1. Bits and Bytes
There are 10 types of people in this world: those that understand binary numbers, and those that don’t. . .
We’ve talked a lot about how computers work so far. Let’s recap a little – what have we talked about so far in this class? (hardware, booting, OS, XP, viruses) Well, the whole thing turns out to be a very complicated system for fiddling with ones and zeroes. It turns out that you can do a whole lot with ones and zeroes. Words, pictures, sounds, movies – can all be turned into numbers. And anything you can turn into a number, you can turn into ones and zeroes.
Two values instead of our decimal digits, which can have 10
A switch has two states – on and off. Computer memory is basically made up of switches -- easy to represent 1s and 0s.
But nobody is very interested in a blob of 1s and 0s – need to assemble them into larger and more interesting things.
How do we use 1s and 0s to represent numbers? Words? Music? Pictures? Movies?

2. Bits and Binary Numbers
A bit is a Binary digIT – a digit that can have two values: one or zero.
Binary numbers are composed of bits, just like decimal numbers are composed of digits 0-9
Binary numbers can be used to represent letters, numbers, pictures, sound, or just about anything.
The exact same pattern of 1s and 0s might represent many different things, depending on where it is and what program is using it.
For instance, the bits b could represent the number 66, or the letter ‘A’, or part of a picture, or part of a sound, or. . .
The interpretation depends on the context. This includes things like the file extension, and the program that's reading the data.
What does coin mean?
coin – money in english, corner in french

3. Decimal Numbers and Place Value
Base 10 is our old friend:
5473 = (5 * 1000) + (4 * 100) + (7 * 10) + (3 * 1)
Can also be written as:
(5 * 103) + (4 * 102) + (7 * 101) + (3 * 100)
Why do we use base 10? Because we have 10 fingers (what do the Simpsons use?)
10 is the radix – a fancy name for the base of a number system.
You need 10 different symbols to represent the values in base 10. We use the digits 0-9
The rest of the meaning is conveyed by place – the 1s, 10s, 100s, etc.
We could use any number as a base – 10 just happens to be convenient if you have 10 fingers.

4. Binary Numbers
Base two only uses 0 and 1. The place values are powers of 2:
1, 2, 4, 8, 16, 32, 64, 128, and so forth, ad infinitum.
Example: b
= (1*27) + (0*26) + (0*25) + (1*24) + (0*23) + (1*22) + (1*21) + (1*20) =
128's
64's
32's
16's
8's
4's
2's
1's 1
= 151

5. A Binary Odometer Binary: Decimal: And so on, ad infinitum. . . 1 1 21
10's
1's 1 2 3 4 5 6 7 8 9
have kids with 1/0 paper – count 0 – 32
go around room from there
And so on, ad infinitum. . .

6. Example 198 = 11000110b Convert 198 to binary: 1
198 / 128 = 1, remainder is 70
70 / 64 = 1, remainder is 6
6 / 32 = 0, remainder = 6
6 / 16 = 0, remainer = 6
6 / 8 = 0, remainder = 6
6 / 4 = 1, remainder is 2
2 / 2 = 1, remainder is 0
0 / 1 = 0
128
128's
64's
32's
16's
8's
4's
2's
1's 1
198 = b

7. Bytes A byte is 8 bits. 8 bits can represent 256 different values.
Computers generally use a byte or a series of bytes to represent information.
16 bits will represent 256*256=65536 different values.
24 bits will represent 256*256*256=16,777,216 different values.
and so on. . .
How many different values can be represented in 16 bits? 256 * 256 = 65536
Why 8 bits in a byte? Like asking why is 12 a dozen. Just something that worked out well over the years.
Really became set in stone with IBM 360 computers in 1965

8. Counting Bytes: Kilo, Mega, Giga, Tera, and Lotta
Prefix
Approximately?
Abbrev.
Exactly?
How many bytes?
Kilo
Thousand (103) KB
1024
Mega
Million (106) MB
1024 KB
1,048,576
Giga
Billion (109) GB
1024 MB
1,073,741,824
Tera
Trillion(1012) TB
1024 GB
1,099,511,627,776
Peta
Quadrillion (1015) PB
1024 TB
1,125,899,906,842,624
Exa
Quintillion (1018) EB
1024 PB
A whole lot. . .
Zetta
Sextillion (1021) ZB
1024 EB
Even more. . .
Yotta
Septillion (1024) YB
1024 ZB
Fuggedabaddit. . .
Giga comes from greek work gigas, for giant
Tera comes from greek word teras, for monster
A typical video store contains about 8 terabytes of video (Wikipedia)
San Diego Supercomputer Center has a petabyte of hard disk storage and 6 petabytes of tape storage
There are about 7 sextillion grains of sand on all the worlds deserts and beaches
If you had a 1 kbps connection to the internet, how long would it take to transmit 1 KB of data?
When you talk about speed,you talk about Bits per second, and 1 Kb isn’t 1024 bits, its So 1 KB = 1024 * 8 = 8192
Would take seconds to send 1KB
If you have 20 people waiting in line at a bank, and the tellers can handle 2 people per minute, how long will it take to handle the line?

