1. Quiz on Ch.2
Convert to decimal

2. Chapter 03 Data Representation
At beginning of class we have 2 review activities for Ch.2 – see separate text file!

3. Data and Computers
Computers are multimedia devices,dealing with a vast array of information categories
Computers store, present, and help us modify
Numbers
Text
Audio
Images and graphics
Video

4. Data compression
Reduction in the amount of space (memory) needed to store the data
Measured by the Compression ratio = The size of the compressed data divided by the size of the original data
Example: Two files are compressed with the ZIP utility:
One is originally 200 MB, and becomes 150 MB after compression
The other is originally 15 MB, and 11 MB after compression
Which file is better compressed?

5. Quiz
A video file is originally 3.5 GB long. We compress it to 490 MB. What is the compression ratio?

6. Data compression
The Compression ratio is always between 0 and 1 (between 0% and 100%) Compression techniques can be Lossless → the data can be retrieved without any loss of the original information Lossy → some information may be lost in the process (but it doesn’t matter for the purposes of the intended application)

7. Analog vs. Digital
Many quantities of interest in the real-world are infinite and continuous
Zeno’s paradox: “That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ” — Aristotle, Physics VI:9
But computers are finite and discrete!
How do we represent an infinite and continuous quantity in a computer?
Answer: We approximate → represent only enough to satisfy our computational needs and our senses of sight and sound.

8. Information can be represented in one of two ways: analog or digital
Analog data
A continuous representation, similar to the actual information it represents
Digital data
A discrete representation, breaking the information up into separate elements

9. Analog and Digital Information
Computers cannot work well with analog data, so we digitize the data
Digitizing = Breaking data into pieces and representing those pieces separately, by using a finite number of binary digits
Fine distinction: there are two operations performed:
one in time (a.k.a. sampling)
the other in amplitude (a.k.a. quantizing)
Explain on thermometer, using a waveform example!

10. Quiz
A digital thermometer has a scale from 50 to 100 degrees (F). The temperature is represented on 7 bits. What is the smallest temperature difference that it can measure?

11. Analog and Digital Information
Why do we use binary to represent digitized data?
Price: transistors are cheap to produce
Remember Babbage!
Reliability: transistors don’t get jammed

12. Electronic Signals Important facts about electronic signals
An analog signal continually fluctuates in voltage up and down
A digital signal has only a high or low state, corresponding to the two binary digits
Figure An analog and a digital signal

13. The difference is that digital signals can be regenerated!
All electronic signals (both analog and digital) degrade due to absorption in transmission lines
The amplitude (voltage) of electronic signals (both analog and digital) fluctuates due to environmental effects, a.k.a. noise
Figure Degradation of analog and digital signals
The difference is that digital signals can be regenerated!

14. Binary Representations
One bit can be either 0 or 1
One bit can represent two things
Two bits can represent four things (Why?)
How many things can three bits represent?
How many things can four bits represent?

15. Why does the number of combinations double with every extra bit?

16. Binary Representations
How many things can n bits represent? What happens every time you increase the number of bits by one?
EoL

17. Binary Representations
How many things can n bits represent?
Reversing the problem: How many bits are needed to represent N things?
All desktops in this lab

18. QUIZ
How many bits are needed to represent all the courses you’re planning to take in college?

19. 3.2 Representing Numeric Data
Negative integers Signed-magnitude representation The sign represents the ordering, and the digits represent the magnitude of the number

20. Negative Integers
There is a problem with the sign-magnitude representation: plus zero and minus zero.
More complex hardware is required!
Solution: Let’s not represent the sign explicitly!
“Complement” representation

21. Ten’s complement Using two decimal digits, represent 100 numbers
If unsigned, the range would be 0…?
let 1 through 49 represent 1 … 49
let 50 through 99 represent -50 … -1

22. Top: representations (the “label on the jar)
Ten’s complement
Top: representations (the “label on the jar)
Bottom: the actual numbers that are being represented (the “content of the jar)

23. QUIZ
Given the following representations, find in each case what actual number is being represented: 51 52 96 47

24. EXTRA-CREDIT QUIZ
If the representation is 76, what actual number is being represented?

25. Why the “complement” in ten’s complement?
100 – 50 = 50
100 – 49 = 51
……………………..
100 – 1 = 99
In general:
100 – I = representation of – I

26. QUIZ What is the representation for each of these actual numbers? -48
-40
-30 -5

27. Let’s use ten’s complement!
To perform addition, add the numbers and discard any carry
Now you try it
48 (signed-magnitude) 1 47
How does it work in
the new scheme?

