Quiz on Ch.2

Convert 201023 to decimal

Quiz on Ch.2

Convert the decimal number 51 to binary using repeated division.

Quiz on Ch.2

3

Carry values

go here

1 1 0 1 1 1 0 0

+0 1 0 0 0 1 0 1

14

Binary addition (Check in base ten!)

Chapter 03

Data Representation

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

• Haptics (touch)

• Smell (machine olfaction: see link on our webpage)

• Chess positions (computer chess: see link on our webpage)

• Etc. etc. etc.

5

Positive integers

Negative integers

Real (various precisions)

Complex

Data compression

Reduction in the amount of space (memory) needed to store or transmit 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?

6

Quiz

A video file is originally 3.5 GB long.

We compress it to 350 MB. What is the compression ratio?

7

Quiz

A video file is originally 3.5 GB long.

We compress with a compression ratio of 0.2

What is the final size of the file?

8

Data compression

The Compression ratio is always between 0 and 1

(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)

9

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.

10

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

11

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)

12

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)

13

Quiz

A digital compass reads the position of a robot 20 times a second.

What is the time elapsed between two consecutive readings?

Is this a sampling error or quantization error?

14

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!

15

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

16

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

17

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?

18

19

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?

20

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?

Is this a sampling error or quantization error?

21

Solution

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?

7 bits → 27 = 128 values → 127 intervals

(100-50)/127 = 0.394 deg/interval

Is this a sampling error or quantization error?

Quantization, since it’s in the vertical direction 22

Beware of the “fencepost error”!

Image source: http://en.wikipedia.org/wiki/Fencepost_error

Quiz

A digital volt-meter has a scale from 10 to 30 volt (V).

The voltage is represented on 9 bits. What is the smallest voltage difference that it can represent?

Is this a sampling error or quantization error? 23

For next time:

• Read pp.54-57 of text

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

24 EOL1

Quiz

A weather log file is originally 4.2 GB long.

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

25

Quiz

An Analog-to-Digital Converter (ADC) accepts an input voltage between -3 and +12 V, and uses 10 bits to represent it digitally.

What is the precision?

26

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?

27

How many bits are needed to represent all 45 desktops in this lab?

The inverse of the power (2n) is the logarithm:

28

What’s wrong with this answer?

Base is 2

How many bits are needed to represent all 45 desktops in this lab?

29

How many bits are needed to represent all 45 desktops in this lab?

30

Alternative solution:

What’s the smallest power of 2 that is ≥ N?

QUIZ

How many bits are needed to represent all the

days you’re planning to spend in college for your

undergraduate degree?

– Use either method!

31

Remember: 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!)

• Haptics (touch)

• Etc. etc. etc. 32

Positive integers

Negative integers

Real (various precisions)

Complex

3.2 Representing Numeric Data

Negative integers

Signed-magnitude representation

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

33

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

34

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

35

Ten’s complement

36

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

37

EXTRA-CREDIT QUIZ

If the representation is 76, what actual number

is being represented?

38

Why the “complement” in ten’s complement?

100 – 50 = 50

100 – 49 = 51

……………………..

100 – 1 = 99

In general:

100 – i is the representation of – i

39

Positive number

Negative number

QUIZ

What is the representation for each of these actual numbers?

• -48

• -40

• -30

• -5

40

Let’s use ten’s complement!

To perform addition, add the numbers and discard any carry

41

Now you try it

48 (signed-magnitude)

- 1

47

How does it work in

the new scheme?

Adding two negative numbers:

42

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!

43 EOL2

44

Two’s Complement

What do you notice

about the left-most bit

(MSB)?

Important: It’s not

sign-magnitude!!

45

QUIZ: Two’s Complement

John has encountered this two’s

complement number:

1000 0111

He says: The number is negative,

b/c the MSB is one.

The magnitude is just 111, which

means 7.

Therefore the number is -7 in

decimal!

Is John correct?

Two’s complement on 4 bits (k = 4)

46

What is:

• The largest positive number?

• The largest negative number?

• -1?

Repeat the questions above for:

• 5 bits (k = 5)

• 6 bits

• 8 bits

• N bits (general N)

47

http://xkcd.com/571/

Two’s complement on 16 bits

48

“Signposts” for two’s comp.

0000 0000 means ...

0111 1111 means ...

1000 0000 means ...

1111 1111 means ...

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:

7

-7

-110

200 (trick question!)

-129 (trick question!)

0

49

“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 50

This is how subtraction is implemented in

computer hardware! A – B = A + (-B)

QUIZ

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

• -13

• 40

51

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

52

QUIZ

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

40 – 13

53

Trick QUIZ

What decimal number does this binary number represent?

1001 1110

54

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

1 0000 0010

… but adding -1 to 3 doesn’t!

55

Conclusions:

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

Overflow is something we should always expect (and make provisions for) when mapping an infinite world onto a finite machine!

56

SKIP Representing Real Numbers

57

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 58

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?

59

7-bit ASCII Character Set

60

QUIZ

Encode “Hello, world!” in ASCII

Decode 67 83 32 49 48 50 from ASCII

61

Python to the rescue!

Encode “Hello, world!”

in ASCII

62

For next time:

• Read pp.58-62 of text

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

• Solve end-of-chapter ex. 6, 33 in notebook

63 EOL3

QUIZ

What decimal number does this binary number represent?

1011 0010

64

QUIZ

Perform this addition in 8-bit two’s complement:

8 – 11 =

65

The ASCII Character Set

66

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

67

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

Not in text

8-bit (“extended”) ASCII Character Sets

68

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

69

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

Not in text

Extended ASCII: Latin-1

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

Not in text

Trick QUIZ

What do these bits represent?

