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Chapter 1 Digital Systems and Binary Numbers

授課教師 : 張傳育 博士 (Chuan-Yu Chang Ph.D.)E-mail: chuanyu@yuntech.edu.twTel: (05)5342601 ext. 4337Office: EB212http://MIPL.yuntech.edu.tw

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Digital Logic Design • Text Book

– M. M. Mano and M. D. Ciletti, “Digital Design," 4th Ed., Pearson Prentice Hall, 2007.

• Reference– class notes

• Grade– Quizzes:

– Mid-term:

– Final:

– Appearance:

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• Digital age and information age

• Digital computers– general purposes

– many scientific, industrial and commercial applications

• Digital systems– telephone switching exchanges

– digital camera

– electronic calculators, PDA's

– digital TV

• Discrete information-processing systems– manipulate discrete elements of information

Chapter 1: Digital Systems and Binary Numbers

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Signal

• An information variable represented by physical quantity

• For digital systems, the variable takes on discrete values– Two level, or binary values are the most prevalent values 

• Binary values are represented abstractly by:– digits 0 and 1

– words (symbols) False (F) and True (T)

– words (symbols) Low (L) and High (H)

– and words On and Off.

• Binary values are represented by values or ranges of values of physical quantities

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5.0

4.0

3.0

2.0

1.0

0.0

Volts

HIGH

LOW

HIGH

LOW

OUTPUT INPUT

Signal Example – Physical Quantity: Voltage

Threshold Region

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Signal Examples Over Time

Analog

Asynchronous

Synchronous

Time

Continuous in value &

time

Discrete in value & continuous

in time

Discrete in value &

time

Digital

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A Digital Computer Example

Synchronous or Asynchronous?

Inputs: Keyboard, mouse, modem, microphone

Outputs: CRT, LCD, modem, speakers

Memory

Controlunit Datapath

Input/Output

CPU

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Binary Numbers and Binary Coding

• Information Types– Numeric

» Must represent range of data needed

» Represent data such that simple, straightforward computation for common arithmetic operations

» Tight relation to binary numbers

– Non-numeric

» Greater flexibility since arithmetic operations not applied.

» Not tied to binary numbers

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Number of Elements Represented

• Given n digits in radix r, there are rn distinct elements that can be represented.

• But, you can represent m elements, m < rn

• Examples:– You can represent 4 elements in radix r = 2 with n = 2

digits: (00, 01, 10, 11).

– You can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000).

– This second code is called a "one hot" code.

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• Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers.

• Example: Abinary codefor the sevencolors of therainbow

• Code 100 is not used

Non-numeric Binary Codes

Binary Number 000001010011101110111

ColorRedOrangeYellowGreenBlueIndigoViolet

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Binary Numbers

• Decimal number

5 4 3 2 1 0 1 2 35 4 3 2 1 0 1 2 310 10 10 10 10 10 10 10 10a a a a a a a a a

… a5a4a3a2a1.a1a2a3…

Decimal point

3 2 1 07,329 7 10 3 10 2 10 9 10

Example:

jaBase or radix

Power

• General form of base-r system1 2 1 1 2

1 2 1 0 1 2n n m

n n ma r a r a r a r a a r a r a r

Coefficient: aj = 0 to r 1

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Binary Numbers

Example: Base-2 number

2 10

4 3 2 1 0 1 2

(11010.11) (26.75)

1 2 1 2 0 2 1 2 0 2 1 2 1 2

Example: Base-5 number

Example: Base-8 number

Example: Base-16 number

3 2 1 016 10(B65F) 11 16 6 16 5 16 15 16 (46,687)

10

10123

5

4.5115251525054

2.4021

10

1012

8

5.8784878281

4.127

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• To convert to decimal, use decimal arithmetic to form (digit × respective power of 2).

• Example:Convert 110102 to N10:  

Converting Binary to Decimal

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Binary NumbersExample: Base-2 number

2 10(110101) 32 16 4 1 (53)

Special Powers of 2

210 (1024) is Kilo, denoted "K"

220 (1,048,576) is Mega, denoted "M"

230 (1,073, 741,824)is Giga, denoted "G"Powers of two

Table 1.1

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Arithmetic operationArithmetic operations with numbers in base r follow the same rules as decimal numbers.

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Binary Arithmetic

• Single Bit Addition with Carry

• Multiple Bit Addition

• Single Bit Subtraction with Borrow

• Multiple Bit Subtraction

• Multiplication

• BCD Addition

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Single Bit Binary Addition with Carry

Given two binary digits (X,Y), a carry in (Z) we get the following sum (S) and carry (C):

Carry in (Z) of 0:

Carry in (Z) of 1:

Z 1 1 1 1

X 0 0 1 1

+ Y + 0 + 1 + 0 + 1

C S 0 1 1 0 1 0 1 1

Z 0 0 0 0

X 0 0 1 1

+ Y + 0 + 1 + 0 + 1

C S 0 0 0 1 0 1 1 0

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• Extending this to two multiple bit examples:

Carries 0 0

Augend 01100 10110

Addend +10001 +10111

Sum• Note: The 0 is the default Carry-In to the least significant bit.