28. Adding two negative numbers
Try these:

29. Solve in notebook for next time:
End of chapter 27 through 31
All examples and exercises presented in the slides so far (text up to p.61, incl.) – Make sure to ask next time if you have questions!

30. Two’s Complement What do you notice about the left-most bit (MSB)?
Generalization: what is 2’comp range on n bits?

31. Two’s complement on 4 bits (k = 4)

32. Two’s complement Addition and subtraction are the same as in unsigned:
Ignore any Carry out of the MSB:

33. Warning: The example on p.62 has a typo:
Two’s complement
Formula to compute the negative of a number on k digits:
for ten’s comp: Negative(I) → 10k - I
for two’s comp: Negative(I) → 2k - I
Practice: find the 8-bit two’s comp. representations of: -2
-51
+ 128 (trick question!)
- 129 (trick question!)
Warning: The example on p.62 has a typo:
28 is 256, not 128.

34. QUIZ
What is the 8-bit two’s complement representation of these numbers?
-13 40

35. “Fast” two’s complement
Easier way to change the sign of a number: Flip all bits, then add 1 Try it out! Find the negatives of the following two’s complement numbers:
This is how subtraction is implemented in computer hardware!
A – B = A + (-B)

36. QUIZ
Perform the following operation in 8-bit two’s complement:
40 – 13

37. If k = 8 bits, adding 127 to 3 overflows
What happens if the computed value won't fit in the given number of bits k?
Overflow
If k = 8 bits, adding 127 to 3 overflows
… but adding -1 to 3 doesn’t!
Conclusion: overflow is specific to the representation (unsigned, sign-mag., two’s comp., floating point etc.)

38. Overflow is something we should expect when mapping an infinite world onto a finite machine!

39. Trick QUIZ
What decimal number does this binary number represent?

40. Solve in notebook for next time:
End of chapter 33, 40

41. Representing Real Numbers
A number with a whole part and a fractional part
104.32, , 357.0, and
Positions to the right of the decimal point are the fractional part (tenths, hundredths, thousandths etc.):
10-1, 10-2 , 10-3 …

42. Binary fractions
Same rules apply in binary as in decimal Decimal point is actually the radix point Positions to the right of the radix point in binary are 2-1 (one half) = (one quarter) = (one eighth) = …

43. Converting fractions from binary to decimal
Easy! Just multiply with the powers of 2, as we did for unsigned binary. Only difference is that now the powers are negative. Example: = 0. 10

44. QUIZ
Convert: = 0. 10

45. Converting fractions from decimal to binary
Remember the repeated division algorithm? We apply it for the integer part of the number. To covert the fractional part, we use the repeated multiplication algorithm! Example: = 0. 2

46. QUIZ
Convert:
0.310 =

47. QUIZ Finite decimal fractions may have infinite binary representation!
0.310 =
Stop after 8 bits!

48. Representing Real Numbers
In general, we use “sign-mantissa-exponent”
R =  mantissa * 10exp
R =  mantissa * 2exp
Depending on the form of the mantissa, we have:
Floating-point notation
Scientific notation

49. In the mantissa, the radix point is always at the extreme right!
Floating point
In the mantissa, the radix point is always at the extreme right!
This representation is called floating point because, although
the number of digits is fixed (5 in the example above),
the radix point floats (according to the exponent)

50. Floating point in binary?
Not in text!
Floating point in binary?
It does exist - remember floating point numbers in Python!
It can be implemented either through software, or directly in
the hardware of the computer.
Until 1989, all desktops had an optional chip, separate from
the main CPU, called FPU (a.k.a. math coprocessor).
The Intel 80486, introduced that year, had the first integrated
CPU + FPU.
80486 was also the first chip with over 1 million transistors!

