DIGITAL LOGIC DESIGN

BINARY SYSTEMS

DIGITAL SYSTEMS

� Prominent Role in Everyday Life

� Present Technological Period is called Digital Age

� Used in Communication, Business Transactions,Traffic Control, Space Guidance, MedicalTreatment, Weather Monitoring Etc.

� Ex: DTH, DVD’s, Digital Cameras, DigitalPhones, Digital Computers Etc.

� Most important Properties of a Digital Computeris its generality and Flexibility

DIGITAL SYSTEMS

� One Characteristic of digital systems is their

ability to manipulate discrete elements of

information

� Ex: 10 decimal digits, 26 alphabets, 52 Playing

Cards Etc.

� The Term Digital Emerged from the word Digit

� Discrete elements of information are represented

in a digital system by physical quantities called

signals (Voltage & Current)

DIGITAL SYSTEMS

� Signals in present day digital systems use twodiscrete values (0 & 1bits) are said to be binary

� Discrete elements of information are representedwith groups of bits called binary codes

� Digital System is a system that manipulatesdiscrete elements of information that isrepresented internally in binary form

� This discrete information emerges from nature ofdata being processed or may be quantized fromcontinuous process

DIGITAL SYSTEMS

� Best Example for a digital system is “Digital

Computer”

� Fundamental Reasons why Commercial Systems

are made with Digital Circuits –

�Most devices are programmable

�Changing the programming , same H/W can be

used for multiple applications

�Cost reductions due to advances in DIC Technology

�Speed

DIGITAL SYSTEMS

�Operate with extreme reliability by using errorcorrecting codes

�Ex: Recording in a DVD

�Digital Systems also termed as interconnection ofdigital modules

�Can be designed using HDL’s which resemblesprogramming language and is suitable fordescribing digital circuits in textual form, simulatesa digital system to verify its operation and used inconjunction with logic synthesis tools to automatedesign

BINARY NUMBERS

� Decimal point is represented by a series of coefficientsA

5A

4A

3A

2A

1A

0.A

-1A

-2A

-3

� Aj coefficients are any of 10 digits and J gives the place

value and hence the value power of 10 by which the

coefficient must be multiplied, given by 105A5 + 104A4 + 103A3

+ 102A2 + 101A1 + 100A0 + 10-1A-1 + 10-2A-2 + 10-3A-3

� Decimal number system is said to be of base or radix

10 as it uses 10 digits and coefficients are multiplied by

powers of 10

� In Binary System, Coefficients of binary values (0 and

1) Aj is multiplied by 2j

� Ex:24 x 1 + 23 x 1 + 22 x 0 + 21 x 0 + 20 x 0 + 2-1 x 1 + 2-2 x 1 = 26.75

BINARY NUMBERS

� Convert the following to Decimal Numbers

� (4021.2)5

� (127.4)8

� (B65F)16

� (110101)2

BINARY NUMBERS

� In Computer Terminology

�210 Kilo

�220 Mega

�230 Giga

�240 Tera

� Computer Capacity is given in Bytes

� Byte is equal to eight bits and can accommodate

one keyboard character

BINARY NUMBERS

� Arithmetic Operations with numbers in base r

follows the same rules as for decimal numbers

� Ex: Augend: 101101

Addend: 100111

-----------

1010100

COMPLEMENTS

� Used in digital computers for simplifying thesubtraction operation and for logical manipulation

� Two types of complements for each base-r system

�Radix Complement (r’s Complement)

�Diminished Radix Complement (r-1’s Complement)

�DRC - Given a number N in base r having n digits(r-1)’s complement of N is defined as (rn-1) – N

�Ex: 9’s Complement of 546700 is (106 – 1) – 546700= 453299

COMPLEMENTS

� Radix Complement – r’s complement of an n digit

number N in base r is defined as rn – N for N ≠ 0

and 0 for N = 0

� Ex: 10’s Complement of 012398 is 106 – 012398 =

987602

� Complement of a complement restores the original

value

COMPLEMENTS

� Subtraction with Complements

� When normal subtraction is implemented on H/Wit is less significant

�Procedure using Complements

�Add the minuend, Mm to the r’s complement of thesubtrahend, N. This performs M + (rn – N) = M – N+ rn

� If M ≥ N, sum will produce an end carry, rn, whichcan be discarded; what is left is M-N

� If M < N, sum does not produce an end carry and isequal to rn – (N-M), which is complement of N-M

COMPLEMENTS

� Ex: Using 10’s Complement subtract 3250 from72532

M = 72532

10’s Complement of N = 96750

Sum = 169282

Discard end Carry 105 = -100000

Answer = 69282

When subtracting with complements, negativeanswer is recognized from absence of end carryand complemented result