Multiple Bit Binary Addition

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Binary Arithmetic• Addition

Augend: 101101

Addend: +100111

Sum: 1010100

• Subtraction

Minuend: 101101

Subtrahend: 100111

Difference: 000110

• Multiplication

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Number-Base Conversions

Name Radix Digits

Binary 2 0,1

Octal 8 0,1,2,3,4,5,6,7

Decimal 10 0,1,2,3,4,5,6,7,8,9

Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F The six letters (in addition to the 10 integers) in

hexadecimal represent: 10, 11, 12, 13, 14, and 15, respectively.

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Number-Base Conversions

Example1.1 Convert decimal 41 to binary. The process is continued until the integer quotient becomes 0.

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Number-Base Conversions The arithmetic process can be manipulated more conveniently as follows:

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Number-Base ConversionsExample 1.2 Convert decimal 153 to octal. The required base r is 8.

Example1.3Convert (0.6875)10 to binary.

The process is continued until the fraction becomes 0 or until the number of digits has sufficient accuracy.

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Number-Base ConversionsExample1.3

To convert a decimal fraction to a number expressed in base r, a similar procedure is used. However, multiplication is by r instead of 2, and the coefficients found from the integers may range in value from 0 to r 1 instead of 0 and 1.

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Number-Base ConversionsExample1.4Convert (0.513)10 to octal.

From Examples 1.1 and 1.3: (41.6875)10 = (101001.1011)2

From Examples 1.2 and 1.4: (153.513)10 = (231.406517)8

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Octal and Hexadecimal Numbers Numbers with different bases: Table 1.2.

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Octal and Hexadecimal Numbers Conversion from binary to octal can be done by positioning the binary number into

groups of three digits each, starting from the binary point and proceeding to the left and to the right.

Conversion from binary to hexadecimal is similar, except that the binary number is divided into groups of four digits:

(10 110 001 101 011 ． 111 100 000 110) 2 = (26153.7406)8

2 6 1 5 3 7 4 0 6

Conversion from octal or hexadecimal to binary is done by reversing the preceding procedure.

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Complements There are two types of complements for each base-r system: the radix complement and

diminished radix complement.

the r's complement and the second as the (r 1)'s complement.

■ Diminished Radix Complement

Example:

For binary numbers, r = 2 and r – 1 = 1, so the 1's complement of N is (2n 1) – N.

Example:

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Complements (cont.)■ Radix Complement

The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r 1) 's complement, we note that the r's complement is obtained by adding 1 to the (r 1) 's complement, since rn – N = [(rn 1) – N] + 1.

Example: Base-10

The 10's complement of 012398 is 987602The 10's complement of 246700 is 753300

Example: Base-10

The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001

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Complements (cont.)■ Subtraction with Complements

The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:

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Complements (cont.)Example 1.5

Using 10's complement, subtract 72532 – 3250.

Example 1.6

Using 10's complement, subtract 3250 – 72532

There is no end carry.

Therefore, the answer is – (10's complement of 30718) = 69282.

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Complements (cont.)Example 1.7

Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y and (b) Y X by using 2's complement.

There is no end carry. Therefore, the answer is Y – X = (2's complement of 1101111) = 0010001.

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Complements (cont.)

Subtraction of unsigned numbers can also be done by means of the (r 1)'s complement. Remember that the (r 1) 's complement is one less then the r's complement.

Example 1.8

Repeat Example 1.7, but this time using 1's complement.

There is no end carry, Therefore, the answer is Y – X = (1's complement of 1101110) = 0010001.

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Signed Binary Numbers To represent negative integers, we need a notation for negative values. It is customary to represent the sign with a bit placed in the leftmost position of the

number. The convention is to make the sign bit 0 for positive and 1 for negative.

Example:

Table 3 lists all possible four-bit signed binary numbers in the three representations.

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Signed Binary Numbers (cont.)

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Signed Binary Numbers (cont.)■ Arithmetic Addition

The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign if the larger magnitude.

The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits.

A carry out of the sign-bit position is discarded.

Example:

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Signed Binary Numbers (cont.)■ Arithmetic Subtraction

In 2’s-complement form:

1. Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit).

2. A carry out of sign-bit position is discarded.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

A B A B

A B A B

Example:

( 6) ( 13) (11111010 11110011)

(11111010 + 00001101)

00000111 (+ 7)

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Binary Coded Decimal (BCD)

• The BCD code is the 8,4,2,1 code.

• This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9.

• Example: 1001 (9) = 1000 (8) + 0001 (1)

• How many “invalid” code words are there?

• What are the “invalid” code words?

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1010, 1011, 1100, 1101, 1110, 1111

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Binary Codes■ BCD Code A number with k decimal digits will

require 4k bits in BCD. Decimal 396 is represented in BCD with 12bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit. A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9. A BCD number greater than 10 looks different from its equivalent binary number, even though both contain 1's and 0's. Moreover, the binary combinations 1010 through 1111 are not used and have no meaning in BCD.

