Quiz on Ch.2

Convert 201023 to decimal

Quiz on Ch.2

Convert the decimal number 51 to binary using repeated division.

Chapter 03

Data Representation

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

• Smell (machine olfaction!)

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

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)

11

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?

12

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

–Remember Babbage!

13

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

14

Figure 3.2

An analog and a digital signal

• 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

15

Figure 3.3

Degradation of analog and digital signals

The difference is that digital signals can be regenerated!

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?

16

17

Why does the number of combinations double with

every extra bit?

Binary Representations

How many things can n bits represent?

What happens every time you increase the number of bits by one?

18 EoL

For next time:

• Read pp.54-57 of text

• Solve end of chapter ex. 1 through 5, 27, 28 in notebook

19

Quiz

A weather log file is originally 4.2 GB long.

We compress it to 150 KB. What is the compression ratio?

20

Quiz

An Analog-to-Digital Converter (ADC) accepts an input voltage between -3 and +12 V, and uses 9 bits to represent it digitally.

What is the precision?

21

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

22

QUIZ

How many bits are needed to represent all the

days you’re planning to be in college?

23

3.2 Representing Numeric Data

Negative integers

Signed-magnitude representation

The sign represents the ordering, and the digits represent the magnitude of the number

24

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

25

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

26

Ten’s complement

27

Top: representations (the “label on the jar”)

Bottom: the actual numbers that are being

represented (the “content of the jar”)

QUIZ

Given the following representations, find in each case what actual number is being represented:

• 51

• 52

• 96

• 47

28

EXTRA-CREDIT QUIZ

If the representation is 76, what actual number

is being represented?

29

Why the “complement” in ten’s complement?

100 – 50 = 50

100 – 49 = 51

……………………..

100 – 1 = 99

In general:

100 – i is representation of – i

30

QUIZ

What is the representation for each of these actual numbers?

• -48

• -40

• -30

• -5

31

Let’s use ten’s complement!

To perform addition, add the numbers and discard any carry

32

Now you try it

48 (signed-magnitude)

- 1

47

How does it work in

the new scheme?

Adding two negative numbers:

33

Try these:

4 - 4 -4

- 3 +3 + -3

Important conclusions

In the complement representation:

• Positive and negative numbers are treated the same! We can add without knowing if they’re positive or negative!

• Subtraction is performed as addition, by changing signs: a – b = a + (-b). This greatly simplifies the hardware!

34

For next time:

• Read pp.58-61 of text

• Go through all examples presented so far in Ch.3 and make sure you understand them. Ask next time if you have questions!

• Solve end-of-chapter ex. 33, 34, 35 in notebook

35

QUIZ

We presented ten’s complement on 2 digits:

Draw the axis for ten’s comp. on only 1 digit!

36

QUIZ

Perform the following operations in 1-digit ten’s comp.:

• 1 + 2

• 3 + (- 1)

• - 1 – 2

37

Two’s complement on 4 bits (k = 4)

38

39

Two’s Complement

What do you notice

about the left-most bit

(MSB)?

Two’s complement Addition and subtraction are the same as in unsigned:

-127 1000 0001

+ 1 0000 0001

-126 1000 0010

Ignore any Carry out of the MSB:

-1 1111 1111

+

-1 1111 1111

-2 1111 1110

40

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!)

0

41

Warning: The example on p.62 has a typo in the previous (4th) edition:

28 is 256, not 128.

QUIZ

What is the 8-bit two’s complement representation of these numbers?

• -13

• 40

42

“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:

0000 0011

1000 0000

1000 0001

1000 0011

1001 0110

1111 1111 43

This is how subtraction is implemented in

computer hardware! A – B = A + (-B)

QUIZ

Perform the following operation in 8-bit two’s complement:

40 – 13

44 EoL

Solve in notebook for next time:

End of chapter 40

45

Trick QUIZ

What decimal number does this binary number represent?

1001 1110

46

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 1111 1111

+ 0000 0011

0 1000 0010

… but adding -1 to 3 doesn’t!

Conclusion: overflow is specific to the representation (unsigned, sign-mag., two’s comp., floating point etc.)

