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Digital Logic Chapter 2

Digital Logic Chapter 2. Number Conversions Digital Systems by Tocci. Binary  Decimal. Convert a binary number, 100101 2 , to a decimal number by summing the positional weights that contain “1”. 32 + 0 + 0 + 4 + 0 + 1 = 37 10. How about Decimal  Binary?.

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Digital Logic Chapter 2

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  1. Digital Logic Chapter 2 Number Conversions Digital Systems by Tocci

  2. Binary  Decimal • Convert a binary number, 1001012, to a decimal number by summing the positional weights that contain “1”. 32 + 0 + 0 + 4 + 0 + 1 = 3710 How about Decimal  Binary?

  3. Decimal  Binary • Userepeated division: • Divide the decimal number by 2. The remainder of this division is the LSB • Continue dividing the results, adding the remainders to the left of the LSB until a quotient of zero is obtained. • The last division is always two into 1 giving a result of 0 with a remainder of 1. This 1 is the MSB. • Additional zeros can be added padding the binary number so the total digits are some multiple of 8.

  4. Repeated Division: Example 1 Until a quotient of “0” is obtained

  5. Repeated Division: Example 2

  6. Repeated Division: Flow Chart • Similar procedure can be used to convert from decimal to other number systems. Check your solutions by converting back to decimal.

  7. Hexadecimal Number System • Hexadecimal number system uses base-16 • The characters used in hex are: • Digits 0~9 • Letters A, B, C, D, E, F • The digit positions are weighted as powers of 16, rather than as powers of 10 as in the decimal system

  8. Counting in Hex • Why Hexadecimal? • It is useful to represent long strings of bits. Each character in hex can represent 4 bits reducing the length of a number to a quarter of the original size. It makes binary numbers more “readable”. • Counting in hex restarts at zeroand produces a carry after the count reaches F in order to increment to the next value.

  9. Hex  Decimal Conversion Multiplying each hex digit by its positional weight. Example:

  10. Decimal  Hex Conversion • Remember the repeated division? • Divide the decimal number by 16 • The 1st 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~F are used. Until a quotient of “0” is obtained

  11. Decimal  Hex Conversion

  12. Hex  Binary Conversion • Hex  Binary: • Each Hex digit is converted to its four-bit binary equivalent • 9F216 = 9 F 2 • 1001 1111 0010 = 1001111100102 • Binary  Hex: • 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. • Example: 11101001102 = 0011 1010 0110 = 3 A 6 = 3A616 Note the addition of leading zeroes

  13. http://www.learn-programming.za.net/articles_decbinhexoct.htmlhttp://www.learn-programming.za.net/articles_decbinhexoct.html Conversion among Decimal, Binary, Hex Decimal Binary Hexadecimal ? How to do all the conversions

  14. BCD Code • Binary Coded Decimal (BCD) is another way to present decimal numbers in binary form. • BCD is widely used and combines features of both decimal and binary systems. • Each BCD digit is converted to a binary equivalent.

  15. Decimal  BCD • To convert the number 87410 to BCD: 8 7 4 0100 0111 0100 = 010001110100BCD • Each decimal digit is represented using 4 bits. • Each 4-bit group can never be greater than 9. • Reverse the process to convert BCD to decimal

  16. BCD • BCD is NOT a number system. • BCD is a decimal number with each digit encoded to its binary equivalent. • The primary advantage of BCD: easy to convert to and from binary. • A BCD number is NOT the same as a straight binary number.

  17. BCD Review Questions Is “1001 1011 0101” a valid BCD?

  18. BYTE, Nibble, WORD • Byte: • Most microcomputers handle and store binary data in groups of 8 bits. • So, special name is given to a string of 8 bits, called a byte. • Two common questions: • How many bytes in a 32-bit string (a string of 32 bits)? • What is the largest decimal number that can be represented in binary using two bytes?

  19. BYTE, Nibble, WORD • Byte = 8 bits • Nibble = 4 bits • Word: • Word size in a simple system may be one byte (8 bits) • Word size in a PC is 8 bytes (64 bits) • Word size is specific to particular machines.

  20. Alphanumeric Codes – ASCII Code • Represents characters and functions found on a computer keyboard. • ASCII – American Standard Code for Information Interchange. • Seven bit code: 27 = 128 possible code groups • Table 2-4 lists the standard ASCII codes • Applications: • To transfer information between computers, between computers and printers, and for internal storage.

  21. Parity Method for Error Detection • Binary data and codes are frequently moved between locations. • For example: • Digitized voice over a microwave link. • Storage and retrieval of data from hard disks. • Communication between computer systems over telephone lines using a modem. • Electrical noise can cause errors during transmission. • Many digital systems employ methods for error detection (and sometimes correction).

  22. Parity Method for Error Detection • The parity method of error detection requires the addition of an extra bit to a code group. • This extra bit is called the parity bit. • The bit can be either a 0 or 1, depending on the number of 1s in the code group. • There are two methods: even and odd.

  23. Parity Method for Error Detection • Even Parity Method: • The total number of “1”s in a group, including the parity bit, must add up to an even number. • The binary group 1 0 1 1 would require the addition of a parity bit 11 0 1 1 • The parity bit may be added at either end of a group. • Odd Parity Method: • The total number of “1”s in a group, including the parity bit, must add up to an odd number.

  24. Parity Method for Error Detection • The transmitter and receiver must “agree” on the type of parity-checking being used. • Two bit errors would not indicate a parity error. • Both odd and even parity methods are used, but even seems to be used more often.

  25. Schematic for Even Parity Generator

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