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SYEN 3330 Digital Systems

SYEN 3330 Digital Systems. Chapter 1. Digital System. Types of Systems. Digital System Example:. A Digital Computer Example. Synchronous or Asynchronous?. Outputs: CRT, LCD, modem, speakers. Inputs: Keyboard, mouse, modem, microphone. Signals. Physical Signal Example - Voltage.

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SYEN 3330 Digital Systems

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  1. SYEN 3330 Digital Systems Chapter 1 SYEN 3330 Digital Systems

  2. Digital System SYEN 3330 DIGITAL SYSTEMS

  3. Types of Systems SYEN 3330 DIGITAL SYSTEMS

  4. Digital System Example: SYEN 3330 DIGITAL SYSTEMS

  5. A Digital Computer Example Synchronous or Asynchronous? Outputs: CRT, LCD, modem, speakers Inputs: Keyboard, mouse, modem, microphone SYEN 3330 DIGITAL SYSTEMS

  6. Signals SYEN 3330 DIGITAL SYSTEMS

  7. Physical Signal Example - Voltage Threshold Region SYEN 3330 DIGITAL SYSTEMS

  8. Threshold in the News! • Punched = 1 • Not punched = 0 • What about the rest? SYEN 3330 DIGITAL SYSTEMS

  9. Other Physical Signals • What are other physical signals represented by 1 and 0? • CPU Voltage • Disk • CD • Dynamic RAM Magnetic Field Direction Surface Pits/Light Electrical Charge SYEN 3330 DIGITAL SYSTEMS

  10. Signal Examples Over Time SYEN 3330 DIGITAL SYSTEMS

  11. Number Systems SYEN 3330 DIGITAL SYSTEMS

  12. Powers of Ten SYEN 3330 DIGITAL SYSTEMS

  13. Positive Powers of 2 SYEN 3330 DIGITAL SYSTEMS

  14. Important Powers of 2 SYEN 3330 DIGITAL SYSTEMS

  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 SYEN 3330 DIGITAL SYSTEMS

  16. Converting Binary to Decimal • To convert to decimal, use decimal arithmetic to sum the weighted powers of two: • 110102 => 1 x 24 = 16 + 1 x 23 = 8 + 0 x 22 = 0 + 1 x 21 = 2 + 0 x 20 = 0 __________ = 2610 SYEN 3330 DIGITAL SYSTEMS

  17. Converting Decimal to Binary • 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: 62510 Result from the listed powers: 10011100012 – 29 = 625 – 512 = 113 => 9 113 – 26 = 113 – 64 = 49 => 6 49 – 25 = 49 – 32 = 17 => 5 – 24 = 17 – 16 = 1 => 4 17 – 20 = 1 – 1 = 0 => 0 1 SYEN 3330 DIGITAL SYSTEMS

  18. Commonly Occurring Bases SYEN 3330 DIGITAL SYSTEMS

  19. Numbers in Different Bases SYEN 3330 DIGITAL SYSTEMS

  20. General Base Conversion SYEN 3330 DIGITAL SYSTEMS

  21. Term Actual Values Product Radix 10 Example 2,345 6710 => a3a2a1a0 a-1a-2 = 2*1000 + 3*100 +4*10 + 5 + 6*(1/10) + 7*(1/100) (Integer part) + (Fraction part) a3*r3 = 2 * 1000 = 2000 a2*r2 = 3 * 100 = 300 a1*r1 = 4 * 10 = 40 a0*r0 = 5 * 1 = 5 a-1*r-1 = 6 * 01 = 06 a-2*r-2 = 7 * 001 = 007 Sum => 2,34567 SYEN 3330 DIGITAL SYSTEMS

  22. Conversion Between Bases SYEN 3330 DIGITAL SYSTEMS

  23. Conversion Details SYEN 3330 DIGITAL SYSTEMS

  24. Convert 46.687510 To Base 2 SYEN 3330 DIGITAL SYSTEMS

  25. Convert Integer 46 To Base 2 Step 1 46 / 2 = 23 remainder = 0 Step 2 23 / 2 = 11 remainder = 1 Step 3 11 / 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 4610= 23 => 1011102 SYEN 3330 DIGITAL SYSTEMS

  26. Convert Fraction 0.687510 to Base 2 Step 1 0.6875 * 2 = 1.3750 integer = 1 Step 2 0.3750 * 2 = 0.7500 integer = 0 Step 3 0.7500 * 2 = 1.5000 integer = 1 Step 4 0.5000 * 2 = 1.0000 integer = 1 Step 5 0.0000 * 2 = 0.0000 integer = 0 Result 0.687510=> 0.101102 SYEN 3330 DIGITAL SYSTEMS

  27. Join Integer and Fraction SYEN 3330 DIGITAL SYSTEMS

  28. Checking the Conversion SYEN 3330 DIGITAL SYSTEMS

  29. Octal to Binary and Back SYEN 3330 DIGITAL SYSTEMS

  30. Octal to Hexadecimal via Binary SYEN 3330 DIGITAL SYSTEMS

  31. A Final Conversion Note SYEN 3330 DIGITAL SYSTEMS

  32. Binary Numbers and Coding SYEN 3330 DIGITAL SYSTEMS

  33. Enumerating elements SYEN 3330 DIGITAL SYSTEMS

  34. Example: Radix 2, 3 digits SYEN 3330 DIGITAL SYSTEMS

  35. Representing Numeric Information SYEN 3330 DIGITAL SYSTEMS

  36. Binary Coding SYEN 3330 DIGITAL SYSTEMS

  37. Number of Bits Required SYEN 3330 DIGITAL SYSTEMS

  38. Min. and Max. Digits Required SYEN 3330 DIGITAL SYSTEMS

  39. Binary Codes for Decimal Digits SYEN 3330 DIGITAL SYSTEMS

  40. Binary Coded Decimal (BCD) SYEN 3330 DIGITAL SYSTEMS

  41. Other Decimal Codes SYEN 3330 DIGITAL SYSTEMS

  42. Warning: Conversion or Coding? SYEN 3330 DIGITAL SYSTEMS

  43. Binary Addition SYEN 3330 DIGITAL SYSTEMS

  44. Binary Addition (Extended) • Extending this to multiple digits: Carries 00 Augend 01100 10110 Addend +10001+10111 Sum Note: The underlined “0” is a Carry-In to the least digit. 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 1 SYEN 3330 DIGITAL SYSTEMS

  45. Z Z 0 1 0 1 1 0 1 0 X X 0 0 0 0 1 1 1 1 BS BS 11 0 0 1 1 1 0 0 1 0 0 0 0 1 1 - Y - Y -0 -0 -1 -1 -0 -0 -1 -1 Binary Subtraction • 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: SYEN 3330 DIGITAL SYSTEMS

  46. Binary Subtraction (Extended) • Extending this to multiple digits: Borrows00 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. 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 SYEN 3330 DIGITAL SYSTEMS

  47. Binary Multiplication SYEN 3330 DIGITAL SYSTEMS

  48. BCD Arithmetic SYEN 3330 DIGITAL SYSTEMS

  49. BCD Addition Example • Add 1897BCD to 2905BCD 1 0 1 1 0001 1000 1001 0111 + 0010100100000101 1100 0100 10010 1010 + 0110 + 0110 + 0000 + 0110 0100 0010 1000 0000 SYEN 3330 DIGITAL SYSTEMS

  50. Error-Detection Codes SYEN 3330 DIGITAL SYSTEMS

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