47

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

48

Representing Real Numbers

Real numbers

A number with a whole part and a fractional part

104.32, 0.999999, 357.0, and 3.14159

Positions to the right of the decimal point are the fractional part (tenths, hundredths, thousandths etc.):

10-1, 10-2 , 10-3 …

49

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) = 0.5

2-2 (one quarter) = 0.25

2-3 (one eighth) = 0.125

…

50

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: .10012 = 0. 10

51

QUIZ

Convert:

.10112 = 0. 10

52

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: 0.43510 = 0. 2

53

QUIZ

Convert:

0.310 = 0. 2

54

QUIZ

Finite decimal fractions may have infinite binary representation!

0.310 = 0. 0100110011 2

55

Stop after 8 bits!

Solve in notebook for next time:

End of chapter 7, 40, 44

56

QUIZ

Convert:

0.410 = 0. 2

Use at most 8 bits.

57

Representing Real Numbers

58

In general, we use “sign-mantissa-exponent”

N = mantissa * 10exp

N = mantissa * 2exp

Depending on the form of the mantissa, we have:

• Floating-point notation

• Scientific notation

Floating point

59

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)

The radix point is always at the

extreme right, e.g. 12001.0

Scientific notation

Particular form of floating point, in which the

decimal point is kept to the right of the leftmost

digit.

12001.32708 is represented 1.200132708 E+4

60

Exactly one digit to the left of the radix point

QUIZ

Convert to floating point and to scientific notation:

123.332 =

-0.0034 =

0.0 =

61

Floating point in binary?

62

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.

Not in text!

80486 was also the first chip with over 1 million transistors!

Floating point in binary?

63

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

Not in text!

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 64

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?

65

7-bit ASCII Character Set

66

QUIZ

Encode “Hello, world!” in ASCII

Decode 67 83 32 49 48 50 from ASCII

67

Python to the rescue!

Encode “Hello, world!”

in ASCII

68

The ASCII Character Set

69

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

Control Character: newline

70

Source: http://en.wikipedia.org/wiki/Newline#Unicode

Not in text

8-bit (“extended”) ASCII Character Sets

71

By using 8 bits instead of 7, the number of codes extends from 128 to 256.

Extended ASCII is always a superset of 7-bit ASCII:

• The first 128 characters correspond exactly to 7-bit ASCII

Problem ???

Not in text

Extended ASCII: IBM code page 437

72

http://en.wikipedia.org/wiki/Code_page_437

Not in text

Extended ASCII: Latin-1

73 http://en.wikipedia.org/wiki/ISO/IEC_8859-1

Not in text

Trick QUIZ

What do these bits represent?

1101 1110

74

Trick QUIZ

What do these bits represent?

1101 1110

75

Unsigned integer: …

Signed integer (2’s complement): …

Fraction: …

IBM 437 character: …

Latin-1 character: …

The Unicode Character Set

None of the Extended ASCII character sets were enough for international use (256!)

Unicode uses 16 bits per character

How many characters can UNICODE represent?

Unicode is a superset of Latin-1: The first 256 characters correspond exactly to Latin-1 characters (http://unicode.org/charts/PDF/U0080.pdf )

76

Simplified Chinese has

6500!

Unicode examples

77 Figure 3.6 A few characters in the Unicode character set

Romanian Characters in Unicode

78

Part of the Latin-Extended-B

character sub-set

Miscellaneous Characters in Unicode

79

See more online at the official Unicode site

For next time:

Read pages 62-69 of text

Homework: 8, 9, 29, 30, 31, 41, 44, 48, 49

Due Monday at the beginning of lab, before

the midterm!

80

QUIZ:

Your boss tells you to develop a webpage using the extended ASCII character set. What do you reply?

81

QUIZ Select all that apply:

The Latin-1 character set:

• Is a 5-bit representation

• Is a 7-bit representation

• Is a 16-bit representation

• Is an extension of ASCII

• Is an extension of Unicode

• Contains letters used in European languages

82

QUIZ Select all that apply:

The Unicode representation:

• Uses 16 bits

• Is an extension of ASCII

• Is an extension of Latin-1

• Is an extension of IBM-437

• Contains letters used in all world languages

• Can accommodate over 65,000 characters

• Is used in the majority of web pages today

83

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

84

Keyword Encoding

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

85

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.

86

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.

