Information Representation

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Information Representation
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What is the Digital Domain?

• computers process discrete or digital data
• data is information represented by a digital
  symbol system
• all forms of information must be converted to a
  digital form for processing

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Analog vs. Digital Information

• natural forms of information are analog
• analog information is continuous, e.g., wave
• waveforms are measured by amplitude and frequency

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Analog vs. Digital Information

• digital information is discrete
• analog information may be digitized, i.e,
  converted to a digital representation
• digitized data may be stored and processed by
  computers

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Advantages of the Digital Domain

• greater precision
• ordinality (built-in ordering scale)
• efficient storage
• fast transfer
• scalability (resolution independence)
• unlimited absolute replication
• selective and random access
• compression
• content analysis and synthesis

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Five Types of Information

• Numbers
• Symbols - Alphabetic letters, punctuation,
  numerals
• Visual - pictures
• Audio - voices, music
• Instructions - program commands

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Binary Numbers

• Binary numbers may be used to store information,
  in the computer memory or in storage devices.
• Binary Numbers
• Are used to encode information in computer
  readable form.
• Are a representation of the binary form which is
  actually electronic states in the computer and
  magnetic states in auxiliary storage devices.

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Binary Bits and Bytes

• binary numbering system is a base-2 positional
  numbering system
• in binary, each digit is either 0 or 1 and is a
  product of powers of two
• binary digits are called bits
• bits are organized into groups, e.g., 8 byte
• digital data have finite precision (fixed number
  of bits) because of the finite size of memory in
  the computer.

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Number Bases

• Decimal (Base ten)
• Symbols used 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Example 27110
• Binary (Base two)
• Symbols used 0, 1 (Binary digit Bit)
• Example 1001012
• Hexadecimal (Base sixteen)
• Symbols used 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,
  B, C, D, E, F
• Example 15B316

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Convert Base ten to Base two

• Need to find what digits (0 or 1) to locate in
  which places (with which placeholder)
• I.e. convert 3910 to ?2
• We know the placeholders are located by powers of
  two so we extract powers of two from the base 10
  number to get the placeholder digits. We must
  start with the largest power of two since
  extracting the smallest (1) would be useless.
• From 39 we extract 32 (25), leaving 7, which has
  0 - 16s and 0 - 8s, but 1 - 4 leaving us with 3
  which has 1 - 2 and 1 1
• The base 2 equivalent of 3910 may be written as
• 25 24 23 22 21 20
• 1 0 0 1 1 1 2

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Digitizing Data

• the process of converting information to a binary
  form is called digitization
• both discrete and analog forms of information may
  be digitized

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Digitizing Data

• Discrete forms of information (e.g., numbers,
  text) are encoded by numbering schemes, e.g.,
  binary coded decimal numbers

• Analog forms of information are digitized in two
  steps
• sampling (discrete samples representing the
  continuous data)
• quantizing (samples are converted to numeric form)

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Digital Number Representations

• Integers
• infinite discrete subset of the number line
• are represented with a limited range
• Decimal numbers (real numbers)
• infinite and continuous
• are represented with limited range and limited
  precision

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Integer Storage

• All integers between two values (one negative and
  one positive) are stored with exact precision
• The specific values marking the range limits
  depend on the particular computer system being
  used
• If calculations with integers give rise to
  numbers outside the allowable range, we say that
  an integer overflow error has occurred

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Real (Decimal) Number Storage

• Real numbers are stored in floating point
  representation
• a sign
• an exponent
• a mantissa (normalized decimal fraction)
• no digits to the left of the decimal
• first digit to the right of the decimal is
  nonzero
• Limited precision because most real numbers have
  an infinite decimal expansion (this holds no
  matter what number base is used in the
  representation)

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Real Number StorageLimited Range and Precision

• There are three categories of numbers left out
  when floating point representation is used
• numbers out of range because their absolute value
  is too large (similar to integer overflow)
• numbers out of range because their absolute value
  is too small (numbers too near zero to be stored
  given the precision available
• numbers whose binary representations require
  either an infinite number of binary digits or
  more binary digits than the bits available

