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DIGITAL ELECTRONICS - Basics

Review of number systems-representation-conversions, Review of Boolean algebra- theorems, sum of product and product of sum simplification, canonical forms min term and max term, Simplification of Boolean expressions-Karnaugh map, Implementation of Boolean expressions using universal gates ,Tabulation methods.<br>

Helen56
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DIGITAL ELECTRONICS - Basics

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  1. DIGITAL SYSTEMS DESIGN-Basic concepts Dr. C.HelenSulochana Professor/ECE St.Xaviers Catholic College of Engg. Chunkankadai

  2. Digital Electronics • Any Quantities can be measured, monitored, recorded, manipulated Their values are represented as 1. Analog(continuous) eg. voltage, current, power, energy, temp. variation 2. Digital(discrete- symbols called digits) eg. Digital clock Analog Digital • Analog to digital (A/D)convertor

  3. Advantages • economical and easy to design. • very well suited for both numerical and non-numerical information processing. • high speed and high noise immunity • easy to duplicate similar circuits and complex digital ICs are manufactured Applications • Communication • Business transactions • traffic control • spacecraft guidance • medical treatment, weather monitoring, • Internet • Commercial industry- digital telephones, digital televisions, digital cameras,

  4. BASIC CONCEPTS(DIGITAL FUNDAMENTALS) • Number Systems • Boolean Algebra • Karnaugh map Minimization • Tabulation Method

  5. Number Systems • Types of number systems -mathematical notation for representing numbers radix(r) or base -number of unique digits, including the digit zero, used to represent numbers.  1. Decimal number system (Base - 10) 2. Binary number system (Base - 2) 3. Octal number system (Base - 8) 4. Hexadecimal number system (Base – 16)

  6. Common Number system 10 11 15

  7. Ten digits : 0, 1, 2, 3……9 -Base or radix is 10 -coefficients are multiplied by powers of 10 -every digit position has a weight which is a power of 10 1.Decimal number system (weight) Example 7392 =7x 103 + 3x102 + 9x 101 + 2x 100 257.45 = 2x 102 + 5x101 + 7x 100 + 4x10-1 +5x 10-2

  8. 2. Binary number system • digital systems use just two discrete values • Two digits 0 and 1 • -every digit position has a weight which is a power of 2 • -Base or radix is 2 A binary digit is called as a bit eight bits- one byte • Computers use binary numbers(it has many digits than decimal Example (110)2= 1x 22 + 1x21 + 0 x 20 (101.01)2= 1x 22 + 0x21 + 1x 20 + 0 x 2-1 +1x 2-2

  9. 3. Octal number system Eight digits : 0, 1, 2, 3……7 -every digit position has a weight which is a power of 8 -Base or radix is 8 • Octal is the Shorthand for binary • Large base(not too many digit) • Easy to convert between octal and binary • Not an efficient representation of byte(7 is 111) Example (127.4)8= 1x 82 + 2x81 + 7x 80 + 4x8-1

  10. 4. Hexadecimal number system A compact way to represent binary numbers -Group of four binary digits are represented by a hexadecimal digit -Base or radix is 16 -Hexa decimal degits are 0 to 9, A to F • Preferred Shorthand for binary • Efficiently represent the byte Example (B65F)16= 11 x 163 + 6x162 + 5x 161 + 15x160

  11. Number base Conversion to Representations of a number in a different radix • DecimalBinary conversion • Decimal • Binary Octal Octal Hexadecimal Hexadecimal

  12. Numbers with Different Bases

  13. 1. Decimal Binary(base 2) conversion Decimal to Binary Binary to Decimal • expanding the number in a power series of base r (2) • adding all the terms • dividing the number and all successive quotients by base r(2) • accumulating the remainders.

