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EKT 121 / 4 ELEKTRONIK DIGIT 1

EKT 121 / 4 ELEKTRONIK DIGIT 1. CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC. Number & codes (1). Digital vs. Analog Numbering systems Decimal (Base 10) Binary (Base 2) Hexadecimal ( Base 16) Octal ( Base 8) Number conversion Binary arithmetic

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EKT 121 / 4 ELEKTRONIK DIGIT 1

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  1. EKT 121 / 4ELEKTRONIK DIGIT 1 CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC

  2. Number & codes (1) • Digital vs. Analog • Numbering systems • Decimal (Base 10) • Binary (Base 2) • Hexadecimal (Base 16) • Octal (Base 8) • Number conversion • Binary arithmetic • 1’s and 2’s complements of binary numbers

  3. Number & codes (2) • Signed/Unsigned numbers • Arithmetic operations with signed numbers • Coded • Binary-Coded-Decimal (BCD)/ 8421 • ASCII • Gray • Excess-3) • Error Detecting and Correction Codes • Floating Point Numbers

  4. Digital vs. Analog • Two ways of representing the numerical values of quantities : i) Analog (continuous) ii) Digital (discrete) • Analog : a quantity represented by voltage, current or meter movement that is proportional to the value that quantity. • Digital : the quantities are represented not by proportional quantities but by symbols called digits (0/1).

  5. Digital vs. Analog (cont.) • Digital system: • combination of devices designed to manipulate logical information or physical quantities that are represented in digital forms • Analog system: • contains devices manipulate physical quantities that are represented in analog forms

  6. Digital vs. Analog (cont.)

  7. Digital vs. Analog (cont.) • Why digital ? • Problem with all signals – noise • Noise isn't just something that you can hear - the fuzz that appears on old video recordings also qualifies as noise. In general, noise is any unwanted change to a signal that tends to corrupt it. • Digital and analogue signals with added noise: Digital : easily be recognized even among all that noise : either 0 or 1 Analog : never get back a perfect copy of the original signal

  8. Digital Techniques • Advantages: • Easier to design • Information storage is easy • Accuracy and precision are greater • Operation can be programmed - simple • Digital circuits less affected by noise • More digital circuitry can be fabricated on IC chips • Limitations: • In real world there are analog in nature and these quantities are often I/O that are being monitored, operated on, and controlled by a system. Thus, conversion and re-conversion in needed

  9. Analog Waveform

  10. Digital Waveform

  11. Introduction to Numbering Systems • We are familiar with decimal number systems for daily used such as calculator, calendar, phone or any common devices use this numbering system : Decimal = Base 10 • Some other number systems: • Binary = Base 2 • Octal = Base 8 • Hexadecimal = Base 16

  12. Numbering Systems • Decimal • Binary • Octal • Hexadecimal • 0 ~ 9 • 0 ~ 1 • 0 ~ 7 • 0 ~ 9, A ~ F

  13. Dec Hex Octal Binary 000000010010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 0123456789ABCDEF 0123456789101112131415 00010203040506071011121314151617 Numbering Systems (cont.) N U M B E R S Y S T E M S

  14. Significant Digits Binary : 1 0 1 1 0 1 Most Significant Bit Least Significant Bit (MSB) (LSB) Hexadecimal: 1 D 6 3 A 7 Most Significant Digit Least significant Digit (MSD) (LSD)

  15. Decimal numbering system (base 10) Weights for whole numbers are positive power of ten that increase from right to left , beginning with 100 • Base 10 system: (0,1,2,3,4,5,6,7,8,9) • Example : 39710 3 9 7 + 7 X 100 3 X 102 9 X 101 + => 300 + 90 + 7 39710 =>

  16. Binary Number System (base 2) • Base 2 system: (0 , 1) • used to model the series of computer electrical signals represent the informations. • 0 represents the no voltage or an ‘off’ state • 1 represents the presence of voltage or an ‘on’ state • Example: 1012 1 0 1 Weights in a binary number are based on power of two, that increase from right to right to left, beginning with 20 + 0 X 21 + 1 X 20 1X 22 => 4 + 0 + 1 => 510