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Limited Range and Precision Some Consequences

• Limited range will invalidate certain
  calculations
• If integers are involved, this can often be
  avoided by switching to real numbers
• For real number calculations, this problem arises
  infrequently and in those cases can sometimes be
  handled by special methods. It is not a common
  occurrence in non-scientific work.
• Limited precision for real numbers is very
  pervasive
• Assume that most decimal calculations will, in
  fact, be in error!
• Evaluate and use computer calculations with this
  in mind

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Digital Representation of Text

• Languages with relatively few distinct characters
  are the best candidates for text processing
• Such small character sets are represented by a
  one-to-one assignment to binary codes
• Using an eight bit code (one byte) to represent a
  single character allows the representation of 256
  distinct characters -- sufficient for English and
  many other languages
• ASCII (American Standard Code for Information
  Exchange) is the most commonly used representation

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English Character Set

• All uppercase and lowercase letters
• Punctuation symbols like ! . , ? etc.
• Digits 0, , 9
• Arithmetic symbols - / lt gt
• Assorted special symbols like _at_
  ( ) etc.
• Invisible formatting characters

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Representing Pictures

• Raster Graphics
• Bitmap graphics represent each pixel with a
  numeric value, the value being dependant on the
  bit depth or colour depth.
• Monochrome has bit depth of one (1).
• Grayscale graphics has a bit depth of 4 to 8
• Colour graphics has a bit depth of 4 to 24
• Vector Graphics
• The representation and storage of images by
  mathematical equations or functions.

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What is a Pixel?

• Pixel is short for picture element
• It is the smallest possible unit of a digital
  image.
• It is the atom of digital images

• Its two dimensional
• Its square
• Its all one colour
• It can be any colour
• It has no inherent height or width. It can be any
  height or width.

111111110100000001000000
FF8080
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Example Digitizing Images

• images are digitized using a two step process
• sampling the continuous tone image for pixels
• quantizing pixels

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Representing Pictures

• The amount of memory required to store a picture
  is a function of
• The resolution of the image
• The bit depth - the amount of colour we wish to
  reproduce
• For example with a 1024 X 768 pixel image with 32
  bit true colour (bit depth 32) we would
  require
• (1024 X 768) pixels 32 bits/pixel 25,165,824
  bits
• and since there are 8 bits / byte 3,145,728
  bytes
• or 3.1 MB

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Digitizing Data

• the sampling rate affects the fidelity of
  digitization
• the quantizing scale (dynamic range) affects the
  sensitivity of digitization

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Representing Sound

• A standard music CD is sampled at a rate of 44.1
  Khz / channel and each sample is stored in either
  one or two bytes.
• One hour of CD quality stereo with each sample
  stored in 2 bytes will require
• 44100 sample/sec 3600 sec/hr 2 channels 2
  bytes/sample 635 MB

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Digital Camera Needs

• A 2 mega pixel digital camera has a resolution of
  1792 X 1200 pixels. If it outputs true (24 bit)
  colour how much storage is required for each
  picture?
• The camera must store 1792 X 1200 pixels
  2150400 pixels
• Each pixel will use 24 bits (3 bytes) of storage,
  using a total of
• 2150400 pixels X 3 bytes/pixel 6,451,200 bytes
  6.45 MB
• How many pictures can be stored on a 20 MB
  storage device?
• 20 MB / 6.45 MB 3.1 or 3 pictures

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Digital Video Needs

• Newer digital video cameras have a frame
  resolution of 680,000 pixels, with 24 bit colour,
  running at 30 frames / sec.
• They use CD quality sound sampled at 44.1 KHz
  with 16 bit resolution.
• Video requirements are
• 680,000 pixels x 3 bytes x 30 frames
  61,200,000 Bps
• frame pixel second
• Audio requirements are
• 2 bytes x 44,100 samples x 2 channels 176,000
  Bps
• sample sec
  
• Total for system 61.2 MBps 0.2 MBps 61.4
  MBps