Digital Fundamentals

Digital Fundamentals CHAPTER 1 (CONT…) Number Systems, Operations, and Codes

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1. Number Systems 1.1 Decimal Numbers • The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 • The decimal numbering system has a base of 10 with each position weighted by a factor of 10:

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1.2 Binary Numbers • The binary number system has two 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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• Conversion 2.1 Binary to Decimal Conversion • Convert binary to decimal by summing the positions that contain a 1.

1

0

0 1 0 1

2

2 + 2 + 2 + 2 + 2 + 2 = 32 + 4 + 1 = 3710 5

4

3

2

1

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0

2.2 Decimal to Binary Conversion •

Two methods to convert decimal to binary: – Reverse process described in 2.1 – Use repeated division

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Decimal to Binary Conversion (cont…) •

Reverse process described in 2.1 – Note that all positions must be accounted for

3710 = 2 + 0 + 0 + 2 + 0 + 2 5

1

2

0

0

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1

0

0

12

Decimal to Binary Conversion (cont…) • 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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Decimal to Binary Conversion (cont…) • Repeated division – This flowchart describes the process and can be used to convert from decimal to any other number system.

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3. Hexadecimal Number System • Decimal, binary, and hexadecimal numbers

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Hexadecimal Number System (cont….) • 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

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Hexadecimal Number System (cont….) • Convert from hex to decimal by multiplying each hex digit by its positional weight. Example:

16316 = 1× (16 2 ) + 6 × (161 ) + 3 × (160 ) = 1× 256 + 6 ×16 + 3 ×1 = 35510 Slide 11

Hexadecimal Number System (cont….) • Convert from decimal to hex by using the repeated division method used for decimal to binary and decimal to octal conversion. • Divide the decimal number by 16 • 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. Slide 12

Hexadecimal Number System (cont….) Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

• Example of hex to binary conversion: 9F216 = 9 F

2

1001 1111

0010 = 1001111100102

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Hexadecimal Number System (cont….) • 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. Slide 14

Hexadecimal Number System (cont….) Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

•

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. Slide 15

Hexadecimal Number System (cont….) • Hexadecimal is useful for representing long strings of bits. • Understanding the conversion process and memorizing the 4 bit patterns for each hexadecimal digit will prove valuable later.

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4. Binary Coded Decimal (BCD) • 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 digit is converted to a binary equivalent. Slide 17

Binary Coded Decimal (cont…) Decimal and BCD digits

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Binary Coded Decimal (cont…) • 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. Slide 19

Binary Coded Decimal (cont…) • BCD is not a number system. • BCD is a decimal number with each digit encoded to its binary equivalent. • A BCD number is not the same as a straight binary number. • The primary advantage of BCD is the relative ease of converting to and from decimal.

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5.

Digital Codes

• Gray code:  The gray code is used in applications where numbers change rapidly.  In the gray code, only one bit changes from each value to the next.

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Digital Codes (cont…) • ASCII – American Standard Code for Information Interchange. – Seven bit code: 27 = 128 possible code groups – Table 2-4 lists the standard ASCII codes – Examples of use are: to transfer information between computers, between computers and printers, and for internal storage.

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Digital Codes (cont…) • ASCII code (control characters)

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Digital Codes (cont…) • ASCII code (graphic symbols 20h – 3Fh)

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Digital Codes (cont…) • ASCII code (graphic symbols 40h – 5Fh)

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Digital Codes (cont…) • ASCII code (graphic symbols 60h – 7Fh)

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Digital Codes (cont…) Extended ASCII code (80h – FFh) • Non-English alphabetic characters • Currency symbols • Greek letters • Math symbols • Drawing characters • Bar graphing characters • Shading characters Slide 27

6.

Putting It All Together Decimal Binary Hexadecimal BCD 0 0 0 0 1 1 1 0001 2 10 2 0010 3 11 3 0011 4 100 4 0100 5 101 5 0101 6 110 6 0110 7 111 7 0111 8 1000 8 1000 9 1001 9 1001 10 1010 A 0001 0000 11 1011 B 0001 0001 12 1100 C 0001 0010 13 1101 D 0001 0011 14 1110 E 0001 0100 15 1111 F 0001 0101 Slide 28

Gray 0 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

7. Complements of Binary Numbers • 1’s complements • 2’s complements

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Complements of Binary Numbers (cont…)