51. Floating point in binary?
Not in text!
Floating point in binary?
Motorola launched their first integrated CPU+FPU, the
68040 the next year, in 1990.
The detailed binary implementation of floating point is
beyond the scope of our class.
If you want to learn more:
Search for “IEEE 754” on the web
Take CS 343 or CS 344

52. Scientific notation
Particular form of floating point, in which the decimal point is kept to the right of the leftmost digit.
is represented E+4

53. QUIZ
Convert to floating point and to scientific notation: = = 0.0 =

54. For next time: Read the entire section 3
For next time: Read the entire section 3.2 Solve in the notebook problems 33, 34, 40

55. 3.3 Representing Text Basic idea:
There are finite number of characters to represent, so list them all and assign each a (binary) number, a.k.a. code.
Character set
A list of characters and the codes used to represent each one
Computer manufacturers (eventually) agreed to standardize
Read “Character Set Maze” on p.67

56. The ASCII Character Set
ASCII = American Standard Code for Information Interchange
ASCII originally used seven bits to represent each character, allowing for 128 unique characters
Later extended ASCII evolved so that all eight bits were used
How many characters can be represented?

57. 7-bit ASCII Character Set

58. 8-bit ASCII Character Set
Extended ASCII is a superset of 7-bit ASCII: The first 128 characters correspond exactly to 7-bit ASCII

59. QUIZ
Encode “Hello, world!” in ASCII Decode from ASCII

60. Python to the rescue!
Encode “Hello, world!” in ASCII

61. The ASCII Character Set
The first 32 characters in the ASCII character chart do not have a simple character representation to print to the screen.
They are called control characters

62. ASCII Control Characters

63. The Unicode Character Set
Extended ASCII is not enough for international use Unicode uses 16 bits per character How many characters can UNICODE represent? Unicode is a superset of ASCII: The first 256 characters correspond exactly to the extended ASCII character set

64. Unicode examples
Figure 3.6 A few characters in the Unicode character set

65. Romanian Characters in Unicode
Part of the Latin-Extended-B character sub-set

66. Miscellaneous Characters in Unicode
See more online at the official Unicode site

67. Text Compression
Sometimes, assigning 8 or 16 bits to each character in a document uses too much memory
We need ways to store and transmit text efficiently
Text compression techniques:
keyword encoding
run-length encoding
Huffman encoding

68. Keyword Encoding
Replace frequently used words with a single character, for example here’s a substitution chart:

69. Keyword Encoding Given the following paragraph:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

70. Keyword Encoding The encoded paragraph is:
We hold # truths to be self-evident, $ all men are created equal, $ ~y are endowed by ~ir Creator with certain unalienable Rights, $ among # are Life, Liberty + ~ pursuit of Happiness. $ to secure # rights, Governments are instituted among Men, deriving ~ir just powers from ~ consent of ~ governed, $ whenever any Form of Government becomes destructive of # ends, it is ~ Right of ~ People to alter or to abolish it, + to institute new Government, laying its foundation on such principles + organizing its powers in such form, ^ to ~m shall seem most likely to effect ~ir Safety + Happiness.

71. Keyword Encoding
How much did we compress? Original paragraph 656 characters Encoded paragraph 596 characters Characters saved 60 characters Compression ratio 596/656 = Could we use this substitution chart for all text?

72. Keyword Encoding
More advanced idea: Encode parts of words, like common prefixes and suffixes, e.g. “pre”, “ex”, “ing”, “tion”

73. Run-Length Encoding *n8
A single character may be repeated over and over again in a long sequence.
Replace a repeated sequence with
a flag character
repeated character
number of repetitions
*n8
* is the flag character
n is the repeated character
8 is the number of times n is repeated

74. Run-Length Encoding Encoding example: Original text is
bbbbbbbbjjjkllqqqqqq+++++
Encoded text is
*b8jjjkll*q6*+5
(Why isn't l encoded? J?)
The compression ratio is 15/25 = .6
Q: This type of repetition doesn’t occur in English text; can you think of a situation where it is very likely to occur?

