Real Time Systems

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Chapitre 2 Représentations Internes des données
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Binary Numbers
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary 0000 0001 0010 0011 0100 0101 0110 0111 10
00 1001 1010 1011 1100 1101 1110 1111
Octal 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17
Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
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Binary NumbersSome useful values
128 256 1 024 32 768 65 536 2 147 483 148 4 294
967 296
27 28 210 215 216 231 232
? 103
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Data Representations

• Text
• Baudot - ASCII - UNICODE
• Numbers
• Decimal - Binary / Integer - Floating point
• Images
• Bitmaps - Structured

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Text Encoding

• Telex public network
• 5 bit/char BAUDOT code (Alph. Num. Mode)
• Older computers (fifties and sixties)
• 6 bit proprietary codes (no lower case)
• IBM mainframes and compatibles
• 8 bit EBCDIC code
• Vast majority of modern computers
• 7 bit ASCII (no national characters)
• 8 bit ASCII with proprietary extensions
• Microsoft NT systems
• 16 bit Unicode for all known character-sets
• ASCII is a subset of Unicode

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Baudot Character Set (CCITT Nr 2)Alphabetical
State
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Baudot Character Set (CCITT Nr 2)Numerical State
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Printable ASCII Character Set
047
/
063
?
079
O
095
_
111
o
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ASCII special characters
128
DEL
Delete
10
ASCII extensions for pc

_

207
223
239
255
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Unicode
FFFF
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Decimal Numbers
ASCII Characters 8 bit / digit. BCD Characters
4 bit / digit.
Example In a 32 bit word
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Absolute Value Sign

• Conventions
• n number of bits in representation
• M value to be represented
• M' value of representation
• Absolute Value Sign definition
• M gt 0 M' M
• M lt 0 M 2n-1 M
• Range
• - 2 n-1 lt M lt 2 n-1

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Absolute Value Sign
0
1
-7
0000
0001
1111
2
-6
0010
1110
3
-5
0011
1101
1100
0100
4
-4
n 4
0101
1011
5
-3
0110
1010
6
-2
1001
0111
1000
-1
7
-0
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Binary Offset

• Conventions
• n number of bits in representation
• M value to be represented
• M' value of representation
• Binary offset definition
• M 2n-1 M
• Range
• - 2 n-1 lt M lt 2 n-1

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Binary Offset
-8
-7
7
0000
0001
1111
-6
6
0010
1110
-5
5
0011
1101
1100
0100
-4
4
n 4
0101
1011
-3
3
0110
1010
-2
2
1001
0111
1000
1
-1
0
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Twos complement

• Conventions
• n number of bits in representation
• M value to be represented
• M' value of representation
• Two's complement definition
• M' (M 2n) MOD 2n
• Range
• - 2 n-1 lt M lt 2 n-1

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Twos Complement
0
1
-1
0000
0001
1111
2
-2
0010
1110
3
-3
0011
1101
1100
0100
4
-4
0101
1011
5
-5
0110
1010
6
-6
1001
0111
1000
-7
7
-8
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Some Twos Complement Numbers
4 bit
value
8 bit
16 bit
-32768
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Twos complement Arithmetic
All arithmetic operations can be performed on
the twos complement representations, using
positive, modulo 2n arithmetic.
A 3 B -5
A 3 B 27
0 0 0 1 1 1 1 0 1 1
0 0 0 1 1
0 0 0 1 1
n 5
0 0 0 0 0
0 0 0 1 1
0 0 0 1 1
0 0 1 0 1 0 0 0 1
A B - 15
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Twos complement Arithmetic
All arithmetic operations can be performed on
the twos complement representations, using
positive, modulo 2n arithmetic.
S A B with A gt 0 and B gt 0
A A B B
S (A B) MOD 2n A B S
Provided, of course, that S lt 2 n-1
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Twos complement Arithmetic
All arithmetic operations can be performed on
the twos complement representations, using
positive, modulo 2n arithmetic.
S A B with A gt 0 and B lt 0
A A B B 2 n 2 n - B
S (A B) MOD 2n (A - B 2 n) MOD
2n Suppose A gt B (? S gt 0) then A - B
2 n gt 2n and S A - B S Suppose A lt B
(? S lt 0) then A - B 2 n lt 2n and S A
- B 2n S 2n
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Twos complement Arithmetic
All arithmetic operations can be performed on
the twos complement representations, using
positive, modulo 2n arithmetic.
S A B with A lt 0 and B lt 0
A A 2n 2n - A B B 2n 2n - B
S (A B) MOD 2n (2.2n -(A B)) MOD
2n Where 2 lt A B lt 2n-1 therefore
2n1-2 gt (2.2n -(A B)) gt 3.2n-1 and
(2.2n -(A B)) MOD 2n (2.2n- 2n
-(A B)) or S (2n -(A
B)) 2n S
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Twos complement Arithmetic
All arithmetic operations can be performed on
the twos complement representations, using
positive, modulo 2n arithmetic, provided that the
result is in range.
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Twos complement Overflow

• With a 16 bit representation
• 30 000 10 000 - 7 232 !!!

