DCD Lecture 01

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Digital Circuit Design

Digital Systems andBinary Numbers

Lan-Da Van (), Ph. D.Department of Computer Science

National Chiao Tung UniversityTaiwan, R.O.C.

Fall, 2010

[email protected]

http://www.cs.nctu.edu.tw/~ldvan/


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Lecture 1


Lan-Da Van DCD-01-2

Digital Systems

Binary Numbers

Number-Base Conversion

Octal and Hexadecimal Number

Signed Binary Numbers

Binary Codes

Binary Storage and Registers

Binary Logic

Outline


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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

Digital System

Di i l Ci i D i


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A Digital Computer Example

Synchronous or

Asynchronous?

Inputs: Keyboard,

mouse, modem,

microphone

Outputs: CRT,

LCD, modem,

speakers

Memory

Control

unit

Datapath

Input/Output

CPU

Di it l Ci it D i


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Signal

An information variable represented by physical quantityFor digital systems, the variable takes on digital 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)

words On and Off.

Binary values are represented by values or ranges of

values of physical quantities



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Signal

Analog signal

t->y: y=f(t), y:C, n:C

Discrete-time signal

n->y: y=f(nT), y:C, n:Z

Digital signal

n->y: y=D{f(nT)}, y:Z,n:Z

)3(

)1(

)2(

)1(

2)1110(

2)1000(

2)1011( 2)1000(

t n n

y y y

Analog Signal Discrete-Time Signal Digital Signal



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Binary Numbers

Decimal number

5 4 3 2 1 0 1 2 3

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

a5a4a3a2a1a0.a1a2a3

Decimal point

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

Example:

ja

Base or radix

Power

General form of base-r system

1 2 1 1 21 2 1 0 1 2

n 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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Example: Base-2 number

Binary Numbers

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


5

3 2 1 0 110

(4021.2)

4 5 0 5 2 5 1 5 2 5 (511.5)



3 2 1 0

16 10(B65F) 11 16 6 16 5 16 15 16 (46,687)

4

101012

8

)5.87(84878281

)4.127(



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Binary Numbers


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"



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Arithmetic operations with numbers in base rfollow the same rules as decimalnumbers.

Binary Numbers



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Binary Arithmetic

Addition

Augend: 101101

Addend: +100111

Sum: 1010100

Subtraction

Minuend: 101101

Subtrahend: 100111

Difference: 000110

Multiplicand 1011

Multiplier 101Partial Products 1011

0000 -1011 - -

Product 110111

Multiplication



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Octal and Hexadecimal Numbers

Numbers with different bases: Table 1.2.



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Example1.1

Convert decimal 41 to binary. The process is continued until the integer quotientbecomes 0.

Number-Base Conversions

10/2

5/2

2/2

1/2

5

2

1

0



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The arithmetic process can be manipulated more conveniently as follows:


Answer=(101001)2



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Example 1.2

Convert decimal 153 to octal. The required base ris 8.




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Example1.3

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

Convert (0.6875)10

to binary.

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




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Example1.4

Convert (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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Conversion from binary to octal can be done by positioning the binary number intogroups 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 isdivided 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 precedingprocedure.




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There are two types of complements for each base-rsystem: the radix complement anddiminished 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 Nis (2n 1) N.

Example:

Complements



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Radix ComplementThe r's complement of an n-digit number Nin base r is defined as rn Nfor N 0and as 0 for N= 0. Comparing with the (r 1) 's complement, we note that the r'scomplement is obtained by adding 1 to the (r 1) 's complement, since rn N= [(rn1) N] + 1.

Example: Base-10

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

Example: Base-2

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

Complements



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Subtraction with ComplementsThe subtraction of two n-digit unsigned numbers M Nin base rcan be done as follows:

Complements

an



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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.

Complements



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Example 1.7

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

There is no end carry.Therefore, the answer is

Y X= (2's complementof 1101111) = 0010001.

Complements



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Complements

Subtraction of unsigned numbers can also be done by means of the (r 1)'scomplement. Remember that the (r 1) 's complement is one less than 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 isY X= (1's complementof 1101110) = 0010001.


Si d Bi N b


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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:


Si d Bi N b


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


Si d Bi N b


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Arithmetic Addition

The addition of two signed binary numbers with negative numbers represented insigned-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:

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 andgive the sum the common sign. If the signs are different, we subtract the smallermagnitude from the larger and give the difference the sign if the larger magnitude.




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Arithmetic Subtraction In 2s-complement form:

1. Take the 2s 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)


BCD Code


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BCD Code

BCD Code A number with kdecimal digits willrequire 4kbits in BCD. Decimal 396is 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 combinations1010 through 1111 are not used and

have no meaning in BCD.


BCD Code


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BCD Code

Example:

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

BCD Addition


BCD Code


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BCD Code

Example:

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

Decimal Arithmetic

1


Other Decimal Codes


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Other Decimal Codes

Other Decimal Codes


Gray Code


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Gray Code

Gray Code


Gray Code


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Does this special Gray code property have

any value?An Example: Optical Shaft Encoder

B 0

111

110

000

001

010

011100

101

B 1

B 2

(a) Binary Code for Positions 0 through 7

G 0G 1

G 2

111

101

100 000

001

011

010110

(b) Gray Code for Positions 0 through 7

Gray Code


ASCII Character Code


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American Standard Code for Information Interchange (Refer

to Table 1.7)A popular code used to represent information sent ascharacter-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 markingand flow control (e.g. STX and ETX start and end text areas).




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ASCII has some interesting properties:

Digits 0 to 9 span Hexadecimal values 3016 to 3916 .

Upper case A - Z span 4116to 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!


Error Detection Code


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Error-Detecting Code

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

A paritybit 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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Error-Detecting CodeRedundancy (e.g. extra information), in the form of extra bits,can be incorporated into binary code words to detect andcorrect errors.

A simple form of redundancy is parity, an extra bit appendedonto the code word to make the number of 1s odd or even.

Parity can detect all single-bit errors and some multiple-biterrors.

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

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


Conversion or Coding?


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Conversion or Coding?

Do NOT mix up conversion of a decimal number to a

binary number with coding a decimal number with aBINARY CODE.

1310 = 11012 (This is conversion)

13 0001|0011 (This is coding)




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Registers A binary cellis a device that possesses two stable states and is capable of storing

one of the two states.

A registeris a group of binary cells. A register with ncells can store any discrete

quantity of information that contains nbits.

ncells 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


Transfer of information


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Example of BinaryInformation Processing


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Information Processing


Binary Logic


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y g

Binary logic consists of binary variables and a set of logical operations. The variablesare 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 threebasic logical operations: AND, OR, and NOT.


Binary Logic


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


Binary Logic


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Logic gates Example of binary signals


Binary Logic


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g

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

Input-Output signals

for gates


Binary Logic


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


Conclusion


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Conclusion

You have learned the following terms:

Binary number Number Conversion

Complement

Simple arithmetic

Binary codes

Storage and register Binary logic

DCD Lecture 01

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