1. Chapter 3: Data Representation

2. Types of Data Numbers Characters and symbols Images Audio Video
2324, ,
Characters and symbols
A, B, C, … Z, a, b, c,… z,
0, 1, 2, 3 … 9, +, -, ), (, *, &, etc
Images
Photos, charts, drawings
Audio
Sound, music, etc
Video
Video clips and movies
Instructions
Computer instructions are coded in sequences of 0’s and 1’s

3. Binary Number System Cheapest and simplest in design and engineering
Switch: on  1; off  0
Circuit: voltages
1.7 volts – higher  1
0.0 volts volts  0
Voltages (1.3 to 1.7) are avoided in design
Mathematics: binary numbers
Using digits 0 and 1 only.

4. Decimal vs. Binary Decimal # system Binary # system
10 symbols: 1, 2, 3,…9, 0
Base = 10 (We have 10 fingers)
Decimal number 2324 reads “2 thousands 3 hundreds twenty four”.
Binary # system
2 symbols: 0 and 1
Base = 2
Binary number 1101 = ?

5. Decimal vs. Binary Decimal # System: 2 3 2 4 . Each digit represents:
2*1000
3*100
2*10
4*1
Position values:
1000
100 10 1
Position values (base):
103
102
101
100
Value in Decimal:
2*1000+3*100+2*10+4*1 = 2324D
Binary # System: 1 1 1 .
Position values (base): 23 22 21 20
Position values: 8 4 2 1
Each digit represents:
1*8
1*4
0*2
1*1
Value in Decimal:
1*8+1*4+0*2+1*1 = 13D

6. Storage Units Binary digits – bits 8 bits = 1 byte
210 bytes = 1024 bytes =1 kilobytes = 1KB
220 bytes = 210 KB = 1 megabytes = 1MB
230 bytes = 210 MB = 1 gigabytes = 1GB
240 bytes = 210 GB = 1 terabytes = 1TB

7. Representation of Numbers
Fixed-size-storage approach:
Computers allocate a specified amount of space for a number
Integers
1 bit: 0 to 1
2 bits: 00, 01, 10, 11  0 to 3
4 bits: 0000, 0001, 0010, … 1111  0 to 15
1 byte: 0 to 255
2 bytes: to
4 bytes: -2,147,483,648 to +2,147,483,647
Note: with 4 bytes for integers, any number smaller than -2,147,648 or larger than 2,147,483,647 would be incorrectly represented.,

8. Representation of Numbers
Binary representation of real numbers
Binary # System: 1 . 1 1 1
Position values (base): 21 20
2-1
2-2
2-3
Position values: 2 1
1/2
1/4
1/8
Each digit represents:
1*2
0*1
1*0.5
1*0.25
1*0.125
Value in Decimal:
2 + ½ + ¼ + 1/8 = 2.875D

9. Representation of Numbers
Floating-point numbers for real numbers
Three parts of representation:
Sign (always 1 bits: 0 for + and 1 for -)
Significant digits (e.g., six bits)
the power of 2 for the leftmost digit (e.g., 3 bits)
Example for binary
Sign: 1 (negative)
Significant digits: B
Power of 2: 011B
Example for binary B
Sign: 0 (positive)
Significant digits: B
Note: the last digit is lost, which is 1/16 in decimal
Power of 2: 010B

10. Representation of Numbers
Single-precision floating-point numbers
Sign (always 1 bits: 0 for + and 1 for -)
Significant digits: 23 bits
exponent: 8
Double-precision floating-point numbers
Significant digits: 52 bits
exponent: 11
What you should know?
Computers can represent numbers only in limited accuracy.
E.g., when you enter a 20 digit decimal # into a program that uses single-precision, only about 7 digits are actually stored, the rest are lost.
Real examples:
Designing aircraft on p.35
The Vancouver Stock Exchange Index on pp

11. Representation of Numbers
// file: public_html/2005f-html/cil102/accuracy.c
#include <stdio.h>
int main() {
int x, y, result; // x, y, and result all use 32 bits to represent integers (-2,147,648 to +2,147,483,647)
char op;
int i;
for (i = 0; i < 100; i++) {
printf("please enter an expression:\n");
scanf("%d %c %d", &x, &op, &y);
if (op == '+')
result = x + y;
else if (op == '-')
result = x - y;
else {
printf("Invalid operator!!");
break; }
printf("%d %c %d = %d\n", x, op, y, result);
// When you enter , the result is

12. Representation of Numbers
Variable-size-storage approach:
Allow a wide-range of numbers to be stored accurately
Needs significant more time to process
Fixed-size approach is used more common than variable-size approach.

13. Representation of characters
There are no visual letters A, B, C, etc stored in computers like we have in mind.
Letters and symbols are encoded in 8 bits – one byte - of 0’s and 1’s.
Keyboard converts keys A, B, C etc to their corresponding codes and
monitor converts the code into visual letters A, B, C etc on screen.
Two commonly used coding schemes:
ASCII: American Standard Code Information Interchange
EBCDIC: Extended Binary Coded Decimal Interchange Code

14. Representation of characters
EBCDIC
ASCII A B a b 1 2
, (comma)
- (dash)

15. Representation of characters
Foreign characters – two approaches
Use one byte per char
Ex.,
ISO for Western (Roman)
ISO for Greek
ISO-2022-CN for simplified Chinese
Webpage: using “META charset=…” to specify which encoding is used.
Use two bytes per char/symbols
16 bits have 65,536 combinations (characters)
Unicode coding system