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With thanks to Ariane for the score
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Twos complement Overflow
No carry from n-2 to n-1, nor from n-1 to n.
Carry from both n-2 to n-1, and n-1 to n.
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Extended Multiplication
In general, two n bit factors give a 2n bit
product ! Simplest algorithm to compute a n bit
product Sign extend both factors to 2 n
bit Perform the multiplication modulo 2 2n
A 3 B -5
A 3 B 27
0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 1
n 4
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 1 1
1 1 1 1 0 0 0 1
A B - 15
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Ones complement

• Conventions
• n number of bits in representation
• M value to be represented
• M' value of representation
• Two's complement definition
• M gt 0 M' M
• M gt 0 M' M 2n - 1
• Range
• - 2 n-1 lt M lt 2 n-1

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Ones Complement
0
1
-0
0000
0001
1111
2
-1
0010
1110
3
-2
0011
1101
1100
0100
4
-3
0101
1011
5
-4
0110
1010
6
-5
1001
0111
1000
-6
7
-7
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REAL TypesApproximate representation for real
numbers

• Possible values and accuracy
• implementation dependant
• majority of computer systems IEEE754.
• Single precision (32 bit) float
• Smallest value 10-38
• Largest value 1038
• Relative error lt 3 10-8
• Double precision (64 bit) double
• Smallest value 10-308
• Largest value 10308
• Relative error lt 10-16

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Floating Point Numbers
base predefined constant (2 or 16)
Normalized Floating Point Number
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Normalizationon a scale...
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Relative erroron Normalized Floating Point
Numbers
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Gradual Underflow
Normalized Underflow
0
1000.2
min1
1001.2
min
min
1000.2
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Floating Point NumbersIEEE 754, Short Real
Sign
Significand
Exponent
Value
(-1)Sign 1.Significand 2 Exponent - 127 ?
0
(-1)Sign 0.Significand 2 - 126 0
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Floating Point NumbersIEEE 754, Short Real
Sign
Significand
Exponent
Smallest 10 - 38 Largest 10 38 Rel.Error
3.0 10 - 8
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Floating Point NumbersIEEE 754, Long Real
Sign
Significand
Exponent
Value
(-1)Sign 1.Significand 2 Exponent - 1023
? 0
(-1)Sign 0.Significand 2 - 1022 0
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Floating Point NumbersIEEE 754, Long Real
Sign
Significand
Exponent
Smallest 10 - 308 Largest 10
308 Rel.Error 1.1 10 - 16
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Floating Point NumbersIEEE 754, Temporary Real
Sign
Significand
Exponent
Value
(-1)Sign Significand 2 Exponent - 16383 ?
0
(-1)Sign Significand 2 - 16382 0
Significand 1.XXXX or 0.XXXX
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Floating Point NumbersIEEE 754, Temporary Real
Sign
Significand
Exponent
Smallest 10 - 4932 Largest 10
4932 Rel.Error 5.4 10 - 20
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Floating Point NumbersIEEE 754, Special values

• Zero
• Exponent all 0s
• Significand all 0s
• Infinity
• Exponent all 1s
• Significand all 0s
• Not a Number
• Exponent all 1s
• Significand any non zero value

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Encoding of Images

• By means of bit maps
• Each pixel is represented by one or several
  bytes.
• By means of image components
• ASCII codes for letters
• Coordinates of end points for vector.
• Coordinates of center radius for circle.

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A bit map
Size 10 MBytes
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An other bit map
Size 10 MBytes
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A geometric construct
Size 13 KBytes
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Alphanumeric Display
Number of character positions 80 24
1920. Number of different characters 256,
encoded by 8 bit. Number of bytes for entire
screen 1920. Number of pixels per character
8 20 160. Number of pixels on screen
1920 160 307 200 38 400 bytes
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Data Compression
Four colours encoding 1000 colours, non-uniform
distribution
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Data Compression
Four colours encoding 1000 colours, uniform
distribution
frequency
code
nbr of bits
code
nbr of bits
red
25
00
500
0
250
blue
25
01
500
10
500
green
25
10
500
110
750
yellow
25
11
500
111
750
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Data CompressionShannons Law
Minimal Number of bits per symbol
nmin - ?ipi log2 pi
In the four colours example
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Data Compression

• Entropy coding
• based on symbol frequency
• No loss of information
• used for data compression and storage
• Examples
• Huffman coding variable length codes
• See colors example
• run length encoding repetition factors
• 123333320000000123333333311111112234
• 12532701283712234
• Example Zip - Unzip

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Data Compression

• Source coding
• based on meaning of data
• acceptable loss of information
• used for data compression and storage
• spectacular compression rates
• Examples
• GSM vocoders only for human voice
• 64 Kb/s gt 13 Kb/s or 6.5 Kb/s
• JPEG still images, compression 120
• MPEG TV images, compression 1100