  14. Decimal fraction to binary • Decimal to Binary • multiplying the number and all successive fractionby the base r(2) until the fraction becomes 0 • integers are accumulated Convert decimal number 112 into binary number Decimal fractional number 0.8125 into binary Ans : (112)2 = (1110000)2

  15. Binary to decimal Binary fraction to Decimal converting the integer and the fraction separately and then combining the two answers Example 3

  16. 2. Decimal Octal conversion Decimal to octal Octal to Decimal

  17. Decimal fraction to octal

  18. 3. Decimal Hexadecimal conversion Hexadecimal to Decimal Decimal to Hexadecimal

  19. Hexadecimal fraction to Decimal

  20. 4. Binary octal conversion Binary to octal octal to binary • partitioning the binary into groups of three digits from the binary point to the left and to the right. • The corresponding octal digit is then assigned to each group Each octal digit is converted to its three‐digit binary equivalent

  21. Hexadecimal to binary Binary to Hexadecimal • partitioning the binary into groups of fourdigits from the binary point to the left and to the right. • The corresponding Hexadecimal digit is then assigned to each group • Each octal digit is converted to its three‐digit binary equivalent 4. Binary Hexadecimal conversion computer manuals use either octal or hexadecimal numbers to specify binary quantities

  22. Representation of numbers • Unsigned integers • - n bit number has 2n distinct combinations • For n=3 , 23 =8 combinations • 000, 001, 010, 011, 100, 101, 110,111 • Signed integers • -Positive or negative number • Three possible methods of representation • 1. Sign magnitude representation • 2. 1’s compliment representation • 3. 2’s compliment representation

  23. 2’s compliment representation 2. 1’s compliment representation • Computation of 2’s complement • – Perform 1’s Complement • - then add one to the resulting number. Invert all bits. Each 1 becomes a 0 and 0 becomes 1 -Negative numbers are represented as 1’s complement form • Example,n=4 1000 --- -7 1001 --- -6 1010 --- -5 1011 --- -4 1100 --- -3 1101 --- -2 1110 --- -1 1111 --- -0 0000 --- +0 0001 --- +1 0010 --- +2 0011 --- +3 0100 --- +4 0101 --- +5 0110 --- +6 0111---- +7 0000 --- +0 0001 --- +1 0010 --- +2 0011 --- +3 0100 --- +4 0101 --- +5 0110 --- +6 0011---- +7 1000 --- -8 1001 --- -7 1010 --- -6 1011 --- -5 1100 --- -4 1101 --- -3 1110 --- -2 1111 --- -1 +4 = 0100 -4 = 1’s complement of 0100 = 1011 +4 = 0100 -4 = 2’s complement of 0100 = 1011+1 = 1100

  24. Subtraction via addition using the 1’s complement

  25. Example 2 : 3 - 5 binary of 5 = 0101 1’s complement of 5 =1010 3 : 0011 -5 : 1010 1101 = -2 Example 1: 6 - 2 binary of 2 = 0010 1’s complement of 2 =1101 6 : 0110 -2 : 1101 1 0011 1 0100 = +4 • Using 1’s complement 0010=-2 • No Carry is added back to the result • Result is negative • Take 1’s complement • Consider 4 bit representation • Carry is added back to the result • Result is positive

  26. Subtraction via addition using the two’s complement

  27. Example 1 : 6-2 Example 2:3- 5 • 0001+1=0010=-2

  28. CODES Coding is the process of altering the characteristics of information to make it more suitable for intended applications. Weighted Codes : each binary digit is assigned a specific weight eg. BCD code (8421 code) and 1 2 4 8. Non-Weighted Code - no positional weighting system eg. ex- Excess-3, Gray code, ASCII Code, EBCDIC

  29. BINARY CODES • Digital systems use two distinct values (two stable states-0 & 1) • Any discrete element is represented with unique binary code • Binary code -combinations of 0’s and 1’s(used in computer). n‐bit binary code(minimum) - group of n bits 2ndistinct combination

  30. 1.Binary-Coded Decimal Code(8421 code) • Computer use binary system, people use the decimal system. computer A binary code that distinguishes among 10 elements - contain at least four bits,(6 unassigned combinations) Binary code Arithmetic operation Decimal Decimal • BCD- replace a decimal digit(number) with an individual binary code of 4 bits Decimal to BCD • BCD to Decimal

  31. BCD code has weights of 8, 4, 2, and 1, correspond to each bit. • eg. (0110)BCD= 8 * 0 + 4 * 1 + 2 * 1 + 1 * 0 = 6. • for one decimal digit- 4 bit code • k decimal digit- 4k bit code • Decimal 396 is represented with 12 bits • Decimal number in BCD = its binary , if the number is between 0 and 9. • binary combinations 1010 to 1111 are not used in BCD