1101 1110

71

Trick QUIZ

What do these bits represent?

1101 1110

72

Unsigned integer: …

Signed integer (2’s complement): …

Fixed-point fraction

IBM 437 character: …

Latin-1 character: …

Floating-point fraction

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 )

73

Simplified Chinese has

6500!

Unicode examples

74 Figure 3.6 A few characters in the Unicode character set

Romanian Characters in Unicode

75

Part of the Latin-Extended-B

character sub-set

Miscellaneous Characters in Unicode

76

See more online at the official Unicode site

QUIZ:

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

77

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

78

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

79 EOL4

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

80

Keyword Encoding

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

81

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.

82

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.

83

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?

84

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 85

Run-Length Encoding Encoding example:

Original text is

bbbbbbbbjjjkLLqqqqqq+++++

Encoded text is

*b8jjjkLL*q6*+5

(Why isn't L encoded? J?)

Compression ratio: 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?

86

Run-Length Encoding

Decoding example:

Encoded text is

*x4*p4l*k7

Original text is

xxxxpppplkkkkkkk

87

QUIZ Run-Length Encoding

Decode using RLE:

*a4*A4HIJ*Z5

Encode using RLE:

Hummm, Burrrrr, OOOPS!

88

Huffman Codes Letter & Word Frequency distributions:

89

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!

90

Huffman encoding example

“ballboard” would be 1010001001001010110001111011

compression ratio

28/63 (7-bit ASCII)

QUIZ:

Encode “roadbed”

91

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

92

QUIZ Huffman decoding

93

Decode:

1011111001010

QUIZ Huffman decoding

94

Decode:

1001101111011

QUIZ: Decipher the coded text using the Huffman table:

95

0010110101001110100110011011010011000111

01111001110100111

3.4 Representing Audio Data

96

We perceive sound when:

• a series of air waves cause to vibrate a membrane in

our ear (eardrum), which

• is connected to the malleus, incus, and stapes

(hammer, anvil, and stirrup), which

• are connected to the cochlea, which

• sends nerve signals to our brain.

Anatomy of the middle ear

97

Not in text

Source of figures: http://en.wikipedia.org/wiki/Evolution_of_mammalian_auditory_ossicles

Correspondence discovered in 1837 (!)

through embriology

Anatomy of the middle ear

98

Not in text

Source of figures: http://en.wikipedia.org/wiki/Evolution_of_mammalian_auditory_ossicles

Morganucodon, a.k.a. Morgie

discovered in the1950s

Correspondence discovered in 1837 (!)

through embriology

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.

99

Remember: Sampling and Quantizing

100

Some information

is lost, but a

reasonable

sound is

reproduced

Not in text

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

QUIZ: Sampling

A telephone voice channel is designed to allow frequencies up to 4,000 Hz (4 kHz).

How many samples must be collected every second?

102

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

QUIZ: Audio Formats

MP3

– analyzes the frequency spread and discards information that can’t be heard by most humans (>16 kHz)

How many MP3 samples are there in a 3-minute song?

108

QUIZ: Audio Formats

MP3

– analyzes the frequency spread and discards information that can’t be heard by most humans (>16 kHz)

How many MP3 samples are there in a 3-minute song?

If each sample is represented as one Byte, what is the total size of the file?

109

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

110

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.

111

112

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>

113

RGB Color Chart in hex

QUIZ

114

Explain the similarities and differences between 00FF00 and 008800

The color cube

115 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

116

Extra-credit question

117

QUIZ

118

Are these HiColor

or TrueColor?

EOL5

119

Tarleton Purple

120

The correct "Tarleton Purple" color codes:

• Hex: 0x4F 2D 7F

• RGB: 79 45 127

Source: http://www.tarleton.edu/webservices/guidelines.html

Indexed Color

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

121

Figure 3.11

The Netscape color palette

QUIZ

How many bits are needed to represent this palette?

Show your work.

122

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

123

Digitized Images and Graphics

124

Figure 3.12 A digitized picture composed of many individual pixels

Whole

picture

Digitized Images and Graphics

125

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

QUIZ: Images

A low-res image has 200 rows and 300 columns of pixels.

• What is the resolution?

• If the pixels are represented in True-Color, what is the size of the file?

• Same question in High-Color

126

Two types of image formats

• Raster Graphics = Storage on a pixel-by-pixel

basis

• Vector Graphics = Storage in vector (i.e. mathematical) form

127

Raster Graphics

GIF format • Each image is made up of only 256 colors (indexed color –

similar to palette!)

• 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

128

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!

• Lossless, but files are large!

JPEG format Averages color hues over short distances

• Lossy compression

Optimal for color photographs

129

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

130

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

131

3.6 Representing Video

132

133

Representing Video

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 compressions to minimize the huge amounts of data associated with video

QUIZ video

A 10-minute videoclip is shot in a resolution of 768x1024 True-Color pixels, and 24 frames/second.

• What is the uncompressed size of the video file?

• How long would it take to transmit it over a 100 Mbps Ethernet connection?

134

135

Representing 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

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

136

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

137

Individual work for Wednesday:

Read pages 69-80 of text.

Solve in the notebook these problems :

● 10 through 20

● 50 through 53

138

Not homework! Do not turn in, it’s for

your own review

For next time:

--Read pages 69-75 of text

--Re-solve all today’s quizzes

Homework: 8,9,29,30,31,41,47,49,51,53, 61

--Due Friday, Sep.25, but recommended as

preparation before the midterm

Midterm next week during lab

--morning lecture on that day is review

139 EOL6