1. ECE/CS 352 Digital System Fundamentals© T. Kaminski & C. Kime 1 ECE/CS 352 Digital Systems Fundamentals Spring 2001 Chapter 1 Tom Kaminski & Charles R. Kime

2. ECE/CS 352 Digital System Fundamentals Chapter 1 2 Digital System

3. ECE/CS 352 Digital System Fundamentals Chapter 1 3 Types of Systems

4. ECE/CS 352 Digital System Fundamentals Chapter 1 4 Digital System Example:

5. ECE/CS 352 Digital System Fundamentals Chapter 1 5 A Digital Computer Example Synchronous or Asynchronous? Inputs: Keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers

6. ECE/CS 352 Digital System Fundamentals Chapter 1 6 Signals

7. ECE/CS 352 Digital System Fundamentals Chapter 1 7 Physical Signal Example - Voltage Threshold Region

8. ECE/CS 352 Digital System Fundamentals Chapter 1 8 Threshold in the News! Punched = 1 Not punched = 0 What about the rest?

9. ECE/CS 352 Digital System Fundamentals Chapter 1 9 What are other physical signals represented by 1 and 0?  CPU Voltage  Disk  CD  Dynamic RAM Other Physical Signals Magnetic Field Direction Surface Pits/Light Electrical Charge

10. ECE/CS 352 Digital System Fundamentals Chapter 1 10 Signal Examples Over Time

11. ECE/CS 352 Digital System Fundamentals Chapter 1 11 Number Systems

12. ECE/CS 352 Digital System Fundamentals Chapter 1 12 Powers of Ten

13. ECE/CS 352 Digital System Fundamentals Chapter 1 13 Positive Powers of 2

14. ECE/CS 352 Digital System Fundamentals Chapter 1 14 Important Powers of 2

15. ECE/CS 352 Digital System Fundamentals Chapter 1 15 Number Digits Decimal number digits are 0 through 9 Binary number digits are 0 through 1 Base (radix) r number digits are 0 through r - 1

16. ECE/CS 352 Digital System Fundamentals Chapter 1 16 To convert to decimal, use decimal arithmetic to sum the weighted powers of two: 11010 2 => Converting Binary to Decimal 1 x 2 4 = 16 + 1 x 2 3 = 8 + 0 x 2 2 = 0 + 1 x 2 1 = 2 + 0 x 2 0 = 0 = 26 10 __________

17. ECE/CS 352 Digital System Fundamentals Chapter 1 17 Method 1 (Method 2, Repeated Division Later)  Subtract the largest power of 2 that gives a positive result and record the power.  Repeat subtracting from the prior result until the remainder is zero.  Place 1’s in the positions in the binary result corresponding to the powers recorded; in all other positions place 0’s. Example: 625 10 Result from the listed powers: 1001110001 2 Converting Decimal to Binary – 29 29 = 625 – 512 = 113 => 9 113 – 26 26 = – 64 = 49 => 6 49 – 25 25 = – 32 = 17 => 5 17 1 – 24 24 = – 16 = 1 => 4 – 20 20 = 1 – 1 = 0 0

18. ECE/CS 352 Digital System Fundamentals Chapter 1 18 Commonly Occurring Bases

19. ECE/CS 352 Digital System Fundamentals Chapter 1 19 Numbers in Different Bases

20. ECE/CS 352 Digital System Fundamentals Chapter 1 20 General Base Conversion

21. ECE/CS 352 Digital System Fundamentals Chapter 1 21 Radix 10 Example a 3 *r 3 = 2 * 1000 = 2000 a -2 *r -2 = 7 * 0  01 = 0  07 a 2 *r 2 = 3 * 100 = 300 a 0 *r 0 = 5 * 1 = 5 a -1 *r -1 = 6 * 0  1 = 0  6 a 1 *r 1 = 4 * 10 = 40 Sum => 2,345  67 2,345  67 10 => a 3 a 2 a 1 a 0  a - 1 a - 2 = 2*1000 + 3*100 +4*10 + 5 + 6*(1/10) + 7*(1/100) (Integer part) + (Fraction part) TermActual Values Product

22. ECE/CS 352 Digital System Fundamentals Chapter 1 22 Conversion Between Bases

23. ECE/CS 352 Digital System Fundamentals Chapter 1 23 Conversion Details

24. ECE/CS 352 Digital System Fundamentals Chapter 1 24 Convert 46.6875 10 To Base 2

25. ECE/CS 352 Digital System Fundamentals Chapter 1 25 Convert Integer 46 To Base 2 Step 146 / 2 = 23 remainder = 0 Step 223 / 2 = 11 remainder = 1 Step 311 / 2 = 5 remainder = 1 Step 4 5 / 2 = 2 remainder = 1 Step 5 2 / 2 = 1 remainder = 0 Step 6 1 / 2 = 0 remainder = 1 Result 46 10 = 23 => 101110 2

