 => P(bibj=10)=1/21/2=1/4 )
011 0.125 2.14 RLL(2, 7)
2.15 RLL(2, 7)
0
0
2.16 MFM RLL(2,7) NRZI OK
MFM 12 RLL 8
RLL OK MFM 50%
RLL
Root
0 1
0 1 0 1
0 1 0 1
0 1
P(10)= P(11)=
P(010)= P(000)= P(011)=
P(0010)= P(0011)=
14
14
18
18
18
116
116
2.14 RLL(2, 7)
2.8
2.7.1 OKAY 4 3
<ETX>( )
Character BitPattern
RLL(2, 7)Code
101100001001100100011
010010000001001001000010000010010000001000
2.15 RLL(2, 7)
1 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1
1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
12 Transitions
8 Transitions
2.16 Ok MFM( ) RLL(2, 7)( )
100
(
(implementation-dependent))
2.8.1 (Cyclic Redundancy Check)
(Checksum)
(International Standard Book Numbers ISBN)
(Cyclic Redundancy
Check, CRC)
CRC
(systematic error detection)
(syndrome)
(cyclic)
2(Arithmetic Modulo 2)
12 11:00 2 1:00 2
2
2
(algebraic field)
0 0 00 1 11 0 11 1 0
2.27 10112 1102 2
1011+110
11012 (mod 2)
2
2 2.28
2.28 10010112 10112
1.
2. 2
3. 1
4. 1
5. 2
6.
7. 1012 10112 1012
10102
2
10112 1 23 0 22 1 21 1 20
X 2 10112
1 X3 0 X2 1 X1 1 X0
2.28
X6 + X3 + X + 1��
X3 + X +1
1011)10010111011 0010
001001 1011 0010 00101
CRCs
CRCs
1. I 10010112 ( )
2. P 10112 (P
1 )
3. I P
I 10010110002
4. I P 2 ( 2.28 )
1002 CRC (checksum)
5. I M
10010110002 1002 10010111002
6. M M P
M
• CRC-CCITT (ITU-T): X16 + X12 + X5 + 1• CRC-12: X12 + X11 + X3 + X2 + X + 1• CRC-16 (ANSI): X16 + X15 + X2 + 1• CRC-32: X32 + X26 + X23 + X22 + X16 + X12 + X11 + X10 + X8 + X7 + X5 + X4
+ X + 1
1010100 1011 ) 1001011100 1011
001001 1011
0010001011
10110000
CRC-CCITT CRC-12 CRC-16 CRC-32
32
CRC 99.8
CRC (bit pattern)
16 32
2.8.2 (Hamming Codes)
(check
bits) (redundant bits) m r
( )
(code word) m r n
n m r
m bits r bits
(Hamming distance)
1 0 0 0 1 0 0 11 0 1 1 0 0 0 1
* * *
3(
)
d d
( )
2 n
(minimum Hamming distance)
D(min) X
Y X D(min)
k ( )
D ( m i n ) k 1 D ( m i n ) 1
|_((D(min) 1)/2)_| 1 k
2k 1
r
( ASCII
)
1 |_ _| |_ 8.3 _| 8|_ 8.9_| 8
2.29 2 ( 1
) ( )
000
011
101
110
0
1
1
0
00
01
10
11
3 3 8
( *)
000*
001
010
011*
100
101*
110*
111
001
011 001
011 000
D(min) 2
D(min)
2.29 D(min)
k
D(min) 2k 1 k
2.30
2.30 (
)
0 0 0 0 0
0 1 0 1 1
1 0 1 1 0
1 1 1 0 1
D ( m i n )
D(min) 3
10000
10000
10000 00000 1 01011
4 10110 2 11101 3
[1, 4, 2, 3] 10000
00000
1
10110 10000
11000
[2, 3, 3, 2]
m r
2m
1
n n
n 1
n 1 ( n
) n n m r 2n
(n + 1) � 2m ≤ 2n
n 1 2 m
2n n m r
(m + r + 1) � 2m ≤ 2m�r
(m + r + 1) ≤ 2r
m r
(
)
m 4
(4 + r + 1) ≤ 2r
r 3 r 3
4 3
1. r n (n m r) 1 (
0)
2. 2
3. b b1 b2 ... bj
b1 b2 ... bj b ( 2 )
2.31 8 ASCII K (
)
K
1 n
m 8 (8 r 1) ≤ 2r r 4 r 4
2 n 1