75. Run-Length Encoding Decoding example: Encoded text is *x4*p4l*k7
Original text is
xxxxpppplkkkkkkk

76. Run-Length Encoding More advanced idea (not required for exam):
The number of repeated characters cannot be 0, 1, 2 or 3 (Why?)
We can shift the range 0 – 255 to represent run lengths between
Perform the encoding and decoding in the prev. examples with the convention above!

77. Huffman Codes
Letter & Word Frequency distributions:

78. Huffman Codes
Conclusion: each language and each topic have specific frequencies of characters and groups of characters (digraphs, trigraphs etc.) Why should the characters “X" or "z" take up the same number of bits as "e" or "t"? Huffman codes use variable-length bit strings to represent each character. More frequently-used letters have shorter strings to represent them, and vice-versa!

79. Huffman encoding example
“ballboard” would be
compression ratio
28/56
QUIZ:
Encode “roadbed”

80. Huffman decoding
In Huffman encoding no character's bit string is the prefix of any other character's bit string. Codes with this property are called prefix codes.
To decode
look for match left to right, bit by bit
record letter when the first match is found
continue where you left off, going left to right

81. Huffman decoding QUIZ
Decode:

82. For next time: Read the entire section 3
For next time: Read the entire section 3.3 Solve in the notebook problems 49, 53a,b

83. 3.4 Representing Audio Data
We perceive sound when a series of air waves cause
to vibrate a membrane in our ear (eardrum), which
sends signals to our brain.

84. Analog Audio
Record players and stereos send analog signals to speakers to produce sound.
These signals are analog representations of the sound waves.
The voltage in the signal varies in direct proportion to the amplitude of the sound wave.

85. From Analog to Digital Audio
Digitize the signal by sampling and quantizing
periodically measure the voltage
record the numeric value
How often should we sample?
A sampling rate of about 40,000 times per second is enough to create a reasonable sound reproduction
44,000 for audio CD, to be exact

86. Sampling and Quantizing
Some information
is lost, but a
reasonable
sound is
reproduced

87. Digital Audio on a CD “pit” “land”
Figure A CD player reading binary information
“land”

88. Digital Audio on a CD
On the surface of the CD are microscopic pits and lands that represent binary digits A low intensity laser is pointed as the disc. The laser light reflects strongly if the surface is smooth and poorly if the surface is pitted ???

89. Pit height is about ¼ the laser’s wavelength
“destructive interference”

90. Both halves of the laser beam reflect off pit or both halves off land.
The two halves are “in phase”.
Half of the laser beam reflects off pit and half off land.
The 2 halves are “out of phase”.

91. Audio Formats Audio Formats
WAV, AU, AIFF, VQF, and MP3
MP3 (MPEG-2, audio layer 3 file) is dominant
analyzes the frequency spread and discards information that can’t be heard by humans (>16 kHz)
bit stream is compressed using a form of Huffman encoding to achieve additional compression
Is this a lossy or lossless compression?

92. 3.5 Representing Images and Graphics
Color Perception of the frequencies of light that reach the retinas of our eyes Retinas have three types of color photoreceptor cone cells that correspond to the colors of red, green, and blue

93. Color is expressed as an RGB (red-green-blue) value = three numbers that indicate the relative contribution of each of these three primary colors An RGB value of (255, 255, 0) maximizes the contribution of red and green, and minimizes the contribution of blue, which results in a bright yellow.

