Digital Logic: Chapter 2

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Digital LogicChapter 2

• Number System, Operation And Codes

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Objectives

• Review the decimal number system
• Count in the binary number system
• Convert from decimal to binary and from binary to
  decimal
• Apply arithmetic operations to binary numbers
• Determine the 1's and 2's complements of a binary
  number
• Express signed binary numbers in sign-magnitude,
  1's complement, 2's complement, and
  floating-point format
• Carry out arithmetic operations with signed
  binary numbers
• Convert between the binary and hexadecimal number
  systems
• Add numbers in hexadecimal form
• Convert between the binary and octal number
  systems
• Express decimal numbers in binary coded decimal
  (BCD) form
• Add BCD numbers
• Convert between the binary system and the Gray
  code
• Interpret the American Standard Code for
  Information Interchange (ASCII)
• Explain how to detect and correct code errors

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Chapter 2 Overview

• Numbers
• Decimal
• Binary
• Hexadecimal
• Octal
• Conversions
• Binary Coded Decimal (BCD)
• Digital Codes and Parity
• Digital System Application

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Decimal Numbers

• The decimal numbering system has 10 digits 0
  through 9
• The decimal numbering system has a base of 10
  with each position weighted by a factor of 10
• Express decimal 47 as a sum of the values of each
  digit
• 47 (4 x 101) (7 x 100) 40 7
  47

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• Express the decimal number 568.23 as a sum of the
  values of each digit.

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Binary Numbers

• The binary numbering system has 2 digits 0 and 1
• The binary numbering system has a base of 2 with
  each position weighted by a factor of 2

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Binary Count
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Binary-to-Decimal Conversion

• Step 1 Get n, total binary numbers
• 1101101 ? n 7 binary numbers
• Step 2 Conversion will stop at 2n-1 26
• Step 3 Multiply all the numbers with its
  appropriate weight
• Step 4 Sum the result from step 3, and binary is
  now decimal.

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• Step 1 Binary number 100101 ? n 6
• Step 2 Stop at 2n-1 25

Step 3 multiply
Step 4 sum
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Decimal to Binary Conversion

• Two methods to convert decimal to binary
• Sum-of-weights method
• Determine the set of binary weights whose sum is
  equal to the decimal number
• Repeated division-by-2 method
• Repeatedly dividing the decimal number by 2 and
  retrieving the remainder

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Sum-of-weight method

• Binary weights
• 128 64 32 16 8 4 2 1
• 357 256 64 32 4 1 101100101
• 28 27 26 22 20
• Binary weights
• 512 256 128 64 32 16 8 4 2 1
• 1937 1024 512 256 128 16
  1 11110010001

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Sum-of-weight method
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• Try this for 25 and 125

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Repeated division-by-2

• Repeated division steps
• Divide the decimal number by 2
• Write the remainder after each division until a
  quotient of zero is obtained.
• The first remainder is the LSB and the last is
  the MSB
• Note, when done on a calculator, a fractional
  answer indicates a remainder of 1.

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• This flowchart describes the process and can be
  used to convert from decimal to any other number
  system.

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Converting Decimal Fractions-Binary

• 2 ways to convert
• Sum-of-weight
• Repeated multiplication of 2

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Sum-of-weights
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Repeated multiplication by 2

• Conversion of decimal-binary
• Whole numbers ? repeated division by 2
• Fractions ? repeated multiplication by 2
• Step 1 multiply number by 2
• Step 2 note the carry (1 or 0)
• Step 3 repeat with new fraction
• Stop when fraction part 0

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Converting Decimal Fractions-Binary

• Using Sum-of-weights
• Binary weights
• 32 16 8 4 2 1 .5 .25 .125 .0625
• 95.6875 64 16 8 4 2 1 .5 .125
  .0625
• 1011111.1011
• Repeated division by 2 yields the whole number
  while repeated multiplication by 2 of the
  fraction yields the binary fraction

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Hexadecimal Number System

• Most digital systems deal with groups of bits in
  even powers of 2 such as 8, 16, 32, and 64 bits.
• Hexadecimal uses groups of 4 bits.
• Base 16
• 16 possible symbols
• 0-9 and A-F
• Allows for convenient handling of long binary
  strings.

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Hex to Decimal Conversion

• Convert from hex to decimal by multiplying each
  hex digit by its positional weight.
• Example

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Decimal to Hex Conversion

• Convert using the repeated division method
• decimal to hex ? divide by 16
• decimal to octal? divide by 8
• The first remainder is the LSB and the last is
  the MSB.
• Note, when done on a calculator a decimal
  remainder can be multiplied by 16 to get the
  result. If the remainder is greater than 9, the
  letters A through F are used.

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Decimal-Hex
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Example

• Example of hex to binary conversion.
• 9F216 9 F 2
• 1001 1111 0010
• 1001111100102

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Hex-Binary
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Binary to Hex Conversion

• Convert from binary to hex by grouping bits in
  four starting with the LSB.
• Each group is then converted to the hex
  equivalent
• Leading zeros can be added to the left of the MSB
  to fill out the last group.

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Example

• Example of binary to hex conversion.
• (Note the addition of leading zeroes)
• 11101001102 0011 1010 0110
• 3 A
  6
• 3A616
• Counting in hex requires a reset and carry after
  reaching F.

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Binary-Hex
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Octal Number

• The Octal Number System
• uses base 8
• includes only the digits 0 through 7
• The Octal system is based on the binary system
  with a 3-bit boundary

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Octal Number

• Binary to Octal Conversion
• (ex 10101111101100102)
• Octal to Binary Conversion
• 1276628

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Binary-Octal
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Octal-Binary
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Octal-Decimal
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Decimal-Octal
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Hexadecimal Number System

• Hex and octal are useful forms of shorthand to
  represent long strings of bits.
• Understanding the conversion process and
  memorizing the 4 bit patterns for each hex digit
  will prove valuable later.

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Binary Coded Decimal (BCD)

• BCD is a way to express each of the decimal
  digits with a binary code.
• 10 code groups in BCD ? represents 0-9
• Each code has 4 binary bits
• There will be invalid codes as 4 bits can
  represent a total of 16
• Invalid codes in BCD ? 1010, 1011, 1100, 1101,
  1110, 1111 (i.e.10-15)
• BCD provides excellent interface to binary
  systems ? e.g. keypad inputs, digital readouts

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Digital Codes and Parity

• There are other specialized codes used in digital
  systems ? e.g. Gray code, ASCII code
• Some codes are
• strictly numeric e.g. BCD
• Alphanumeric to represent numbers, letters,
  symbols, instructions e.g. ASCII

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• Gray Code
• Unweighted and is not an arithmetic code
• Only one bit changes from one code to the next in
  the sequence
• Gray code can be any amounts of bits.
• ASCII
• American Standard Code for Information
  Interchange
• Has 128 characters and symbols represented in
  7-bit binary code
• E.g. A 10000012 a 11000012

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ASCII example 20 PRINT AX
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Parity

• Parity bit used for bit error detection
• Even parity total number of 1s even
• Odd parity total number of 1s odd
• Example (even parity)
• Code transmitted 00101
• ?1s total even parity bit 0
• Code received 00001
• ?1s total odd parity bit 0 ? error

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Try this. Fill in the appropriate parity bit.
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End of Chapter 2