  17. Octal Number System (base 8) • Base 8 system: (0,1,………,7) • multiplication and division algorithms for conversion to and from base 10 • example: 7568 convert to decimal 7 5 6 Weights in a binary number are based on power of eight that increase from right to right to left, beginning with 80 + 7X 82 5 X 81 + 6 X 80 => 448 + 40 + 6 49410 => • Readily converts to binary • Groups of three (binary) digits can be used to represent each octal number • example : 7568 convert to binary • 7 5 6 1111011102

  18. Hexadecimal Number System (base 16) • Base 16 system • Uses digits 0 ~ 9 & letters A,B,C,D,E,F • Groups of four bits represent each base 16 digit

  19. Hexadecimal Number System (2) • Base 16 system • multiplication and division algorithms for conversion to and from base 10 • example : A9F16 convert to decimal A 9 F Weights in a hexadecimal number are based on power of sixteen that increase from right to right to left,beginning with 160 10X 162 9 X 161 + 15 X 160 + => 2560 + 144 + 15 271910 => • Readily converts to binary • Groups of four (binary) digits can be used to represent each hexadecimal number • example : A9F16 convert to binary • A 9 F 1010100111112

  20. Number Conversion • Any Radix (base) to Decimal Conversion

  21. Number Conversion (BASE 2 –> 10) • Binary to Decimal Conversion

  22. Binary to Decimal Conversion • Convert (10101101)2 to its decimal equivalent: Binary 1 0 1 0 1 1 0 1 Positional Values x x x x x x x x 27 26 25 24 23 22 21 20 Products 128 + 0 + 32 + 0 + 8 + 4 + 0 + 1 = 17310

  23. Octal to Decimal Conversion • Convert 6538 to its decimal equivalent: Octal Digits 6 5 3 x x x Positional Values 82 81 80 Products 384 + 40 + 3 = 42710

  24. Hexadecimal to Decimal Conversion • Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F x x x x Positional Values 163 162 161 160 12288 + 2816 +64 +15 Products = 15,18310

  25. Number Conversion • Decimal to Any Radix (Base) Conversion • INTEGER DIGIT: Repeated division by the radix & record the remainder • FRACTIONAL DECIMAL: Multiply the number by the radix until the answer is in integer • example: 25.3125 to Binary

  26. Decimal to Binary Conversion Remainder 2 5 = 12 + 1 2 1 2 = 6 + 0 2 6 = 3 + 0 2 3 = 1 + 1 2 MSB LSB 1 = 0 + 1 2 2510 = 1 1 0 0 1 2

  27. Decimal to Binary Conversion MSB LSB Carry . 0 1 0 1 0.3125 x 2 = 0.625 0 0.625 x 2 = 1.25 1 0.25 x 2 = 0.50 0 0.5 x 2 = 1.00 1 Answer: 1 1 0 0 1.0 1 0 1

  28. Decimal to Octal Conversion Convert 42710 to its octal equivalent: 427 / 8 = 53 R3 Divided by 8; R is LSD 53 / 8 = 6 R5 Divide Q by 8; R is next digit 6 / 8 = 0 R6 Repeat until Q = 0 6538

  29. Decimal to Hexadecimal Conversion Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R 14 51 / 16 = 3 R3 3 / 16 = 0 R3 = E in Hex 33E16

  30. Decimal to Octal Conversion • Binary to Octal Conversion (vice versa) • Grouping the binary position in groups of three starting at the least significant position.

