Computer Memory Design Overview & Number Representation Techniques

1. Data Rep CSCI130 Instructor: Dr. Lynn Ziegler

2. Overview of Computer Design • Memoryis divided into locations/registers/words • Each holds a fixed amount of data • Numbers, characters • Each has an address used to access its contents • P.O. boxes: number is address and the box is the register • Each word is made of a fixed number of memory units called bits (Binary digITs)each having one of two states. (Word sizes for modern computers are almost always 32 or 64 bits.) • 1 means electric current is strong • 0 no/weak electric current • The normal addressable chunk of informations is 8 bits (1 byte) • Data and addresses are represented in binary • Memory sizes are measured in bytes in groups of • 1024 ~ 103(or 1 KB) or 1,048,576=106 (or 1 MB) or 109 (or 1 GB)

3. Overview of Computer Design • The number system that we use daily is called the decimal system • How many different digits? • Why? • In computers, we have only two possibilities (0 or 1)  we can use the binary system • …+ 104+103+102+101+100 becomes … 24+23+22+21+20 = ...16+8+4+2+1 • 37 = 3x101 + 7x100 • 100101 = 1x25+0x24 + 0x23 +1x22+ 0x21 +1x20

4. Binary to Decimal and Vice Versa • Binary  decimal • 1100 10012 = ? • 1001 10012 =? • 0010 10012 = ? • Decimal  Binary • 1410 = ? • 910 = ? • 129 10 = ? • This is how we can represent (positive) numbers

5. Hexadecimal • Binary is not very convenient for humans to use • 1001000010010010 • Instead we use the hexadecimal system (base 16) • Group every four binary digits into a single hexadecimal value starting at the right. • In base 10, we have 10 digits (0-9) • In base 2, we have 2 digits (0-1) • What about base 16 (hexadecimal)? • 0, 1, 2, 3.. 9, A, B, C, D, E, and F • …+163+162+161+160 = …+4096+256+16+1 • A1F = A*162 + 1*161 +F*160 =10*16*16+ 1*16+ 15 = 259110

6. Binary to Hexa and Vice Versa • From binary  hexadecimal: group 4 bits at a time as one hex. digit • 10011111100100102 = 1001 1111 1001 0010= 9F9216 • 110010012 = ? • 100110012 =? • Group starting from left or right? • 01010012 = ? • From hexadecimal  binary: • 9A9216 = 1001 1010 1001 0010 2 • A416 = ? • 916 = ? • BC16 = ?

7. Unsigned Integers • Range of values for 8-bit unsigned? • What if we want to represent higher values? Say up to 300? Up to 600? • How many bytes (or memory locations) would that require?

8. Signed Integers • Easy to store positive numbers • Negative numbers? • Signed-magnitude representation • Use the last (leftmost) bit as the sign bit • 1 for negative and 0 for positive • 1000 0011 = ? • 0100 0011 = ? • 1000 0000 = ? • 0000 0000 = ?

9. Signed Integers • Problems: • Two values for zero! • Problems for comparisons • Addition won’t work well • Adding in binary? • 1000 0011 (-3) + 0100 0011 (67) -----------------1100 0110 (-70) • Obviously, we need something better • 2’s complement representation

10. 2’s Complement Representation • Used by almost all computers today • All places hold the value they would in binary except for the leftmost place which is negative • 8 bit integer: -128 64 32 16 8 4 2 1 • Range???? • If last bit • is 0, then positive  looks just as before • is 1, then negative add values for first 7 digits and then subtract 128 • 1000 1101 = 1+4+8-128 = -115 • What if I wanted to represent numbers up to -200?

11. 2’s Complement Representation • Converting fromdecimal to 2’s complement • For positive numbers: find the binary representation • For negative numbers: • Find the binary representation for its positive equivalent • Flip all bits • Add a 1 • 43 • 43 = 0010 1011 • -43 • 43 = 0010 1011  1101 0100+1  1101 0101 • 1101 0101 = -128+64+16+4+1 = -43 • We can use the usual laws of binary addition

12. 2’s Complement Representation • 125 - 19 = 106 • 0111 1101 1110 1101 ------------- 0110 1010 = 106 ! • What happened to the one that we carried at the end? • Got lost but still we got the right answer! • We carried into and carried out of the leftmost column • Try 125+65 and -125-65

13. 2’s Complement Representation • 125+65 = 190 • 0111 11010100 0001 -------------1011 1110 = - 66 !!! • We only carried intooverflow • Number too big to fit in 8 bits since range=[-128,127] • 125+65 = 190 > 127

14. 2’s Complement Representation • -125-65 = -190 • 1000 00101011 1111 -------------0100 0001 = 65 !!! • We only carried out ofoverflow • -190 < -127 • Solution • use larger registers (more than 8 bits) • Very large positive and very small negative we might still have a problem combine two registers (double precision)

15. Real Number Representation • We deal with a lot of real numbers in our lives (class average) and while using computers • Fractions or numbers with decimal parts • 3/10=0.3 or 82.34 • 82.34 = 8*10 + 2*1 + 3*1/10 + 4*1/100 • In Math: to represent a real number in binary, we use a radix point instead of a decimal point • 1101.11 = 8 + 4 + 1 + ½ + ¼ = 13.75 • On Computers: How can we represent the radix point? 0? 1?

