CS149D Elements of Computer Science

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CS149D Elements of Computer Science
Ayman Abdel-Hamid Department of Computer
Science Old Dominion University Lecture 3
9/3/2002
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Outline

• Fractions in Binary
• Storing Integers
• Sign and magnitude (covered last lecture)
• Twos complement (covered last lecture)
• Excess Notation
• Storing Fractions
• Representing Text
• Representing Images
• Data Compression
• By now, we should have covered sections 1.4-1.8
  from Brookshear Text

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Fractions In Binary1/3

• Use radix point just like decimal
• To the right of radix point positions are
  numbered 1, -2 , .

210 -1-2-3 i 101.101 d
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Fractions In Binary2/3
Express the following in binary notation

• Convert the integer part
• Convert the fraction part. Try to map the
  fraction as a sum of power of 2 fractions using
  the given denominator as a guide
• Put a radix point in between

310 is 112
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Fractions In Binary3/3

• Addition of binary numbers with radix points
• Align radix point
• Apply binary addition process

10.011 100.11
10.011 100.11 __________ 111.001
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Storing Integers1/3

• Binary number system not used as is to actually
  store an integer number
• Use a numeric storage technique
• Sign and Magnitude
• Problems Two zeros (0, and 0), adding 2 and
  2 does not give zero. Straightforward binary
  addition does not apply
• Twos complement
• Advantages deal with subtractions the same as
  additions, simpler hardware to perform integer
  arithmetic operations
• Excess Notation

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Storing Integers2/3
Excess Notation Excess four notation Bit
pattern Value Unsigned Binary 111
3 7 110 2 6 101 1 5 100
0 4 011 -1 3 010 -2 2 001
-3 1 000 -4 0

• All positive numbers begin with 1
• All negative numbers begin with 0
• 0 is represented as 100
• Unsigned binary value exceeds excess notation by
  4, hence the name excess four
• Why 4?
• (2of bits 1) 23-1 4
• Smallest negative number is 000
• Largest positive number is 111

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Storing Integers3/3
What is 101 in excess four notation?
101 if interpreted unsigned is 5 101 in excess
four notation is (5-4) 1
What is 3 represented in excess four notation?
Excess four means we need (4-1) 3 bits for
representation Remember unsigned value exceeds
excess notation by 4 Then get unsigned value
34 7 Represent 7 in 3-bit binary 111 3 in
excess four is 111
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Storing Fractions1/6
Need to represent number and position of radix
point Use floating-point notation (Textbook uses
one-byte as example)
- --- ---- 8 bits 1-byte 1 bit Sign bit (0
positive, 1 negative) 3 bits Exponent (encodes
position of radix point) 4 bits Mantissa
(encodes number)
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Storing Fractions2/6
Bit pattern 01101011 in floating-point notation
is what in decimal?

• First bit is 0, then positive
• Exponent is 110 mantissa is 1011
• Extract mantissa and place a radix point on its
  left side 0.1011
• Extract exponent and interpret as excess four
  notation
• 110 in excess four is 2 (make sure?)
• 2 exponent means move radix point to the right
  by two bits
• (a negative exponent means move radix to left)
• 0.1011 becomes 10.11

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Storing Fractions3/6
Bit pattern 10111100 in floating point is what in
decimal?

• First bit is 1, then negative
• Exponent is 011 mantissa is 1100
• Extract mantissa and place a radix point on its
  left side 0.1100
• Extract exponent and interpret as excess four
  notation
• 011 in excess four is -1 (make sure?)
• -1 exponent means move radix point to the left by
  1 bit
• 0.1100 becomes 0.01100

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Storing Fractions4/6
What is the following number stored in
floating-point?

• Express number in binary to obtain 1.001 (make
  sure?)
• Copy bit pattern into mantissa field from left to
  right starting with the leftmost 1 in binary
  representation
• Mantissa is 1001
• Compute exponent to get 1.001 from .1001 (imagine
  mantissa with radix point at its left)
• need to move radix point to right one bit
• Exponent is 1 expressed in excess four notation
  is 101 (How?)

