Representation

1
Representation

• There are 10 types of people in the worldthose
  who understand binary and those who dont

2
Learning Objectives

• Understand and explain how any information can be
  represented as bits
• Learn more general concept of representation
• Representation and interpretation are inverses
• Alternative representation schemes make some
  transformations easier or harder
• Can have multiple interpretations of same data

3
Key concept Representation

• What does 00100100 represent?

4
Terminology

• Information something that can change a persons
  mind
• Knowledge communicated concerning a fact,
  subject, or event often contrasted to data
• A pattern that conditions interaction
• Degree of choice exercised in selection
• Representation as thing (noun)
• the representation of that is 0001
• Representation as process (noun)
• representation and interpretation must be done
  quickly in order to
• Represent and interpret as verbs
• Represent a complicated name by its acronym
• Representation scheme
• Rules for representing (and interpreting)

5
Representation

• Takes the place of the original
• Equivalent to the original, in the sense that the
  original can be reconstructed from its
  representation
• Often the original can only be approximately
  reconstructed, although it may be
  indistinguishable to the user

Representation
6
Representation in LMC

• Building block 3 digit numbers
• Representing numbers its easy!
• Representing instructions
• each gets a 3-digit code
• What does 901 represent in LMC?

7
Representations in Digital Computers

• Computers represent all data using variable
  voltage electrical signals.
• We can think of them as representing this data
  with on-off switches.
• Bit A binary digit - it has two possible states
• On or off, true or false, 1 or 0
• Bit-string A consecutive string of bits
• Byte An 8-bit bit-string
• Word A bit-string that can be transferred and
  stored as a unit.

8
Decimal notation

• Decimal
• 10 possible values in each place (0-9)
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ???
• What do we do next?
• Right-most place represents how many 1s (up to
  9), next place to the left for 10s (up to 9 tens,
  or 90), next place for 100s (up to 900), etc.
• 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, , 98, 99, 100,
  etc.
• For example, 506
• 6 ones, no tens, 5 hundreds
• 6 x 1 0 x 10 5 x 100
• 6 x 100 0 x 101 5 x 102

9
Binary notation

• Binary
• Only 2 possible values in each place (0 or 1)
• 0, 1, ???
• What do we do next?
• Right-most place represents how many 1s (up to
  1), next place to the left for 2s (up to 1 two,
  or 2), next place for 4s (up to 1 four, or 4),
  etc.
• 0, 1, 10, 11, 100, 101, , 1101, 1110, 1111, etc.
• For example, 110
• 0 ones, 1 two, 1 four (so 110 binary 6 decimal)
• 0 x 1 1 x 2 1 x 4
• 0 x 20 1 x 21 1 x 22

10
Representing Numbers

• Converting between decimal and binary place value
  system
• Padding is a convention
• Why do we pad on the left?

11
Tom Lehrer New Math

• 342
• -173_____
• 169

• 342
• -173_____
• 147

12
ASCII what is it?
Note that this is not the only way we could
represent these symbols .this one happens to
be a standard (ANSI X3.110-1983)

• Symbol Binary
• Decimal
• 7 55 00110111
• 8 56 00111000
• 9 57 00111001
• 58 00111010
• 59 00111011
• lt 60 00111100
• 61 00111101
• gt 62 00111110
• ? 63 00111111
• _at_ 64 01000000
• A 65 01000001
• B 66 01000010
• C 67 01000011

13
International Characters

• Unicode (3.0)
• 49,194 characters
• Mapped to unique codes (numbers)
• UTF-8 encoding
• Codes 0 to 127 (right-most 7 bits) map to one
  byte (8 bits) each
• Same encoding as ASCII
• First bit is always what?
• All other codes map to multi-byte sequences
• First bit of the leading byte in the sequence is
  always what?

14
Character strings

• How to represent character strings?
• A collection of adjacent words (bit-string
  units) can store a sequence of letters
• Notation enclose strings in double quotes
• "Hello world"
• Representation convention null character defines
  end of string
• Null is sometimes written as '\0'
• Its binary representation is the number 0

'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0'
15
Layered View of Representation
Textstring
Sequence of characters
Character
Bit string
16
Working With A Layered View of Representation

• Represent SI540! at the two layers shown on the
  previous slide.
• Representation schemes
• Top layer - Character string to character
  sequence Write each letter separately, enclosed
  in quotes. End string with \0.
• Bottom layer - Character to bit-stringRepresent
  a character using the binary equivalent according
  to the ASCII table provided.
• Work on this individually for 3 minutes.
• Break into groups to discuss.
• Compare bit-string representations
• If you have different representations, try
  interpreting a bit-string thats different from
  the one you wrote.