COMPLEMENTS

� Exercise

�X = 1010100; Y = 1000011; Perform subtraction

(a) X – Y (b) Y – X using 2’s Complements and

repeat with 1’s Complements

�M = 3250; N = 72532; Perform subtraction M – N

using 10’s Complement and 9’s Complement

Signed Binary Numbers

� Positive numbers (including Zero) are represented

as unsigned numbers

� Normal negative numbers are represented by

Minus sign

� Because of hardware limitations, computers must

represent everything with binary digits

� Sign Bit 0 for positive and 1 for negative

� Represented at the left most position

Signed Binary Numbers

� Need to be able to represent both positive andnegative numbers - Following 3 representations

� Example: Represent +9 and -9 in 7 bit-binarynumber

Only one way to represent +9 ==> 0 001001

Signed magnitude representation

Signed 1's complement representation

Signed 2's complement representation

Signed Binary Numbers

� Three different ways to represent -9:

� In signed-magnitude: 1 001001

� In signed-1's complement: 1 110110

� In signed-2's complement: 1 110111

� In general, in computers, fixed point numbers arerepresented either integer part only or fractionalpart only

ARITHMETIC ADDITION: SIGNED MAGNITUDE

� Compare their signs

� If two signs are the same , ADD the two magnitudes -

Look out for an overflow

� If not the same , compare the relative magnitudes of

the numbers and then SUBTRACT the smaller from

the larger --> need a subtractor to add

� Determine the sign of the result

ARITHMETIC ADDITION: SIGNED MAGNITUDE

6 0110

+) 9 1001

15 1111 -> 01111

9 1001

- ) 6 0110

3 0011 -> 00011

9 1001

-) 6 0110

- 3 0011 -> 10011

6 0110

+) 9 1001

-15 1111 -> 11111

6 + 9 -6 + 9

6 + (- 9) -6 + (-9)

Overflow9 1001

+) 9 1001(1)0010

ARITHMETIC ADDITION: SIGNED 2’s COMPLEMENT

� Add the two numbers, including their sign bit, and

discard any carry out of leftmost (sign) bit - Look

out for an overflow

Example6 0 0110

9 0 1001

15 0 1111

-6 1 1010

9 0 1001

3 0 0011

6 0 0110

-9 1 0111

-3 1 1101

-9 1 0111

-9 1 0111

-18 (1)0 1110

overflow9 0 10019 0 1001 2 operands have the same

signand the result sign changesxn-1yn-1s’n-1 + x’n-1y’n-1sn-1 =cn-1⊕⊕⊕⊕ cn

x’n-1y’n-1sn-1

(cn-1 cn)

xn-1yn s’n-1

(cn-1 cn)

18 1 0010

ARITHMETIC ADDITION: SIGNED 1’s

COMPLEMENT

� Add the two numbers, including their sign bits.

� If there is a carry out of the most significant (sign) bit, the result is incremented by 1 and the carry is discarded.

6 0 0110

-9 1 0110

-3 1 1100

-6 1 1001

9 0 1001

(1) 0(1)0010

1

3 0 0011

+) +)

+)

end-around carry

-9 1 0110-9 1 0110

(1)0 11001

0 1101

+)

+)

9 0 10019 0 1001

1 (1)0010

+)

overflow

Example

(cn-1 ⊕ ⊕ ⊕ ⊕ cn)

COMPARISON OF REPRESENTATIONS

� Easiness of negative conversion

�S + M > 1’s Complement > 2’s Complement

� Hardware

�S+M: Needs an adder and a subtractor forAddition

�1’s and 2’s Complement: Need only an adder

� Speed of Arithmetic

�2’s Complement > 1’s Complement(end-around C)

� Recognition of Zero

�2’s Complement is fast

ARITHMETIC SUBTRACTION

� Arithmetic Subtraction in 2’s complement

�Take the complement of the subtrahend (including

the sign bit) and add it to the minuend including

the sign bits.

( ±±±± A ) - ( - B ) = ( ±±±± A ) + B

( ±±±± A ) - B = ( ±±±± A ) + ( - B )

BINARY CODES� BCD CODE

� Convert the decimal numbers to binary, perform

all the arithmetic operations in binary and convert

all the binary results back to decimal

Decimal BCD

Code

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

BINARY CODES

� Decimal 396 is represented in BCD with 12 bits as

0011 1001 0110, with each group of 4bits

representing one decimal digit

� BCD numbers requires more bits than its binary

equivalent

� BCD numbers are decimal numbers and not

binary numbers even though they use bits in their

representation

BINARY CODES

� BCD ADDITION

� Consider addition of two decimal digits in BCD,together with a possible carry from a previous lesssignificant pair of digits

� Since each digit does not exceed 9, sum cannot begreater than 9+9+1 = 19, with 1 in the sum being usedas a previous carry

� If we add BCD digits as if they are binary digits,binary sum will produce a result in the range from 0 to19.

