Chapter 3: Data Representation

Types of Data Numbers Characters and symbols Images Audio Video 2324, -34.35, 34567890123.12345 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

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 - 1.3 volts  0 Voltages (1.3 to 1.7) are avoided in design Mathematics: binary numbers Using digits 0 and 1 only.

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 = ?

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

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

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: -32768 to +32767 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.,

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

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 -1111.01 Sign: 1 (negative) Significant digits: 111101B Power of 2: 011B Example for binary +100.1101B Sign: 0 (positive) Significant digits: 100110B Note: the last digit is lost, which is 1/16 in decimal Power of 2: 010B

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. 38-39

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 2000000000 + 500000000, the result is -1794967296

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.

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

Representation of characters EBCDIC ASCII A 11000001 01000001 B 11000010 01000010 a 10000001 01100001 b 10000010 01100010 11110000 00110000 1 11110001 00110001 2 11110010 00110010 , (comma) 01101011 00101100 - (dash) 01100000 00100101

Representation of characters Foreign characters – two approaches Use one byte per char Ex., ISO-8859-1 for Western (Roman) ISO-8859-7 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

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

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

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 (00000000 to 11111111) 00000000 black, 01111111 intermediate gray, 11111111 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: 11111111 for red byte, 00000000 for green and blue White: 11111111 for all three bytes Black: 00000000 for all three bytes

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

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.

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.

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

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)

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

Comparison b/w jpg, gif, and ps Comparison of .jpg and .gif http://www.siriusweb.com/tutorials/gifvsjpg/ More on .jpg and .gif http://www.wfu.edu/~matthews/misc/jpg_vs_gif/JpgVsGif.html

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

Analog Audio Sound wave

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

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

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

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

MP3 Quality Bit Rate: # of bits per second encoded in MP3 Bit Rate: 96 - 320 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

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

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

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

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