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CSC212 – Computer Organization and Design

CSC212 – Computer Organization and Design. Digital Electronics – William Kleitz , 9 th Ed. Chapter 01: Number Systems and Codes Chapter 03: Basic Logic Gates Chapter 05: Boolean Algebra Chapter 06: Exclusive-OR and NOR Gates Chapter 07: Arithmetic Operations & Circuits

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CSC212 – Computer Organization and Design

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  1. CSC212 – Computer Organization and Design Digital Electronics – William Kleitz, 9th Ed. Chapter 01: Number Systems and Codes Chapter 03: Basic Logic Gates Chapter 05: Boolean Algebra Chapter 06: Exclusive-OR and NOR Gates Chapter 07: Arithmetic Operations & Circuits Code Converters, Muxers and Demuxers

  2. Chapter 1 Number Systems and Codes 1

  3. Chapter Objectives • You should be able to: • Determine the weighting factor of each digit position in the decimal, binary, octal, and hexadecimal numbering systems. • Convert any number among the four number systems, and its equivalent value in any of the remaining three numbering systems. • Describe binary coded decimal (BCD) numbers. • Translate alphanumeric data to and from ASCII using the ASCII code translation table. 2

  4. Digital versus Analog • Digital • OFF and ON states that can be represented using 0s and 1s (respectively). • Analog • Continuously varying • Examples: temperature, pressure, velocity 4

  5. Discussion Points • Explain the difference between analog and digital signals. • Describe some applications for digital technology. • What are the benefits of using digital systems? • Are there any problems associated with digital systems? 6

  6. Digital vs. Analog 5

  7. Digital Representations of Analog Quantities • Audio Recording • Audio CD and MP3 players/recorders • Video Recording • DVDs store digital representations of analog video and audio signals 7

  8. Analog Signal Voltages and Their Digital Equivalents

  9. Digital-to-Analog and Back Again 8

  10. Why Digital Systems Are Immune to Analog Noise 9

  11. Digital Representations of Alternative Energy Sources • Energy technicians must keep track of the efficiency of their energy collection systems. • Naturally occurring quantities like solar, wind, and temperature are analog quantities and must be digitized before a computer can understand them.

  12. A Solar Radiation Data-logger System

  13. Decimal Numbering System (Base 10) • 10 possible digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 • Least-significant position is on the right end • Most-significant position is on the left end • Weighting factor of 10 10

  14. Binary Numbering System (Base 2) • Only two possible digits: 0 and 1 • Weighting factor of 2 • Conversion techniques • Digit times weighting factor • Successive division 11

  15. Decimal-to-Binary Conversion • Subtracting weighting factors (Example 1-4) • Successive division (Example 1-5) • First remainder is the Least-Significant Bit (LSB) • Last remainder is the Most-Significant Bit (MSB) 12

  16. Octal Numbering System (Base 8) • Eight allowable digits: 0, 1, 2, 3, 4, 5, 6, and 7 • Weighting factor of 8 13

  17. Octal Conversions • Binary to octal • Group binary positions in groups of three • Write the octal equivalent • Octal to binary • Reverse the process • Octal to decimal • Multiply by weighting factors • Decimal to octal • Successive division 14

  18. Hexadecimal Numbering System(Base 16) • 16 allowable digits. • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F • Each hex digit represents a 4-bit group • See Table 1-3 • Two hex digits are used to represent 8 bits • 8 bits are called a byte • 4 bits are called a nibble 15

  19. Hexadecimal Numbering System

  20. Hexadecimal Conversions • Binary-to-hexadecimal conversion • Group the binary in groups of four • Write the equivalent hex digit • Hexadecimal-to-binary conversion • Reverse the process 16

  21. Hexadecimal Conversions • Hexadecimal-to-decimal conversion • Multiply by weighting factors • Decimal-to-hexadecimal conversion • Successive division 17

  22. Binary-Coded-Decimal System(BCD) • Each of the 10 decimal digits is represented by its 4-bit binary equivalent. • Decimal-to-BCD conversion • Convert each decimal digit to its 4-bit binary code • BCD-to-Decimal conversion • Reverse the process 18

  23. The ASCII Code • American Standard Code for Information Interchange (ASCII) • Represents alphanumeric data • Uses 7 bits • 128 different code combinations (see Table 1-5) • 3-bit group is most significant • 4-bit group is least significant 20