� Binary result will be from 0000 to 10011 and for BCDit is 0000 to 1 1001

BINARY CODES

� Example: Add BCD numbers 184 and 576

1 1

187 = 0001 1000 0111

576 = 0101 0111 0110

____ ____ ____

0111 1111 1101

0110 0110

------ --------------------

760 0111 0110 0000

Invalid BCD Number

BINARY CODES

� Decimal Arithmetic

� Representation of signed decimal numbers in BCD

is similar to binary

� Sign Magnitude / Sign Complement methods can

be used

� ‘+’ is represented as 0000 and ‘-’ is represented

with BCD equivalent of 9 i.e., 1001

� To obtain 10’s complement in BCD, we take 9’s

complement first and then add ‘1’ to LSB

BINARY CODES

� Procedures used for signed 2’s complement is alsoapplicable for signed 10’s complement

� Addition is done adding all the digits includingsign bit and discarding end carry which assumesthat all –ve numbers are in 10’s complement form

� Ex: 0 375

+9 760

----------------

0 135

BINARY CODES

� Other Decimal Codes

� Codes can be formulated by arranging four bits in10 distinct possible combinations

� Weighted Codes: BCD, 2421 code

� Non Weighted Codes: Excess-3, 8 4 -2 -1 code

� In Weighted code, each bit position is assigned aweighting factor in such a way that each digit canbe evaluated by adding the weights of all 1’s in thecoded combination

� 2421 and Excess-3 are examples of selfcontemplating codes

BINARY CODES

� Self Contemplating codes has the property that 9’s

complement of a decimal number is obtained

directly by changing 1’s to 0’s and 0’s to 1’s in the

code

� Ex: 395 – 0110 1001 0101 (BCD Equivalent)

� Excess-3 Equivalent of 395 = 0110 1100 1000

� 9’s Complement of 395 = 1001 0011 0111

BINARY CODES

Decimal BCD(8421) 2421 84-2-1 Excess-3

0 0000 0000 0000 0011

1 0001 0001 0111 0100

2 0010 0010 0110 0101

3 0011 0011 0101 0110

4 0100 0100 0100 0111

5 0101 1011 1011 1000

6 0110 1100 1010 1001

7 0111 1101 1001 1010

8 1000 1110 1000 1011

9 1001 1111 1111 1100

BINARY CODES

� Gray Code

� Sometimes beneficiary to use to represent digitaldata when it is converted from analog data

� Advantage is that only one bit in the code groupchanges when going from one number to the next

� Ex: When going from 7 to 8, the gray codechanges from 0100 to 1100. only first bit changesfrom 0 to 1

� Used where the normal sequence of binarynumbers may produce an error or ambiguitywhen transiting from one number to next

BINARY CODES

Decimal

number

Gray Binary

g3 g2 g1 g0 b3 b2 b1 b0

0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 1

2 0 0 1 1 0 0 1 0

3 0 0 1 0 0 0 1 1

4 0 1 1 0 0 1 0 0

5 0 1 1 1 0 1 0 1

6 0 1 0 1 0 1 1 0

7 0 1 0 0 0 1 1 1

8 1 1 0 0 1 0 0 0

9 1 1 0 1 1 0 0 1

10 1 1 1 1 1 0 1 0

11 1 1 1 0 1 0 1 1

12 1 0 1 0 1 1 0 0

13 1 0 1 1 1 1 0 1

14 1 0 0 1 1 1 1 0

15 1 0 0 0 1 1 1 1

BINARY CODES

� ASCII Character Code

ASCII (American Standard Code for Information Interchange) Code

BINARY CODES

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

NUL

SOH

STX

ETX

EOT

ENQ

ACK

BEL

BS

HT

LF

VT

FF

CR

SO

SI

SP

!

“

#

$

%

&

‘

(

)

*

+

,

-

.

/

0

1

2

3

4

5

6

7

8

9

:

;

<

=

>

?