9. Hexadecimal – Base 16
An 8 bit byte is frequently shown as a pair of hexadecimal (hex) digits.
4 bits: 16 different values
dec
bin
hex
0000b
0x0 8
1000b
0x8 1
0001b
0x1 9
1001b
0x9 2
0010b
0x2 10
1010b
0xA 3
0011b
0x3 11
1011b
0xB 4
0100b
0x4 12
1100b
0xC 5
0101b
0x5 13
1101b
0xD 6
0110b
0x6 14
1110b
0xE 7
0111b
0x7 15
1111b
0xF
one byte
two hex digits:
0x6A
Hex is widely used to represent colors in HTML tags: color=“#00FF0E”
6 hex digits are enough to specify 16.7 million different combinations of R,G,B

10. Decimal to Hex: Hex to Decimal:
Convert decimal 173 to hex:
173 / 16 = 10 (0xA), remainder = 13 (0xD)
173 = 0xAD
16's
1's A D
Hex to Decimal:
Convert 0xBE to decimal:
0xBD = (11 * 16) + (14 * 1)
0xBE = 190
16's
1's B E

11. Binary to Hex Example 0xDC 1 D C
128's
64's
32's
16's
8's
4's
2's
1's 1 D C
0xDC
Starting from the right-most bit, break the binary number into groups of 4 bits, and convert each group to the equivalent hex digit.

12. Encoding Text
Fundamentally, computers just deal with numbers. They store letters and other characters by assigning a number for each one. Characters are generally represented in a computer as ASCII or Unicode values, where numbers are assigned to letters, digits, punctuation, etc.
Binary
Decimal
Symbol 48 49 1 50 2 … 65 A 66 B 67 C
Sample ASCII:
The idea is the same as if you made up a code that says that 1 = a, 2 = b, etc
Note anything interesting?
Digits map to lower 4 bits
Upper case, lower case 1 bit different for alphabetization
What about non-western languages? Unicode – 16 bits allows 64K symbols – when upper byte 0, lower byte treated as if ASCII

13. Analog Digital
We saw yesterday how you could express any number in terms of 1s and 0s – so that means if we can express anything as numbers, we can express it as 1s and 0s.
So now the question is, how do we express things in terms of numbers?
The analog clock is continuous. It can’t get from 1:00 to 1:01 without passing through every point between the two. It can display an infinite number of points, even if you can’t see them all. If you used a magnifying glass, you’d see it sweep past every point – it can’t skip over anything
The digital clock is an approximation of the time. It goes directly from 1:00 to 1:01 – there is no in-between. (draw step graph on board)
You can continue to add digits – seconds, tenths, etc – the more digits, the closer you can get. But you always lose a little information.
The values are discrete – you can count how many different values it can display.
An analog clock can display an infinite number of values, even if you can't see them. It's the hands that tell the time, not the numbers.
shows minutes – 0-23, 0-59 – 24 * 60 -> 1440 minutes per day, sec

14. What is Sound?
Sound is variations in air pressure – air moving back and forth quickly. Your ear and your brain turn these variations in air pressure into sound.
Can think of sound as sculpting the air
Is it discrete or continuous?

15. How Do Speakers and Microphones Work?
A speaker converts an electrical signal into movement of air, using magnets to push the air back and forth. When it pushes forward, the air is compressed. When it pulls back, the air is rarified, or decompressed.
A microphone does the opposite – it converts the movement of air into an electrical signal.
So, now we’ve gotten sound into electricity (mike) and electricity into sound (speaker)
The electrical signal from a mike is still continuous -- we haven’t gotten it into 1s and 0s yet
How a speaker works:
windings of wire around iron core – pass current through, 1 end will be N
Near another magnet – will be attracted or repulsed – switch current direction, magnet polarity reversed, switches attraction/repulsion

16. Digital-to-Analog and Analog-to-Digital Conversion
Converter
Analog to digital converter: converts a continuous signal into a sequence of discrete values
-> takes frequent measurements
digital to analog converter: converts a sequence of discrete values into a continuous signal
In-between – series of numbers – can save it on a disk, copy it, it, etc
Digital-to-analog
Converter

