Quiz on Ch.2 Convert 201023 to decimal

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

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

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?

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

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)

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.

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

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!

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?

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

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 3.2 An analog and a digital signal

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 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?

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? EoL

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

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

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

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

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

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)

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

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

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

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

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?

Adding two negative numbers Try these: 4 - 4 -4 - 3 +3 + -3

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!

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

Two’s complement on 4 bits (k = 4)

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

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.

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

“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 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

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 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.)

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

Trick QUIZ What decimal number does this binary number represent? 1001 1110

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

Representing 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 …

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 …

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

QUIZ Convert: .10112 = 0. 10

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

QUIZ Convert: 0.310 = 0. 2

QUIZ Finite decimal fractions may have infinite binary representation! 0.310 = 0. 0100110011 2 Stop after 8 bits!

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

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)

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!

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

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

QUIZ Convert to floating point and to scientific notation: 123.332 = -0.0034 = 0.0 =

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

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

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?

7-bit ASCII Character Set

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

QUIZ Encode “Hello, world!” in ASCII Decode 67 83 32 49 49 48 from ASCII

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

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

ASCII Control Characters

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

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

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

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

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

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

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.

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.

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?

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

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

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?

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

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 4 - 259 Perform the encoding and decoding in the prev. examples with the convention above!

Huffman Codes Letter & Word Frequency distributions:

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!

Huffman encoding example “ballboard” would be 1010001001001010110001111011 compression ratio 28/56 QUIZ: Encode “roadbed”

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

Huffman decoding QUIZ Decode: 1011111001010

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

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.

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.

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

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

Digital Audio on a CD “pit” “land” Figure 3.9 A CD player reading binary information “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 ???

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

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”.

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?

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

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.

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

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

QUIZ Explain the similarities and differences between 00FF00 and 008800

The color cube 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

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.

QUIZ 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 Figure 3.11 The Netscape color palette EOL

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

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

Digitized Images and Graphics Whole picture Figure 3.12 A digitized picture composed of many individual pixels

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 3.12 A digitized picture composed of many individual pixels

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

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

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

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

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

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

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.

To read:

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?

Who is Bob Bemer? I love ASCII !

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?

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

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

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