16. Real Number Representation • We resort to the floating-point representation • We represent the number without the radix point!!! • Based on a popular scientific notation that has three parts: • Sign • Mantissa(one non-zero digit before the decimal point, and two or more after), and • Exponent(written with an Efollowing by a sign and an integer which represents what power to raise 10 to) • 6.124E5 • = 6.124 * 105 = 612,400 • Sign? Mantissa? Exponent? • Scientific to decimal • -9.14E-3 = -9.14 * 10-3 = -9.14/1000 = - 0.00914

17. Real Number Representation • Decimal to scientific • Divide (or multiply) by 10 until you have only one digit (non-zero) before the decimal point • Add (or subtract) one to the exponent every time you divide (or multiply) • 123.8 = ? • 1.238E2 • 0.2348 = ? • 2.348E-1

18. Real Number Representation • Floating-point numbers are very similar • But use only 0s and 1s instead • +/- 1.xxxx*2exponent • For an eight bit number • 00000000 • Sign, exponent, mantissa (not the standard …lab 2) • Sign = 0 for positive and 1 for negative • The exponent has three digits [0,7], but can be positive or negative: • shift by 4 [-4,3] • 000-4, 001-3, 010-2, …, 1113 • i.e. subtract 4 from the corresponding decimal value • Use base two now (i.e. 2 raised to the power of the exponent value)

19. Real Number Representation • The mantissa has one (non-zero) place before the decimal point and a number of places after it • in binary our digits are 0 and 1 and so we always have 1.xxxx • No need to represent the one (add one to the result) • 11100010  1 110 0010 • Negative • 110=6, -4  2, exponent=22 • Mantissa = 0010 = 0*1/2 + 0*1/4 + 1*1/8 + 0*1/16 = 1/8 • without the initial 1 which we’ve omitted 1+1/8 • Or 1 + decimal equivalent/16 = 1 +2/16 • - (2*2)(1+1/8) = -4 1/2

20. Real Number Representation • Floating point to decimal conversion • 1: break bit pattern into sign (1 bit), exponent (3 bits), mantissa (4 bits) • 2: sign is – if 1 and + otherwise • 3: exponent = decimal equivalent -4 • 4: mantissa = 1 + decimal equivalent/24 • 5: number = (sign) mantissa*2exponent • Try 01100011

21. Real Number Representation • What about from decimal to floating point? • Format the number into the form 1.xxxx*2exponent • Multiply (or divide) by 2 until we have a 1 before the decimal point • Subtract (or add) 1 from (to) the exponent for every such multiplication (or division) • Add 4 to result (in floating-to-decimal conversion we subtracted 4) • Convert exponent to binary • Subtract 1 from the resulting mantissa (in floating-to-decimal conversion we added 1) • Multiply mantissa by 16 (in floating-to-decimal conversion we multiplied by 16) • Round to the nearest integer • Convert mantissa to binary

22. Real Number Representation • -8.48: • Sign is negative  1 • 8.48  4.24  2.12 1.06  3 divisions • exponent=3+4 =7 or 1112 • mantissa = mantissa -1 .06 • multiply mantissa by 16 (write it in terms of 16ths)  0.96 ~ 1 0r 0001 • 1 111 0001

23. Real Number Representation • Decimal to floating-point conversion • Sign bit = 1 if negative and 0 otherwise • Exponent = 0, mantissa = absolute(number) • While mantissa < 1 do • mantissa = mantissa * 2 • exponent = exponent -1 • While mantissa > 2 do • mantissa = mantissa / 2 • exponent = exponent + 1 • Exponent = exponent + 4 • Change to binary • Mantissa = (mantissa -1)*16 • Round off to nearest integer and change to binary • Assemble number: sign | exponent | mantissa

24. Real Number Representation • 0.319 • Positive  sign bit = 02 • mantissa = 0.319 * 2 = 0.638  exponent = -1 • mantissa = 0.638 * 2 = 1.276  exponent = -2 • exponent = -2 + 4 = 2 = 0102 • mantissa = (mantissa -1)*16 = 4.416 ~ 4 = 01002 • 0 010 0100 • - 0.319 • Negative  sign bit = 12 • exponent = 0102 • mantissa = 01002 • 1 010 0100 • 0 • Sign bit = 02 • mantissa? Not going to change to 1.xxxx no matter what!