  32. Used in seven segment display • Disadvantage • needs more bits than its equivalent binary value Advantage use of decimal numbers(input and output data are generated as decimal )

  33. BCD Addition • BCD sum cannot be greater than 9 + 9 + 1 = 19(range from 0 to 19, In binary- 0000 to 10011, BCD- 0000 to 1 1001) • binary sum is to 1001 (without a carry), the BCD digit is correct. • when the binary sum is to 1010 or with a carry, the result is an invalid BCD digit. • Add 6 = (0110)2 to the binary sum converts to correct digit with required carry

  34. 2. Excess‐3 code • unweighted code • obtained by adding 3 to the binary value • self‐complementing codes (9’s complement of a decimal number is obtained by changing 1’s to 0’s and 0’s to 1’s)

  35. 3. Gray Code • continuous quantities must be converted into digital by analog‐to‐digital converter. • For digital representation Gray code is used • H Called as unit distance code • Advantage • Only one bit changes from one number to other • Reduces the error during the transition Binary to Gray Gray to Binary

  36. Binary to Gray procedure MSB of binary = MSB of gray To get the next bit of gray code, add MSB of binary with the next bit of binary Repeat step 2 until last bit of binary Gray to Binary procedure • MSB of gray =MSB of binary • To get the next bit of binary code, add MSB of binary with the next bit of gray • Repeat step 2 until last bit of gray

  37. 4. Alphanumeric codes eg. a.American Standard Code for Information Interchange (ASCII) b. Extended binary coded decimal interchange code (EBCDIC) • Binary code for alphanumeric characters

  38. a.American Standard Code for Information Interchange (ASCII) • seven bit code - b1(LSB) to b7(MSB) • ASCII code contains • 94 graphic characters(26 uppercase letters (A through Z), the 26 lowercase • letters (a through z), the 10 numerals (0 through 9), and 32 special printable characters, such as %, *, and $). • control characters -control functions for printing and nonprinting Three types of control characters: Format effectors -control the layout of printing-backspace (BS), horizontal tabulation (HT), and carriage return (CR). Information separators - to separate the data into divisions such as paragraphs and pages-record separator (RS) and file separator (FS). Communication‐control characters - useful during the transmission of text between remote device-STX (start of text) and ETX (end of text),

  39. 1000001 (column 100, row 0001)

  40. b. Extended binary coded decimal interchange code (EBCDIC) • 8-bit binary code for numeric and alphanumeric characters. • 128 characters in ASCII, 256 in EBCDIC • Binary form in ASCII, BCD form in EBCDIC

  41. Boolean algebra • Developed by George Boole • Algebra of binary logic or algebra of two values i.e True/False or Yes/ No Applications of Boolean Algebra • Used to perform logical operations in computer • In digital computer True represented by 1(high voltage) and False represented by 0(low voltage) • Logical operations are performed by logical operators • Fundamental logical operators 1. AND (conjuction) 2. OR (disjuction) 3. NOT(negation/complement)

  42. 1. AND operator x and y - input variable(takes value 0 or 1) z - output variable z = x.y This is called Boolean expression or Boolean function- containsvariables and constants(eg. A=B+1) Represented in a truth table Truth tables - Contains all possible combinations of variables present in the Boolean expression 2 variables– 22=4 combinations n variables – 2n combinations • similar to multiplication

  43. 1. AND operator • similar to multiplication x and y - input variable(takes value 0 or 1) z - output variable z = x.y This is called Boolean expression or Boolean function Truth tables - Contains all possible combinations of variables present in the Boolean expression 2 variables– 22=4 combinations n variables – 2n combinations

  44. 2. OR operator • similarities to Addition x and y - input variable(takes value 0 or 1) z - output variable z = x+y • binary logic binary arithmetic • in • binary arithmetic, 1 + 1 = 10(2) • Binary logic, 1 + 1 = 1

  45. 3. NOT operator • Performs logical Negation • Operates on single variable A- input variable(takes value 0 or 1) C - output variable (complement of A)

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