12 11 10 9 8 7 6 5 4 3 2 1
3
2
1 = 1 5 = 1 + 4 9 = 1 + 82 = 2 6 = 2 + 4 10 = 2 + 83 = 1 + 2 7 = 1 + 2 + 4 11 = 1 + 2 + 84 = 4 8 = 8 12 = 4 + 8
1 1 3 5 7 9 11 1
1 3 5 7 9 11 2 2 3 6 7 10 11
2 2 3 6 7 10 11 4 4 5
6 7 12 8 8 9 10 11 12
12 11 10 9 8 7 6 5 4 3 2 10 1 0 0 1 1 0 1 0 1 1 0
K 010011010110
b9 010111010110
1 1 3 5 7 9 11
2 2 3 6 7 10 11
4 4 5 6 7 12
8 8 9 10 11
1 8 9 11
9 11 2
11 9 (
)
1 8 1 8 9
2.8.3 Reed-Soloman
100 3
(burst errors)
Reed-Soloman(RS)
CRC RS CRC
RS(n, k)
• s ( )
• k ( s )
• n
k RS(n, k)
RS(255, 223) 223 8 32
(syndrome) 255
16
Reed-Soloman
(
)Reed-Soloman
g(x) = (x � ai)(x � ai�1) . . . (x � ai�2t)
t n k x ( ) g(x ) GF(2 s)
( Galois
(integer field) )
n RS
c(x) = g(x) � i(x)
i(x)
Reed-Soloman
(core dump)
(n � k)�
2
ASCII EBCDIC
(Unicode) Java Windows
ASCII EBCDIC
ASCII EBCDIC
Bunt(1988)
Kunth(1998) Computer Algorithm
(
Kunth )
Goldberg(1991) Schwartz (1999) IBM
S y s t e m / 3 9 0 I E E E
Soderquist Leeser(1996)
www.unicode.org
The Unicode Standard (2000)
www.iso.ch
www.ansi.org
Mee Daniel(1988)
Arazi (1988)
Reed-Soloman Galois
Pretzel (1992)
Galois Artin(1998) Warner(1990) (
) Warner
( )
Arazi, Benjamin. A Commonsense Approach to the Theory of Error Correcting Codes. Cambridge,MA: The MIT Press, 1988.
Artin, Emil. Galois Theory. New York: Dover Publications, 1998.
Bunt, Lucas N. H., Jones, Phillip S., & Bedient, Jack D. The Historical Roots of Elementary Math-ematics. New York: Dover Publications, 1988.
Goldberg, David. “What Every Computer Scientist Should Know About Floating-Point Arith-metic.” ACM Computing Surveys 23:1 March 1991. pp. 5–47.
Knuth, Donald E. The Art of Computer Programming, 3rd ed. Reading, MA: Addison-Wesley,1998.
Mee, C. Denis, & Daniel, Eric D. Magnetic Recording, Volume II: Computer Data Storage. NewYork: McGraw-Hill, 1988.
Pretzel, Oliver. Error-Correcting Codes and Finite Fields. New York: Oxford University Press,1992.
Schwartz, Eric M., Smith, Ronald M., & Krygowski, Christopher A. “The S/390 G5 Floating-Point Unit Supporting Hex and Binary Architectures.” IEEE Proceedings from the 14th Sympo-sium on Computer Arithmetic. 1999. pp. 258–265.
Soderquist, Peter, & Leeser, Miriam. “Area and Performance Tradeoffs in Floating-Point Divideand Square-Root Implementations.” ACM Computing Surveys 28:3. September 1996. pp.518–564.
The Unicode Consortium. The Unicode Standard, Version 3.0. Reading, MA: Addison-Wesley,2000.
Warner, Seth. Modern Algebra. New York: Dover Publications, 1990.
1. bit
2. bit( ) byte( ) nibble( ) word( )
3.
4. (radix)
5. 2.1 ( )
6.
7.