87

Keyword Encoding

How much did we compress?

Original paragraph

656 characters

Encoded paragraph

596 characters

Characters saved

60 characters

Compression ratio

596/656 = 0.9085

Could we use this substitution chart for all text?

88

Keyword Encoding

More advanced idea:

Encode only some parts of the words, like common prefixes and suffixes, e.g. “pre”, “ex”, “ing”, “tion”

89

Run-Length Encoding

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 90

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?

91

Run-Length Encoding

Decoding example:

Encoded text is

*x4*p4l*k7

Original text is

xxxxpppplkkkkkkk

92

QUIZ Run-Length Encoding

Decode using RLE:

*a4*A4HIJ*Z5

Encode using RLE:

Hummm, Burrr, OOOPS!

93

Huffman Codes Letter & Word Frequency distributions:

94

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!

95

Huffman encoding example

“ballboard” would be 1010001001001010110001111011

compression ratio

28/56

QUIZ:

Encode “roadbed”

96

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

97

QUIZ Huffman decoding

98

Decode:

1011111001010

3.4 Representing Audio Data

99

We perceive sound when a series of air waves cause

to vibrate a membrane in our ear (eardrum), which

sends signals to our brain.

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.

100

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

101

44,000 for audio CD, to be exact

Remember: Sampling and Quantizing

102

Some information

is lost, but a

reasonable

sound is

reproduced

Not in text

Digital Audio on a CD

103

Figure 3.9

A CD player reading

binary information

“pit”

“land”

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 ???

(p.75 of text)

104

105

Pit height is about ¼ the

laser’s wavelength

“destructive interference”

FYI: How the pits and lands are actually read

Not in text

106

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”.

FYI: How the pits and lands are actually read

Not in text

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?

107

3.5 Representing Images and Graphics

Color

Perception of the frequencies of light that reach

the retinas of our eyes

Human retinas have three types of color

photoreceptor cone cells that correspond to the

colors of red, green, and blue

108

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.

109

110

Dark means low number, light

means high.

Look at the snow and the black

side of the barn! Source: Wikipedia – RGB color model

Representing Images and Graphics

Can you understand this HTML code?

<font color="#FF0000"> Blah blah …

</font>

111

RGB Color Chart in hex

QUIZ

112

Explain the similarities and differences between 00FF00 and 008800

The color cube

113 Figure 3.10 Three-dimensional color space

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

114

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.

115

QUIZ

116

Are these HiColor

or TrueColor?

Indexed Color

A browser may support only a certain number of specific colors, creating a palette from which to choose

117

Figure 3.11

The Netscape color palette

EOL

Extra-credit question

How many bits are needed to represent this palette?

Please show your work.

118

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

119

Digitized Images and Graphics

120

Figure 3.12 A digitized picture composed of many individual pixels

Whole

picture

Digitized Images and Graphics

121

Figure 3.12 A digitized picture composed of many individual pixels

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

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

122

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

123

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

124

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

125

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

126

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

127

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.

128

Individual work for Monday:

Read pages 69-80 of text.

Solve in the notebook these problems :

10 through 20

50 through 53

129

Not homework! Do not turn in, it’s

for your own review

Chapter Review Questions

• Distinguish between analog and digital information

• Explain data compression and calculate compression ratios

• Explain the binary formats for negative (two’s complement), fractional, and floating-point values

• Describe the characteristics of the ASCII and Unicode character sets

• Perform various types of text compression with pencil and paper: Keyword, Run-length, Huffman

130

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

131

Don’t forget!

Homework: 8, 9, 29, 30, 31, 41, 44, 48, 49

Due Monday at the beginning of lab, before

the midterm!

Midterm is Monday in the lab.

Bring:

-- notebook (for grading)

-- cheat-sheet (max. 6 pages total, including ASCII table)

132

How to study for the midterm:

-- read and understand your notes (from the notebook – this

is a good opportunity to improve the notebook, if needed)

-- go over the quizzes we solved together in class (you can find them in your notebook, and in the instructor’s notes posted on the

webpage)

-- solve the homework for Ch.3

-- solve the end-of-chapter quizzes for Chs. 1

and 2 (webpage)

-- write your cheat-sheet (max. 6 pages total, including ASCII

table)

133