16. Representation of Images
A picture is treated as a matrix of dots, called pixels.

17. Representation of Images
The pixels are so small and close together we cannot really see them as separate dots.
Resolution: dots per inch (dpi)
72 dpi for Web images
600 or 1200 dpi for professional printers or home photo printers

18. Representation of Images
The color of each pixel is represented using bits.
Black/White: one bit per pixel
1-white and 0-black
Gray scale: one byte per pixel
256 different degrees of gray ( to )
black, intermediate gray, white
Color: three bytes per pixel
Red, green, blue color
One byte for the intensity of each of the three color
256 possible red, 256 green, 256 blue
Pure red: for red byte, for green and blue
White: for all three bytes
Black: for all three bytes

19. Representation of Images
Image storage -- size
Gray scale: one byte per pixel
E.g., A 3 X 5 picture with 300 dpi resolution
3 * 300 = 900 pixels per column
5 * 300 = 1500 pixels per row
900 * 1500 = 1,350,000 pixels/picture
Needed storage = 1,350,000 bytes/picture = 1MB/picture
Color: three bytes per pixel
Needed storage = 3 (bytes per pixel) * 1,350,000
= 4,050,000 bytes/picture
= 4MB/picture --- TOO BIG

20. Representation of Images
Image compression
Color table
Most pictures contain a small # of different colors
Use a table to define colors that are actually used in the picture
Each pixel has an index to the color table.
Each image contains a color table and table indices
Example
For a picture with 100 different colors, the color table would contain 100 entries, three bytes each entry for each color. One byte can be used as index to the table for each pixel.

21. Representation of Images
Drawing commands
Draw picture using basic commands
Just as artists draws using a pencil or a brush and other basic movements
Example,
A house is drawn by sketching various elements (doors, windows, walls), adding color to them, and moving to the desired position.

22. Representation of Images
Data averaging or sampling
Condense the size by selecting a smaller collection of information to store.
Many different ways of sampling and data averaging
An example: choose to store only every other pixel in an image (sampling)– reducing the size to half. To display the full picture, the computer need to fill in the missing data with, for example, the average of neighboring pixels (data averaging)
The resulting picture cannot be as sharp as the original
Lossy data compression

23. Image Formats Commonly used image file formats -1 Bitmap (.bmp)
Pixel-by-pixel storage of all color information for each pixel.
Lossless representation
Files are huge.
Graphics Interchange Format (.gif)
Use one or more color tables – the color table technique
Each table contains 256 colors.
Suitable for pictures with a small # (<256) of different colors (e.g., organization charts)
Not suitable for pictures with shading (e.g., photos)

24. Image Formats Commonly used image file formats - 2 PostScript (.ps)
Employ the drawing commands technique
“moveto” draws a line from current position to a new one and “arc” draws an arc given its center, radius, etc
General shapes can be used in multiple places
Fonts can be reused.
Useful when the picture can be rendered as a drawing or its contains many of the same elements (e.g., text of the same fonts)
Joint Photographic Experts Group (JPEG) (.jpg)
use the data averaging and sampling on 8*8 pixel blocks
User determines the level of details and clarity
High-quality image – 8*8 blocks maintain their contents
Low-quality image – info in 8*8 blocks is discarded  smaller files

25. Comparison b/w jpg, gif, and ps
Comparison of .jpg and .gif
More on .jpg and .gif

26. Summary of Image Representations
Other commonly used formats
Tiff: Tagged Image File Format
PNG: Portable Network Graphics
New formats will emerge
Understand the format and know the pros and cons
To learn: Google the format
Use programs (GIMP) to convert b/w formats

27. Analog Audio
Sound wave

28. Digital Recording - 1 Digital Recording at low sample rate
Digital Replaying

29. Digital Recording - 2 Digital Recording at low high sampling rate
Digital Replaying

30. Music CD Sample rate: 44,100 samples/second
#of bits for height: 16 bits
# of channel: 2
Total of bytes/sec:
44,100 samples/s x 2 bytes/sample x 2 channels
= 176,400 bytes/second
Total of bytes on a 74 minute CD
176,400 bytes/sec * 70 minutes * 60 seconds/minute
= 783,216,000 => 783 MB

31. MP3 Format Compress the audio based on the following: Lossy Format
People cannot hear sound at very low and very high frequencies
People hear loud sound, not the softer one when there are two sounds
There are sounds humans hear better.
Lossy Format

32. MP3 Quality Bit Rate: # of bits per second encoded in MP3
Bit Rate: bit rate
Quality
320 bit rate  humans cannot tell difference from original music CD
120 bit rate  like hearing music on radio
160 bit rate or higher  for better experience

33. Music CD to MP3 Files Music CD Finest Quality PC Data CD Hard disk MP3
MP3 Encoder Or
Compresser
Ripper

34. Listening to Music and MP3
Music CD
Finest Quality
Data CD
MP3
Music CD
Player
MP3
Player

35. Summary – chapter 3 Computers work in binary
Integers may be constrained in size
Real numbers may have limited accuracy
Computations may produce roundoff errors, affecting accuracy
Characters and languages are encoded in binary
Pictures are displayed pixel by pixel
Color table, draw commands, and data averaging and sampling compression techniques
.bmp, jpg, .gif, .ps formats
Audio presentation: Music CD and MP3

36. Terminology Binary vs. decimal Position value The base of a # system
Bit/byte/KB/MB/GB/TB
Integer binary #s
Real # in binary
Floating point numbers
Representational error
Roundoff errors
ASCII/EBCDIC/Unicode
Pixels
Dots per inch (dpi)
Bitmap
Color table
Data averaging
Data sampling
Data compression
.jpg, .bmp, .gif, .ps