  31. Octal to Binary Conversion • Each octal number converts to 3 binary digits To convert 6538 to binary, just substitute code: 6 5 3 110 101 011

  32. Example : Number Conversion • Convert the following binary numbers to their octal equivalent (vice versa). • 1001.11112 • 47.38 • 1010011.110112 Answer: • 11.748 • 100111.0112 • 123.668

  33. Binary to Hexadecimal Conversion • Binary to Hexadecimal Conversion (vice versa) • Grouping the binary position in 4-bit groups, starting from the least significant position.

  34. Binary to Hexadecimal Conversion • The easiest method for converting binary to hexadecimal is using a substitution code • Each hex number converts to 4 binary digits

  35. Number Conversion • Example: • Convert the following binary numbers to their hexadecimal equivalent (vice versa). • 10000.12 • 1F.C16 Answer: • 10.816 • 00011111.11002

  36. Substitution Code (1) Convert (010101101010111001101010)2 to hex using the 4-bit substitution code : 0101 0110 1010 1110 0110 1010 5 6 A E 6 A = 56AE6A16

  37. Substitution Code (2) Substitution code can also be used to convert binary to octal by using 3-bit groupings: 010 101 101 010 111 001 101 010 2 5 5 2 7 1 5 2 = 255271528

  38. Binary Addition 0 + 0 = 0 Sum of 0 with a carry of 0 0 + 1 = 1 Sum of 1 with a carry of 0 1 + 0 = 1 Sum of 1 with a carry of 0 1 + 1 = 10 Sum of 0 with a carry of 1 Example: 11001 111 + 1101 + 11 100110 ???

  39. Application of counting in binary (pg 50-51 textbook)

  40. Application of counting in binary (pg 50-51 textbook)

  41. Binary Subtraction 0 - 0 = 0 1 - 1 = 0 1 - 0 = 1 10 -1 = 1 0 -1 with a borrow of 1 Example: 1011 101 - 111 - 11 100 ???

  42. Simple Arithmetic Example: 5816 + 2416 7C16 • Addition Example: 100011002 + 1011102 101 1 10102 • Substraction Example: 10001002 - 1011102 101102

  43. Binary Multiplication 0 X 0 = 0 0 X 1 = 0 Example: 1 X 0 = 0 100110 1 X 1 = 1 X 101 100110 000000 + 100110 10111110

  44. Binary Division • Use the same procedure as decimal division

  45. 1’s complements of binary numbers • Changing all the 1s to 0s and all the 0s to 1s Example: 1 1 0 1 0 0 1 0 1 Binary number 0 0 1 0 1 1 0 1 0 1’s complement ****** same as applying NOT gate ******

  46. 2’s complements of binary numbers • 2’s complement • Step 1: Find 1’s complement of the number Binary # 11000110 1’s complement 00111001 • Step 2: Add 1 to the 1’s complement 00111001 + 1 00111010

  47. Information about signed binary numbers (pg 62 textbook) • A signed binary number consists of both sign and magnitude information • The sign indicates whether a number is positive or negative • The magnitude is the value of number • There are 3 forms in which a sign number can be represented • Sign magnitude (least used) • 1’s complement • 2’s complement (most important because computer use 2’s complement for negative number in arithmetic operation) • F.Y.I->Non-integer and very large or very small number can be expressed in floating point form

  48. Signed Magnitude Numbers 110010.. …00101110010101 Sign bit 31 bits for magnitude 0 = positive 1 = negative ***** This is your basic Integer format

  49. Sign numbers • Left most is the sign bit • 0 is for positive, and 1 is for negative • Sign-magnitude 0 0 0 1 1 0 0 1 = +25 sign bit magnitude bits 10 0 1 1 0 0 1 = -25 sign magnitude bits • 1’s complement • The negative number is the 1’s complement of the corresponding positive number • Example: +25 is 00011001 -25 is 11100110

  50. Sign numbers • 2’s complement • Example: +25 is 00011001 • The negative number is the 2’s complement of the corresponding positive number. • Thus, -25 in 2’s complement form is 11100111 Example: Express +19 and -19 in i. sign magnitude ii. 1’s complement iii. 2’s complement

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