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Warning: Conversion or Coding?

• Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE. 

• 1310 = 11012 (This is conversion) 

• 13 0001|0011 (This is coding)

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BCD Arithmetic

Given a BCD code, we use binary arithmetic to add the digits:8 1000 Eight

+5 +0101 Plus 5 13 1101 is 13 (> 9)

Note that the result is MORE THAN 9, so must be represented by two digits! To correct the digit, subtract 10 by adding 6 modulo 16.

8 1000 Eight +5 +0101 Plus 5 13 1101 is 13 (> 9)

+0110 so add 6 carry = 1 0011 leaving 3 + cy

0001 | 0011 Final answer (two digits) If the digit sum is > 9, add one to the next significant digit

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Binary CodesExample:

Consider decimal 185 and its corresponding value in BCD and binary:

■ BCD Addition

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Binary CodesExample:

Consider the addition of 184 + 576 = 760 in BCD:

■ Decimal Arithmetic

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BCD Addition Example

• Add 2905BCD to 1897BCD showing carries and digit corrections.

0001 1000 1001 0111+ 0010 1001 0000 0101 0011 10001 1001 1100

+011010010

+0001 1010+011010000

+011010111+000111000

+0001 0100

4

8 0

2

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Binary Codes■ Other Decimal Codes

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Binary Codes■ Gray Code

0000000100110010011001110101010011001101111111101010101110011000

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• What special property does the Gray code have in relation to adjacent decimal digits?

Gray Code

Decimal 8,4,2,1 Gray 0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101

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• Does this special Gray code property have any value?

• An Example: Optical Shaft Encoder

B0

111

110

000

001

010

011100

101

B1

B2

(a) Binary Code for Positions 0 through 7

G0

G1

G2

111

101

100 000

001

011

010110

(b) Gray Code for Positions 0 through 7

Gray Code (Continued)

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Binary Codes■ ASCII Character Code

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Binary Codes■ ASCII Character Code

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ASCII Character Codes

• American Standard Code for Information Interchange

• A popular code used to represent information sent as character-based data.

• It uses 7-bits to represent:– 94 Graphic printing characters.

– 34 Non-printing characters

• Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return)

• Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).

(Refer to Table 1.7)

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ASCII Properties

ASCII has some interesting properties: Digits 0 to 9 span Hexadecimal values 3016 to 3916 . Upper case A - Z span 4116 to 5A16 . Lower case a - z span 6116 to 7A16 .

• Lower to upper case translation (and vice versa) occurs by flipping bit 6.

Delete (DEL) is all bits set, a carryover from when punched paper tape was used to store messages.

Punching all holes in a row erased a mistake!

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Binary Codes■ Error-Detecting Code

To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity.

A parity bit is an extra bit included with a message to make the total number of 1's either even or odd.

Example:

Consider the following two characters and their even and odd parity:

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Binary Codes■ Error-Detecting Code

• Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.

• A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.

• A code word has even parity if the number of 1’s in the code word is even.

• A code word has odd parity if the number of 1’s in the code word is odd.

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4-Bit Parity Code Example

• Fill in the even and odd parity bits:

• The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data.

Even Parity Odd Parity Message - Parity Message - Parity

000 - 000 - 001 - 001 - 010 - 010 - 011 - 011 - 100 - 100 - 101 - 101 - 110 - 110 - 111 - 111 -

01101001

10010110

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UNICODE

• UNICODE extends ASCII to 65,536 universal characters codes

– For encoding characters in world languages

– Available in many modern applications

– 2 byte (16-bit) code words

– See Reading Supplement – Unicode on the Companion Website http://www.prenhall.com/mano

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Binary Storage and Registers■ Registers

A binary cell is a device that possesses two stable states and is capable of storing one of the two states.

A register is a group of binary cells. A register with n cells can store any discrete qua

ntity of information that contains n bits.

n cells 2n possible states

• A binary cell– two stable state– store one bit of information– examples: flip-flop circuits, ferrite cores, capacitor

• A register– a group of binary cells– AX in x86 CPU

• Register Transfer– a transfer of the information stored in one register to another– one of the major operations in digital system– an example

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Transfer of information

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• The other major component of a digital system– circuit elements to manipulate individual bits of information

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Binary Logic■ Definition of Binary Logic

Binary logic consists of binary variables and a set of logical operations. The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0, There are three basic logical operations: AND, OR, and NOT.

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Binary Logic

■ The truth tables for AND, OR, and NOT are given in Table 1.8.

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Binary Logic■ Logic gates Example of binary signals

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Binary Logic■ Logic gates Graphic Symbols and Input-Output Signals for Logic gates:

Fig. 1.4 Symbols for digital logic circuits

Fig. 1.5 Input-Output signals for gates

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Binary Logic■ Logic gates Graphic Symbols and Input-Output Signals for Logic gates:

Fig. 1.6 Gates with multiple inputs