17. Digitization: Turning Sound into 1s and 0s
Sampling: Converting analog information into a digital representation by measuring the value of the analog signal at regular intervals, and encoding these integer values in digital form – 1s and 0s.
Sampling rate (or frequency): How often the measurements are taken.
Sampling resolution: How many different values you quantize into. 8 bit resolution allows 256 different values, 16 bit resolution allows different values. Another way to look at it is, how many marks are there on your ruler? The finer the gradations, the more accurate the representation can be.
Quantization error: The difference between the actual signal level and the integer value chosen to represent it
have to turn the continuous range into a set of discrete values, like the digital clock – there's no "in-between", we have to pick one of the values on the Y axis to represent the continuous value at that point.
The more often you take measurements and the more precise the measurements are (note Y axis), the closer the digitized version is to the original, and the bigger the file is. . .
Frequency: how often you measure
Resolution: how many different values you have to represent the height
The middle graph resolves the sample into one of ten values 0 – 9
The bottom graph resolves the sample into one of 36 values
You have to decide how good is good enough – voice quality is not the same as music quality
So, once you’ve gotten it into 1s and 0s, what can you do? Mail it, copy it, play it backwards, correct the pitch, etc – we'll see with Audacity
Copy is identical to the original – it’s just a number
You lose some quality when you digitize something, but once it’s digital, you won’t lose any more no matter how many copies you make
it's a number – and a copy of the number is the same number
Compare to a “tape of a tape of a tape”
Clipping and distortion –given the bottom graph, what if a sound wave comes along that's twice as high as what's there? You don't have the range to represent it – so it gets distorted – a.k.a. clipping -- "dial goes into the red" in the recording studio
dynamic range – the range of volume levels that the digitization scheme can accomodate

18. Making Images With Ones and Zeroes

19. Turning Pictures into Numbers
Pixel stands for “picture element”
Pixel should be pickle?
Lay out grid over picture
A pixel is a single colored dot – can’t have two colors in the same pixel
Decide what color best represents that square
Encode color as a number
The finer the grid and the more colors you use, the better the image – and the larger the digital version.
Audio sampling rate -> how fine the grid is, or how many pixels (dpi)
Original Image
Divided into pixels
Each pixel is averaged

20. Turning Pictures into Numbers - 2
Audio sampling resolution -> # of colors in palette, or color depth
Palette: maps colors to numbers

21. Digitizing Gone Bad. . . Not enough colors Resolution too low Original
in palette
Resolution too low
Original

22. A Possible Image File Format
An image file could be constructed like this:
First, include the palette of colors that are used
Second, give the height and width in pixels
Third, give the pixel data
More specifically, if the image used 256 different colors:
palette of 256 colors, each color 3 bytes: 768 bytes total
image width in pixels – 2 bytes
image height in pixels – 2 bytes
1 byte for each pixel – indicates which of the 256 colors goes in that spot
So an image that's 512 x 512 pixels would be:
= bytes total
Think of the palette as the possible colors of all the tiles we're going to use
Different images can use different palettes
picking the best range of colors to represent the image is one of the challenges

23. Run-Length Encoding
There are many ways to compress a bitmapped image. One simple way is run-length encoding, which is effective if an image has a lot of consecutive pixels of the same color. The image is then expressed as a series of pairs, where each pair is a pixel color, and the number of pixels in a row of that color. For the image at left, it could be W, 20, B, 3, W, 1 ,B, 1, W, 11, B, 1, etc.
16 x 16 = 256 bytes uncompressed
76 runs, 76*2 bytes = 152 bytes

24. GIF and JPEG Compression
JPEG (Joint Photographic Experts Group) better for photos, images with complex color schemes
GIF (Graphics Interchange Format) better for line art, images with large areas of single color
Compression:
Since each color takes 24 bits (3 bytes), a 400x400 pixel image would be 480,000 bytes
JPEG – Joint Photographic Experts Group
JPEG – better for photos, more complex images – continuous tone. Not good for line art
GIF – better for line art, images with large areas of single color
GIF – limited to 256 different colors.
JPEG – full color information (24 bits per pixel) – much more complex technique for reducing designed to exploit known limitations of the human eye, notably the fact that small color changes are perceived less accurately than small changes in brightness. Makes images smaller by throwing away some info – “lossy” compression
GIFs can also have transparent areas, animation
(So how do you encode movies?)
Need to find out about PNG

25. Medical Imaging – the CAT Scan
The idea behind Computed Axial Tomography is to take X-rays from many different angles and use the information to construct an image of the object. We are essentially sampling the object with X-rays.
The elements in a 3D image are called voxels.

26. A Thought Experiment. . .
Imagine that you had a BB gun that always shot BBs at the same velocity, and you also had a radar gun that could measure the speed of a BB. . .
You take some measurements, and discover that different substances slow the BB down by different amounts:
Substance (1 cm thick): BB Slows down by:
hamburger 20 m/sec
jello m/sec
slurpee m/sec
water 5 m/sec

27. A Thought Experiment (cont)
Slows down
hamburger 20 m/sec
jello 15 m/sec
slurpee 10 m/sec
water 5 m/sec
100 m/sec
100 m/sec
100 m/sec B
70 m/sec A
100 m/sec C D
70 m/sec
A is slurpee, B is hamburger
C,D are jello
Or A, B are jello
C is slurpee, D is hamburger
What if you took another shot through C, B and A, D? Not 1 cm distance any more. . .
If first solution: AD would be 75, CB would be 65
IF second solution: AD would be 65, CB would be 75
each square is 1 cm2 –
Can you determine what
A, B, C, and D are?
75 m/sec
65 m/sec

28. Computed Axial Tomography
X-ray tube and detectors revolve around patient
Detectors measure intensity of X-rays after passing through the patient
Computer uses information to reconstruct a 3-D image