Digital Logic Chapter 2 Number Conversions Digital Systems by Tocci

Binary  Decimal  Convert a binary number, , to a decimal number by summing the positional weights that contain “1”. How about Decimal  Binary? = 37 10

Decimal  Binary  Use repeated 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.

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

Repeated Division: Example 2

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.

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

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 zero and produces a carry after the count reaches F in order to increment to the next value.

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

Decimal  Hex Conversion  Remember the repeated division?  Divide the decimal number by 16  The 1 st 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

Decimal  Hex Conversion

Hex  Binary Conversion  Hex  Binary: Each Hex digit is converted to its four-bit binary equivalent 9F2 16 = 9 F =  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: = = 3 A 6 = 3A6 16 Note the addition of leading zeroes

Conversion among Decimal, Binary, Hex Decimal BinaryHexadecimal How to do all the conversions ?

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.

 To convert the number to BCD: = BCD  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 Decimal  BCD

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.

BCD Review Questions Is “ ” a valid BCD?

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?

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.

Alphanumeric Codes – ASCII Code  Represents characters and functions found on a computer keyboard.  ASCII – American Standard Code for Information Interchange. Seven bit code: 2 7 = 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.

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).

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.

 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 would require the addition of a parity bit 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. 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.

Schematic for Even Parity Generator