17
Solution

• SI540!
• S I 5 4 0 ! \0
• 01010011010010010011010100110100001100000010000100
  000000
• The colors are intended to help you read it
  computers dont care that all the bits run
  together.

18
Operations on ASCII strings

• Which operations are easy to implement? Why?
• Which operations are hard with this
  representation?
• What representation would make this easier?

19
Multiple Interpretations eBays Currency
Conversion
20
Multiple Interpretations eBays Currency
Conversion
GBP 1.80
2.82
1.80, GBP
1.80, GBP
21
A Picture Is Worth a 1000 Words?
This information is represented in this computer,
but how?
22
Representing a Picture

• Draw a grid of rectangles over the image.
• Represent each cell (pixel) with a bit-string
  that represents the cells color.

23
Example a smaller image
L-R (6x5)
010011111101001010010111100000
010011111101001010010111100000
T-D (5x6)
010000111110010010010010111110
010000111110010010010010111110
24
Standardization
If the representation is not standardized,
the information is garbled!
Represent as 6x5
Interpretas 3x10
01001111101001010010111100000
01001111101001010010111100000
25
Integrity
If the integrity of the data is not maintained,
the information is also garbled!
01001111101001010010111100000
10111111101001010010000000000
26

• How to represent colors?
• 2-bits per pixel
• 422 choices
• 00 (off, off)white
• 01 (off, on)lite grey
• 10 (on, off)dark grey
• 11 (on, on)black

(This slide courtesy of Karen Markey)
27
What Do You Think?

• 256 colors how many bits?
• Hint for calculating
• To figure out how many bits are needed to
  represent a range of values, figure out the
  smallest power of 2 that is equal to or bigger
  than the size of the range.
• That is, find x for 2 x gt 256
• 24-bit color how many colors?
• Hints for calculating
• 210 (10 bits) is about 1000
• 220 is 210 x 210
• 224 24 x 220
• Whats the right number of bits to use?

28
Representation of picture image
Expanding a small portion of the picture, we see
that it is represented by square pixels. .300
tall by 200 wide.. .with a range of 256
intensities per pixel (8 bits 28 256)
Whats the right number of pixels per inch?
300 200 8 bits 480,000 bits (but it can be
compressed) ideas for compression?
29
Vector Representation of Images

• Dont specify each pixel value
• Specify instructions for determining pixel
  values, such as
• Line segment from pixel (3, 7) to (105, 60)
• Triangle with vertices (3,7), (105,60), (200,10)
• Homework exercise
• Think through advantages and disadvantages of
  bitmap vs. vector graphics representation

30
Motion picture
A motion picture can be represented by video a
sequence of images (every tenth shown) Ideas for
compression?
31
Compression

• 3 ideas
• Encode frequent symbols with fewer bits
• Encode frequent symbol sequences
• Summarize sequences inexactly
• Exercise
• How would these work for images/video?
• How would they work for text?

32
Sound

• What is sound?
• What we perceive of as sound is actually our
  sensory systems interpretation of very rapid
  vibration of the ear drum
• Sounds are transmitted as waves, a mechanical
  disturbance propagating through an elastic medium
• For example, sound can reach our ear drum by
  traveling through air or water

See also http//entertainment.howstuffworks.com/he
aring.htm/printable and the Encyclopedia
Britannica Onlines entry for sound
33
Representing Sound Graphically

• X axis time
• Y axis pressure
• A amplitude (volume)
• ? wavelength (inverse of frequency, which
  determines pitch)

34
Representing Sound Graphically
Low Frequency (played on a recorder)
One cycle of dominant frequency
High Frequency (played on a recorder)
One cycle of dominant frequency
35
Digitizing Sound
Capture amplitude at these points
Lose all variation between data points
See Encyclopedia Brittanica article on
Analog-Digital Conversion
Zoomed Low Frequency Signal
36
Reproducing Sound
Generate a wave that includes these points
37
Digitizing Sound

• Exercise
• How many bits would it take to encode a Beethoven
  symphony?
• More or less for a newscast of same length?
• Birds chirping?
• Sample at some frequency (x times per second)
• How frequently?
• For each sample, n-bits encode amplitude
• How many bits per sample?

38
Digitizing Sound

• Sample periodically
• Number of samples per second dictates range of
  pitches that can be recorded
• Nyquist sampling theorem says that sampling rate
  should be twice highest sound frequency
• Humans detect 20 20,000 Hz (or 20KHz)
• Audio CDs sampling rate is 44Kh
• For each sample, n-bits encode amplitude
• Amplitude dictates signal level
• Audio CDs bit rate is 16-bit

39
Some File Formats

• Many representation schemes are associated with a
  file format that we regularly use
• GIF
• JPEG
• MPEG
• BMP
• PostScript
• PDF