8.
9.
10. double-dabble -
11.
12.
13.
14.
15.
16.
17.
18. IEEE-754
19. EBCDIC BCD
20. ASCII
21.
22.
23. (non-return-to-zero)
24. Manchester (non-return-to-zero)
25. (run-length-limited encoding)
26. (cyclic redundancy checks)
27.
28.
29.
30.
31. (burst errors)
32.
◆1.
◆ a) 45810 = ________ 3
◆ b) 67710 = ________ 5
◆ c) 151810 = _______ 7
◆ d) 440110 = _______ 9
2.
a) 58810 = _________ 3
b) 225410 = ________ 5
c) 65210 = ________ 7
d) 310410 = ________ 9
◆3.
◆ a) 26.78125
◆ b) 194.03125
◆ c) 298.796875
◆ d) 16.1240234375
4.
a) 25.84375
b) 57.55
c) 80.90625
d) 84.874023
5. 8 (signed magnitude)
◆ a) 77
◆ b) �42
c) 119
d) �107
6. 3
a) (signed magnitude)
b)
c)
7. 4
a) (signed magnitude)
b)
c)
8. x
a) (signed magnitude)
b)
c)
9. ( ) 6
◆ a)
b)
10.
(Zebronians) 40 (
(zebra) 40 )
40
BCZ(Binary-Coded
Zebronian) BCD
11.
◆ a) 1100
� 101
b) 10101
� 111
c) 11010
� 1100
12.
a) 1011
� 101
b) 10011
� 1011
c) 11010
� 1011
13.
◆ a) 101101 ÷ 101
b) 10000001 ÷ 101
c) 1001010010 ÷ 1011
14.
a) 11111101 ÷ 1011
b) 110010101 ÷ 1001
c) 1001111100 ÷ 1100
◆15. double-dabble 102123 (
)
16.
+ 0 + (�0) =
(�0) + 0 =
0 + 0 =
(�0) + (�0) =
◆17. 4
j
0 → j // Store 0 in j.-3 → k // Store -3 in k.while k ≠ 0j = j + 1k = k - 1
end while
18. 1 3 4
a)
(
0 1 )
b)
◆19.
20. (14 5
16 8 1 )
0 1 0 1 1 0 0 10 1 1 1 1 0 0 0
a) 100.0 0.25
b) 2
c) b
21. (underflow)
22.
23. a 1.0 29 b 1.0 29 c 1.0 21 (14
5 16 8 1
)
b + (a + c) =
(b + a) + c =
24. a) A ASCII 1000001 J ASCII
b) A EBCDIC 1100 0001 J EBCDIC
◆25. 24 24 295
◆ a) 295
◆ b) 8 ASCII 295
◆ c) BCD +295
26. 7 ASCII ASCII
1001010 1001111 1001000 1001110 0100000 1000100 1001111 1000101
◆27.
28. 4 7 ASCII
a) Non-return-to-zero( )
b) Non-return-to-zero-invert( )
c) Mechester
d) Frequency modulation( )
e)
f)
( 1 high 0 low )
29. NRZ
30. 3 1 (
)
31.
◆32.
0011010010111100
0000011110001111
0010010110101101
0001011010011110
33.
0000000101111111
0000001010111111
0000010011011111
0000100011101111
0001000011110111
0010000011111011
0100000011111101
1000000011111110
34. 10
a)
b)
10 1001100110
◆35. 7
4 11
1 0 1 0 1 0 1 1 1 1 0
36.
1 0 1 0 1 0 1 1 1 1 0
37. Reed-Soloman
38. CRC CRC
◆39. 2(modulo 2)
◆ a) 10101112 ÷ 11012
◆ b) 10111112 ÷ 111012
◆ c) 10110011012 ÷ 101012
◆ d) 1110101112 ÷ 101112
40. 2(modulo 2)
a) 11110102 ÷ 10112
b) 10101012 ÷ 11002
c) 11011010112 ÷ 101012
d) 11111010112 ÷ 1011012
◆41. CRC 1011 1011001 CRC
42. CRC 1101 01001101 CRC
◆43. ( 80486 Pentium Pentium IV SPARC Alpha
MIPS)