@

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

[

\

]

m

n

‘

a

b

c

d

e

f

g

h

I

j

k

l

m

n

o

b4b3b2b1b0 0 1 2 3 4 5 6 7

DLE

DC1

DC2

DC3

DC4

NAK

SYN

ETB

CAN

EM

SUB

ESC

FS

GS

RS

US

P

q

r

s

t

u

v

w

x

y

z

{

|

}

~

DEL

b7b6b5

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

BINARY CODESNUL Null

SOH Start of Heading (CC)

STX Start of Text (CC)

ETX End of Text (CC)

EOT End of Transmission (CC)

ENQ Enquiry (CC)

ACK Acknowledge (CC)

BEL Bell

BS Backspace (FE)

HT Horizontal Tab. (FE)

LF Line Feed (FE)

VT Vertical Tab. (FE)

FF Form Feed (FE)

CR Carriage Return (FE)

SO Shift Out

SI Shift In

DLE Data Link Escape (CC)

(CC) Communication Control

(FE) Format Effector

(IS) Information Separator

DC1 Device Control 1

DC2 Device Control 2

DC3 Device Control 3

DC4 Device Control 4

NAK Negative Acknowledge (CC)

SYN Synchronous Idle (CC)

ETB End of Transmission Block (CC)

CAN Cancel

EM End of Medium

SUB Substitute

ESC Escape

FS File Separator (IS)

GS Group Separator (IS)

RS Record Separator (IS)

US Unit Separator (IS)

DEL Delete

BINARY CODES

� Alphanumeric character set is a set of elementsthat includes the 10 digits, 26 letters of alphabetsand a number of special characters

� Such set contains 36-64 elements if only capitalletters are included or 64-128 if both lower andupper case letters

� In first case we need binary code of 6 bits and insecond case we need binary code of 7 bits

� Standard binary code for alphanumericcharacters is ASCII

BINARY CODES

� Ex: Letter ‘A’ represented in ASCII as 1000001

(Colum 100 and Row 0001)

� ASCII code contains 94 graphic characters (26

uppercase letters, 26 lowercase letters, 10

numerals, 32 special printable characters like %,

$, * etc.) that can be printed and 34 non printing

characters used for various controls functions

(Abbreviated names)

� Control characters are used for routing data and

arranging printed text into prescribed format

BINARY CODES

� Control Characters – 3Types

�Format Effectors (control layout of printing – Back

Space, Horizontal Tabulation, Carriage Return)

� Information Separators (Separate data into

divisions such as Paragraphs and Pages)

�Communication Control (Useful during

transmission of text between remote terminals)

� ASCII is 7-bit code, but most computers

manipulate 8-bit quantity as a single unit

BINARY CODES

� Error Detecting Code

� Simplest method for error detection

� One parity bit attached to the information

� Even Parity and Odd Parity

� Even Parity

� One bit is attached to the information so that the total number of 1 bits is an even number

� 1011001 0

� 1010010 1

� Odd Parity

� One bit is attached to the information so that the total number of 1 bits is an odd number

� 1011001 1

� 1010010 0

BINARY CODES

� Parity bit is helpful in detecting errors during

transmission of information from one location to

another.

� Detects one, three or any odd combination of

errors in each character that is transmitted. Even

Combination is undetected

Binary Storage and Registers

� Binary cell is a device that possesses two stable

states and is capable of storing one bit of

information

� Register – Group of binary cells

� A register with n cells can store any discrete

quantity of information that contains n bits

� State of register is an n-tuple number of 1’s and

0’s, with each bit designating the state of one cell

Binary Storage and Registers

� Ex: 16 bit register – 1100001111001001

� A register with 16 cells can be in one of the 2n possiblestate

� Register can store discrete elements of information andthat the same bit configuration ma be interpreteddifferently for different type of data

� Register Transfer –

�Basic operation in digital systems consistingtransfer of binary information from one register toanother register.

�Transfer may be direct or indirect

Binary Storage and Registers

Binary Storage and Registers

Binary Logic

� Consists of binary variables and logic operations

� Variables designated by letters of alphabets like A,B, C, x, y, z etc., with each variable having two andonly two distinct possible values: 1 and 0

� 3 Basic Logical operations: AND, OR and NOT

�AND (.) – x.y = z or xy = z (x and y = z)

�OR (+) – x+y = z (x or y =z)

�NOT (‘ or ~) – x’ = z (x is not equal to z)

� Binary Logic is different from binary arithmetic

Binary Logic

� For each combination of x and y, there is a value z

specified by definition of logic operation

� Definitions listed in compact form using Truth

Table

� Truth Table is a Table of all possible combinations

of the variables showing the relation between

values that the variables may take and the result

of operation

Binary Logic

A B A+B

0 0 0

0 1 1

1 0 1

1 1 1

A B A*B

0 0 0

0 1 0

1 0 0

1 1 1

a A

0 1

1 0

Binary Logic

� Logic Gates are electronic circuits that operate on one

or more input signals to produce an output signal

� The input terminals of digital circuits accept binary

signals within allowable range and respond at the

output terminals with binary signals that fall within

specific range

� Timing diagrams illustrate response of each gate to the

four input signal combinations

� Horizontal axis – Time; Vertical Axis – Signal changes

between two possible voltage levels

Binary Logic

Binary Logic

Binary Logic

Binary Logic

� END OF UNIT - I