Sign 0 (positive) Exponent 101 Mantissa 1001
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Storing Fractions5/6
What is the following number stored in
floating-point?

• Express number in binary to obtain 0.01 (make
  sure?)
• Copy bit pattern into mantissa field from left to
  right starting with the leftmost 1 in binary
  representation
• Mantissa is 1000 (you append zeros to fill the
  4-bit mantissa)
• Compute exponent to get 0.01 from 0.1000
• need to move radix point to left one bit
• Exponent is -1 expressed in excess four notation
  is 011 (How?)

Sign 1 (negative) Exponent 011 Mantissa 1000
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Storing Fractions6/6
What is the following number stored in
floating-point?

• Express number in binary to obtain 10.101 (make
  sure?)
• Copy bit pattern into mantissa field from left to
  right starting with the leftmost 1 in binary
  representation
• Mantissa is 1010 (we run out of bits!! Rightmost
  1 is lost (equivalent to 1/8))
• Compute exponent to get 10.101 from 0.1010
• need to move radix point to right two bits
• Exponent is 2 expressed in excess four notation
  is 110

Sign 0 (positive) Exponent 110 Mantissa 1010
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Representing Text1/2

• ASCII (adopted by American National Standards
  Institute ANSI)
• American Standard Code for Information
  Interchange
• 8-bit to represent each symbol
• Upper and lower case letters of English alphabet,
  punctuation symbols, digits 0 to 9, and other
  symbols
• Can represent 256 (28) different symbols
• Unicode
• 16-bit to represent each symbol
• Can represent 65,536 (216) different symbols
• ISO (International Organization for
  Standardization)
• 32-bit to represent each symbol
• Can represent more than 17 million symbols

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Representing Text2/2
ASCII Chart sample

• Upper case A is 6510
• Lower case a is 9710
• Difference between lower case and upper case of a
  letter is always 3210

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Representing Images

• Bitmap techniques
• Image is a collection of pixels (picture element)
• Each pixel can be represented as a number of bits
  (collection of bits is bitmap)
• 1 bit/pixel ? B/W
• 8 bits/pixel ? Gray Scale (different shades of
  gray from black to white)
• 24 bits/pixel ? 1-byte for each of the primary
  colors RGB
• Size? (need for compression)
• Vector techniques
• Image represented as collection of lines and
  curves
• Fonts on printers ? scalable Fonts (True Type)
• CAD (Computer Aided Design)
• Quality problem

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Data Compression1/3

• Data compression techniques
• run-length encoding
• replace repeating sequences with a code
  indicating the value being repeated and the
  number of times it is repeated, e.g., aaaaa is
  replaced by a5
• relative encoding
• record differences between consecutive data
  blocks, e.g., consecutive frames in a motion
  picture
• Frequency-dependent encoding (variable length
  code)
• number of bits to represent an item is inversely
  related to the frequency of the items use, e.g.,
  (e, t, a, and i) in English would be represented
  by short bit patterns, whereas (z, q, and x)
  would be represented by longer patterns
• Adaptive dictionary encoding
• Example is LZ77 (Lempel-Ziv)
• abaabcb (5,4,a) when decompressed is abaabcbaabca
  (see Textbook page 61)

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Data Compression2/3

• Image Compression
• GIF (Graphic Interchange Format)
• 8 bits/pixel (1 byte/pixel) instead of 24
  bits/pixel (3 bytes/pixel)
• Each of 256 potential pixel values are associated
  with a RGB combination by means of table known as
  the palette
• JPEG (Joint Photographic Experts Group)
• Lossless mode
• store difference between consecutive pixels
  rather than pixel intensities themselves
• Differences encoded using variable-length code
• Resulting bit maps not manageable with todays
  technology
• Base standard (lossy)
• Each pixel represented by a brightness component
  and 2 color components

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Data Compression3/3

• Audio and Video compression
• MPEG (Motion Picture Experts Group)
• Start a picture sequence with an image similar to
  JPEG and then represent rest of sequence using
  relative encoding techniques
• MP3 (MPEG-1 Audio Layer-3) audio compression
  ratios of 12 to 1