94. Look at the snow and the barn!
Source: Wikipedia – RGB color model

95. Representing Images and Graphics
Can you understand this HTML code? <font color="#FF0000"> Blah blah … </font>
RGB Color Chart in hex

96. QUIZ
Explain the similarities and differences between 00FF00 and

97. The color cube
Figure Three-dimensional color space

98. Representing Images and Graphics
color depth The amount of data that is used to represent a color HiColor A 16-bit color depth: five bits used for each number in an RGB value with the extra bit sometimes used to represent transparency TrueColor A 24-bit color depth: eight bits used for each number in an RGB value

99. Extra-credit question
TrueColor
A 24-bit color depth: eight bits used for each number in an RGB value
How many different colors can be represented in TrueColor?
Please show your work.

100. QUIZ
Are these HiColor
or TrueColor?

101. Indexed Color
A browser may support only a certain number of specific colors, creating a palette from which to choose
Figure The Netscape color palette
EOL

102. Extra-credit question
How many bits are needed to represent this palette? Please show your work.

103. How to digitize a picture
Sample it → Represent it as a collection of individual dots called pixels
Quantize it → Represent each pixel as one of 224 possible colors (TrueColor)
Resolution = The # of pixels used to represent a picture

104. Digitized Images and Graphics
Whole
picture
Figure A digitized picture composed of many individual pixels

105. Digitized Images and Graphics
Magnified portion
of the picture
See the pixels?
Hands-on: paste the
high-res image from
the previous slide in
Paint, then choose
ZOOM = 800
Figure A digitized picture composed of many individual pixels

106. Raster Graphics = Storage of data on a pixel-by-pixel basis
Bitmap (BMP), GIF, JPEG, and PNG are the most popular raster-graphics formats
GIF format
Each image is made up of only 256 colors (indexed color)
But they can be a different 256 for each image!
Supports animation! Example
Optimal for line art
PNG format (“ping” = Portable Network Graphics)
Like GIF but achieves greater compression with wider range of color depth
No animations

107. Bitmap format
Contains the pixel color values of the image from left to right and from top to bottom
Great candidate for run-length compression!
Losless
JPEG format
Averages color hues over short distances
Lossy compression
Optimal for color photographs

108. Vector Graphics
A format that describes an image in terms of lines and geometric shapes A vector graphic is a series of commands that describe a line’s direction, thickness, and color The file sizes tend to be smaller because not every pixel is described Example: Flash

109. Vector Graphics
The good side: Vector graphics can be resized mathematically and changes can be calculated dynamically as needed The bad side: Vector graphics are not good for representing real-world images

110. 3.6 Representing Video The problem: huge amount of data!
Example: In HDTV, the Frame size is defined as the number of horizontal pixels × number of vertical pixels:
1280 × 720
1920 × 1080
Calculate: 1] Data rate (bits per second) for 25 fps
2] Size (bytes) of 2-hour movie

111. Video codec = COmpressor/DECompressor Methods used to shrink the size of a movie to allow it to be played on a computer or over a network Almost all video codecs use lossy compression to minimize the huge amounts of data associated with video

112. Compressing Video
Temporal compression A technique based on differences between consecutive frames: If most of an image in two frames hasn’t changed, why should we waste space to duplicate all of the similar information? Spatial compression A technique based on removing redundant information within a frame: This problem is essentially the same as that faced when compressing still images.

113. To read:

114. Ethical Issues
MGM Studios, Inc. v. Grokster, Ltd. Describe the background for this lawsuit What role did Napster play in this case? What was the decision in this case? Has this case affected you personally?

115. Who is Bob Bemer?
I love ASCII !

116. Do you know? What is JamBayes?
How many computer character sets existed
in 1960?
Who described the telegraph as a kind of
very long cat?

117. Chapter Review Questions
Distinguish between analog and digital information
Explain data compression and calculate compression ratios
Explain the binary formats for negative and floating-point values
Describe the characteristics of the ASCII and Unicode character sets
Perform various types of text compression

118. Chapter Review Questions
Explain the nature of sound and its representation
Explain how RGB values define a color
Distinguish between raster and vector graphics
Explain temporal and spatial video compression

119. Homework for Ch.3 Due next Wed, Feb 15
End-of-chapter 3:
Exercises 47, 48, 51, 52, 53, 62, 65, 66,
68 through 72, 74, 76 through 79