26. ECE/CS 352 Digital System Fundamentals Chapter 1 26 Convert Fraction 0.6875 10 to Base 2 Step 10.6875 * 2 = 1.3750 integer = 1 Step 20.3750 * 2 = 0.7500 integer = 0 Step 30.7500 * 2 = 1.5000 integer = 1 Step 40.5000 * 2 = 1.0000 integer = 1 Step 50.0000 * 2 = 0.0000 integer = 0 Result 0.6875 10 => 0.10110 2

27. ECE/CS 352 Digital System Fundamentals Chapter 1 27 Join Integer and Fraction

28. ECE/CS 352 Digital System Fundamentals Chapter 1 28 Checking the Conversion

29. ECE/CS 352 Digital System Fundamentals Chapter 1 29 Octal to Binary and Back

30. ECE/CS 352 Digital System Fundamentals Chapter 1 30 Octal to Hexadecimal via Binary

31. ECE/CS 352 Digital System Fundamentals Chapter 1 31 A Final Conversion Note

32. ECE/CS 352 Digital System Fundamentals Chapter 1 32 Binary Numbers and Coding

33. ECE/CS 352 Digital System Fundamentals Chapter 1 33 Enumerating elements

34. ECE/CS 352 Digital System Fundamentals Chapter 1 34 Example: Radix 2, 3 digits

35. ECE/CS 352 Digital System Fundamentals Chapter 1 35 Representing Numeric Information

36. ECE/CS 352 Digital System Fundamentals Chapter 1 36 Binary Coding

37. ECE/CS 352 Digital System Fundamentals Chapter 1 37 Number of Bits Required

38. ECE/CS 352 Digital System Fundamentals Chapter 1 38 Min. and Max. Digits Required

39. ECE/CS 352 Digital System Fundamentals Chapter 1 39 Binary Codes for Decimal Digits

40. ECE/CS 352 Digital System Fundamentals Chapter 1 40 Binary Coded Decimal (BCD)

41. ECE/CS 352 Digital System Fundamentals Chapter 1 41 Other Decimal Codes

42. ECE/CS 352 Digital System Fundamentals Chapter 1 42 Warning: Conversion or Coding?

43. ECE/CS 352 Digital System Fundamentals Chapter 1 43 Binary Addition

44. ECE/CS 352 Digital System Fundamentals Chapter 1 44 Extending this to multiple digits: Carries 0 0 Augend 01100 10110 Addend +10001 +10111 Sum Note: The underlined “0” is a Carry-In to the least digit. Binary Addition (Extended) 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1

45. ECE/CS 352 Digital System Fundamentals Chapter 1 45 Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B): Borrow in (Z) of 0: Borrow in (Z) of 1: Binary Subtraction Z 1 1 1 1 X 0 0 1 1 - Y -0 -0 BS 11 1 0 0 1 Z 0 0 0 0 X 0 0 1 1 - Y -0 -0 BS 0 1 0 1 0

46. ECE/CS 352 Digital System Fundamentals Chapter 1 46 Extending this to multiple digits: Borrows 0 0 Minuend 10110 10110 Subtrahend - 10010 - 10011 Difference Note: If the Subtrahend is larger than the Minuend, interchange and append a – to the result. The underlined “0” is a Borrow-In to the least digit. Binary Subtraction (Extended) 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0

47. ECE/CS 352 Digital System Fundamentals Chapter 1 47 Binary Multiplication

48. ECE/CS 352 Digital System Fundamentals Chapter 1 48 BCD Arithmetic

49. ECE/CS 352 Digital System Fundamentals Chapter 1 49 BCD Addition Example Add 1897 BCD to 2905 BCD 0001 1000 1001 0111 + 0010 1001 0000 0101 0 1100 101010010 0100 + 0110 1 + 0000 0010 0000 1 1000 0100 1

50. ECE/CS 352 Digital System Fundamentals Chapter 1 50 Error-Detection Codes

51. ECE/CS 352 Digital System Fundamentals Chapter 1 51 3-Bit Parity Code Example

52. ECE/CS 352 Digital System Fundamentals Chapter 1 52 ASCII Character Codes

53. ECE/CS 352 Digital System Fundamentals Chapter 1 53 ASCII Properties

54. ECE/CS 352 Digital System Fundamentals Chapter 1 54 Other Character Codes

55. ECE/CS 352 Digital System Fundamentals Chapter 1 55 Other Character Codes UNICODE extends ASCII to 65,536 universal characters codes  For encoding characters in world languages  Available in many modern applications  2 byte (16-bit) code words  See Supplement in Chapter 1 on Companion Website http://www.prenhall.com/mano if you are interestedhttp://www.prenhall.com/mano