25. Real Number Representation • No representation for 0 • 0 000 0000 = 1.0 * 2-4 (we assumed there is a 1 before the mantissa) • Assume this to be zero! • Another issue, is the method exact? • rounding truncation (close numbers give same floating point values) • Mantissa = 4.416 or 4.0123 or 4.400 are all ~ 4 = 01002 • Problem is also due to using only 8 bits (4-bit mantissa) • Floating-point numbers require 32 or even 64 bits usually • But still we will have to round off and truncate • That happens regularly with us even when not using computers • PI = 22/7 • 2/3 or 1/3 • We approximate a lot … but we should know it ! • Higher precision applications use much larger size registers

26. Non-numeric Data Representation • Text data • The most common type of data people use on computers • How can we transform it to binary --- not as intuitive as numbers! • Words can be divided into characters • Each character can then be encoded by some binary code • Every language has its set of letters  we will limit ourselves to the Latin alphabet • Numbers were (relatively) easy to map to binary • Decimal to binary change • What about letters and other symbols?

27. Non-numeric Data Representation • Many transformations exist and all are arbitrary • Two popular • EBCDIC (Extended Binary Coded Decimal Interchange Code) by IBM • ASCII (American Standard Code for Information Interchange) by American National Standards Institute (ANSI) • Most widely used • Every letter/symbol is represented by 7 bits • How many letters/symbols do we have in total? • A-Z (26) , a-z (26) , 0-9 (10), symbols (33), control characters (33) • If using 1 byte/character  we have one extra bit • Extended ASCII-8 (more mathematical and graphical symbols or phonetic marks)

29. Extended Ascii (Macintosh Courier font)

30. Data types in VB 6

31. Non-numeric Data Representation • Given a memory register with the value 00110100 • 2’s complement 52 • Floating point 0.625 • ASCII character “4” (check ASCII table in book) • There are some code blocks preceding such values informing the computer of the type • Sometimes called meta-data • E.g. Font style

32. Non-numeric Data Representation • Picture/Image Data: • 512*256 image  grid has 512 columns and 256 rows • Divide the screen into a grid of cells each referred to as a pixel • Pixel values and sizes depend on the type of the image • Black & white images: 1 bit for every pixel such that 1 is black and 0 is white • Grayscale images: 1 byte/pixel where 255 is black and 0 is white and anything in between is gray (higher/lower values are closer to black/white) • Color images: Three values per pixel • Red/Green/Blue • 1 byte per color  3 bytes per pixel • For an 512x256 image (512x256x3 = 384K bytes) • For any image, we only store the pixel values

33. Assume all images have 100x80 resolution. Can you find their sizes in bits? bytes?

34. An Image File • //B/W IMAGE • P1 • 15 11 • 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 • 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 • 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 • 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 • 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 • 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 • 0 1 1 1 1 0 0 0 0 1 0 1 0 0 1 • 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 • 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 • //COLORED IMAGE • P3 • 2 7 • 255 • 255 0 0 0 255 0 • 0 0 225 255 255 0 • 255 0 225 0 255 225 • 100 150 200 200 150 100 • 50 50 50 100 100 100 • 0 0 0 255 255 225 • 5 200 225 192 150 225 • http://www.csbsju.edu/computerscience/curriculum/launch/default.htm

35. Non-numeric Data Representation • Image movies are built from a number of images (or frames) that are displayed in a certain sequence at high speeds • 30 images per second • Assume every image has a size of 500KB • 500K * 30 ~ 15 MB • 2-hr movie needs (assume same sample image used) • 15MB * 60 * 120 ~ 108 GB (billion) bytes!

36. Cycle TIME Non-numeric Data Representation Heard sound depends on the amplitudeand frequency • Sound/Audio Data • Produced when objects vibrate in matter (e.g. air) • Sends a wave of pressure fluctuations • Sounds differ because of variations in the sound wave frequency (pitch or speed) • Frequency = # cycles/second • Higher wave frequency pressure fluctuation switches back and forth more quickly during a period of time • We hear this as a higher pitch • Level of air pressure in each fluctuation, the wave's amplitudeorheight, determines how loud the sound is Volts

38. Non-numeric Data Representation • Not all frequencies are audible • Your ears are particularly sensitive to sounds in the middle range, from about 500 Hz to 2 kHz • The hi-fi range is defined as from 20 Hz to 20 kHz • As you get older, you will find it more and more difficult to hear higher frequencies • By the time you are able to afford a decent hi-fi system, you will probably be unable to fully appreciate its performance 

39. Digitizing a sound wave Non-numeric Data Representation • Numbers used to represent the amplitude of sound wave • Analog is continuous and we need digital • Digitize the sound signal • Measure the voltage of the signal at certain intervals (e.g. 44,100 per sec for CDs) • process of sampling • Reconstruct wave • Sound will slightly vary • A sound file is nothing but a sequence of numbers measured at equal intervals

40. Non-numeric Data Representation • Compression can also be used for audio files • MP3: reduces size to 1/10th •  faster transfer over the Internet

41. Non-numeric Data Representation • Digital image and audio have a lot of advantages over non-digital ones • Can easily be modified by changing the bit pattern • Image enhancement, noise/distortion removal, etc … • Superimpose one sound on another or image on another results in newer ones • Courts won’t accept them as evidence

42. Digital alterations