  24. 21

  25. Numbering System Applications • Because digital systems must work with 1s and 0s, learning the different numbering systems is important. • Which system is used is determined by how the data were developed and how they are to be used. • Several numbering system applications follow. 22

  26. Application 1-1 The four chemical storage tanks shown are monitored for temperature (T) and pressure (P). 22

  27. Application 1-1 (continued) • Using the table shown below, interpret the following: • If the computerreads a binary string of 0010 1000 what problems exist? • This indicates that the pressure in tanks C and B are too high. 22

  28. Application 1-1 (continued) • Using the table shown below, interpret the following: • If the computerreads a hex value of 55H what problems exist? • Since 55H =0101 0101 This indicates that all tank temperatures are too high. 22

  29. Application 1-1 (continued) • Using the table shown below, interpret the following: • If the temperature and pressure in tanks B and D are too high, what hex value is read by the computer? • This condition would produce a digital output of 1100 1100 = CCH. 22

  30. Application 1-1 (continued) • Using the table shown below, interpret the following: • Assume that tanks A and B are shut down and all sensors are tied high (1s). What is the lowest decimal value that indicates a problem in the other two tanks? • With the four low-order bits tied high, the lowest value that indicates a problem is 0001 111 or 3110. 22

  31. Application 1-1 (continued) • Using the table shown below, interpret the following: • If only tanks A, B, and C are monitored, what octal value indicates tank B has both temperature and pressure problems? • The binary output would be 001 1002 = 148. 22

  32. Application 1-2 • A CD player converts 12-bit signals from a CD into equivalent analog values. • What are the largest and smallest hex values that can be used in this system? • The largest is FFF16 and the smallest is 00016. • How many different analog values can be represented? • FFF16 = 409510, so including 0 the total is 4096 unique values. 22

  33. Application 1-3 • Typically, digital thermometers use BCD to drive their displays. • How many BCD bits are required to drive a 3-digit display? • 12 bits are required; four for each digit. • What 12 bits represent 147°F? • 0001 (1), 0100 (4), and 0111 (7). 22

  34. Application 1-4 • Most PC-compatible computer systems use a 20-bit address code to identify each of over 1 million memory locations. • How many hex characters are required to identify the address of each memory location? • Five hex characters are required since each hex character represents 4 bits. 22

  35. Application 1-4 (continued) • What is the hex address of the 200th memory location? • 000C8H = 20010, but 00000H is the first memory location, so we must subtract 1. The answer is C8 – 1 = C7. • If 50 memory locations are used for data storage starting at location 000C8H, what is the location of the last data item? • C8H gets the first data item, so the answer is 24910 = F9H. 22

  36. Application 1-5 • The part number 651-M is stored in ASCII in a computer memory. List the binary contents of its memory locations? • 6 = 011 01105 = 011 01011 = 011 0001- = 010 1101M = 100 1101 • Grouping the binary bits in eights, this string represents 5 hex memory locations: 011 0110 011 0101 011 0001 010 1101 100 1101 36 35 31 2D 4D 22

  37. Application 1-6 • A programmer uses a debugging utility to find an error in a BASIC program. The utility shows the ASCII code as hex 474F5430203930. Assume that the leftmost bit of each ASCII string is padded with a zero. • The program segment is translated as GOT0 90. • The error is that a zero (0) was typed instead of the letter O. 22

  38. Summary • Numerical quantities occur naturally in analog form but must be converted to digital form to be used by computers or digital circuitry. • The binary numbering system is used in digital systems because the 1s and 0s are easily represented by ON or OFF transistors, which output 0 V for 0 and +5 V for 1. 28

  39. Summary • Any number system can be converted to decimal by multiplying each digit by its weighting factor. • The weighting factor for the least significant digit in any number system is always 1. • Binary numbers can be converted to octal by forming groups of 3 bits and to hexadecimal by forming groups of 4 bits. 29

  40. Summary • The successive division procedure can be used to convert from decimal to binary, octal, or hexadecimal • The binary-coded-decimal system uses groups of 4 bits to drive decimal displays such as those in a calculator. • ASCII is used by computers to represent all letters, numbers and symbols in digital form. 30

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