• 1’s complement

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Complements of Binary Numbers (cont…)

• 2’s complement

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8. Signed Numbers

• • • • •

Signed-magnitude form 1’s and 2’s complement form Decimal value of signed numbers Range of values Floating-point numbers

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Signed Numbers (cont…) • Signed-magnitude form – The sign bit is the left-most bit in a signed binary number – A 0 sign bit indicates a positive magnitude – A 1 sign bit indicates a negative magnitude

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Signed Numbers (cont…) • 1’s complement form – A negative value is the 1’s complement of the corresponding positive value

• 2’s complement form – A negative value is the 2’s complement of the corresponding positive value

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Signed Numbers (cont…) • Decimal value of signed numbers – Sign-magnitude – 1’s complement – 2’s complement

• Range of Values 2’s complement form:

– (2n – 1) to + (2n – 1 – 1)

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Signed Numbers (cont…) • Floating-point numbers – Single-precision (32 bits) – Double-precision (64 bits) – Extended-precision (80 bits)

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8.1

Arithmetic Operations with Signed Numbers • • • •

Addition Subtraction Multiplication Division

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Arithmetic Operations with Signed Numbers (cont…) Addition of Signed Numbers • The parts of an addition function are: – Addend – Augend – Sum Numbers are always added two at a time.

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Arithmetic Operations with Signed Numbers (cont…) Four conditions for adding numbers: • Both numbers are positive. • A positive number that is larger than a negative number. • A negative number that is larger than a positive number. • Both numbers are negative.

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Arithmetic Operations with Signed Numbers (cont…) Signs for Addition • When both numbers are positive, the sum is positive. • When the larger number is positive and the smaller is negative, the sum is positive. The carry is discarded.

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Arithmetic Operations with Signed Numbers (cont…) Signs for Addition • When the larger number is negative and the smaller is positive, the sum is negative (2’s complement form). • When both numbers are negative, the sum is negative (2’s complement form). The carry bit is discarded.

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Arithmetic Operations with Signed Numbers (cont…) Subtraction of Signed Numbers • The parts of a subtraction function are: – Subtrahend – Minuend – Difference Subtraction is addition with the sign of the subtrahend changed.

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Arithmetic Operations with Signed Numbers (cont…) Subtraction • The sign of a positive or negative binary number is changed by taking its 2’s complement • To subtract two signed numbers, take the 2’s complement of the subtrahend and add. Discard any final carry bit.

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Arithmetic Operations with Signed Numbers (cont…) Multiplication of Signed Numbers • The parts of a multiplication function are: – Multiplicand – Multiplier – Product Multiplication is equivalent to adding a number to itself a number of times equal to the multiplier.

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Arithmetic Operations with Signed Numbers (cont…) There are two methods for multiplication: • Direct addition • Partial products The method of partial products is the most commonly used.

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Arithmetic Operations with Signed Numbers (cont…) Multiplication of Signed Numbers • If the signs are the same, the product is positive. • If the signs are different, the product is negative.

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Arithmetic Operations with Signed Numbers (cont…) Division of Signed Numbers • The parts of a division operation are: – Dividend – Divisor – Quotient Division is equivalent to subtracting the divisor from the dividend a number of times equal to the quotient.

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Arithmetic Operations with Signed Numbers (cont…) Division of Signed Numbers • If the signs are the same, the quotient is positive. • If the signs are different, the quotient is negative.

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9. 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 magnetic and optical 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). Slide 49

Parity Method for Error Detection (cont…) • 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.

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Parity Method for Error Detection (cont…) • Even parity method – the total number of bits 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 1 1 0 1 1 • Note that the parity bit may be added at either end of a group.

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Parity Method for Error Detection (cont…) • Odd parity method – the total number of bits in a group including the parity bit must add up to an odd number. – The binary group 1 1 1 1 would require the addition of a parity bit 1 1 1 1 1

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Parity Method for Error Detection (cont…) • The transmitter and receiver must “agree” on the type of parity checking 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. Slide 53

Parity Method for Error Detection (cont…) • Parity error codes

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Parity Method for Error Detection (cont…) • Hamming error codes – Hamming code words – Hex equivalent of the data bits

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0000000 0000111 0011011 0011110 0101010 0101101 0110011 0110100 1001011 1001100 1010010 1010101 1100001 1100110 1111000 1111111

0 1 2 3 4 5 6 7 8 9 A B C D E F