Chapter 1 Ppt
-
Upload
santhosh-surya-kiran -
Category
Documents
-
view
53 -
download
5
Transcript of Chapter 1 Ppt
![Page 1: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/1.jpg)
A VERY
GOOD
MORNING TO
ALL
OF
U
![Page 2: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/2.jpg)
BINARY SYSTEMS
CHAPTER 1
DIGITAL LOGIC DESIGN
![Page 3: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/3.jpg)
TOPICS
• DIGITAL SYSTEMS• NUMBER SYSTEMS• NUMBER BASE CONVERSIONS• OCTAL AND HEXADECIMAL NUMBERS• COMPLEMENTS• SIGNED BINARY NUMBERS• BINARY CODES• BINARY STORAGE AND REGISTERS• BINARY LOGIC
![Page 4: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/4.jpg)
DIGITAL SYTEMS
• SYSTEM• TYPES OF SYSTEMS• CLASSIFICATION OF SYSTEMS• APPLICATIONS OF DIGITAL SYSTEMS• ADVANTAGES OF DIGITAL SYSTEMS• DISADVANTAGES OF DIGITAL SYSTEMS• DIGITAL COMPUTER• DESIGN OF DIGITAL SYSTEMS• SWITCHING CIRCUITS• WHY BINARY IN DIGITAL SYSTEMS?
![Page 5: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/5.jpg)
SYSTEM
Accepts various Inputs Performs a particular task Generates output
![Page 6: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/6.jpg)
Classification of SystemsAnalog Systems
(Continuous)Digital Systems(Discrete step
by step)
Physical quantities or signals may vary continuously over a specified range.
Physical quantities or signals can assume only discrete values .
Hence represented by continuously variable indicator
Represented by symbols called digits
Thermometer Digital Clock
![Page 7: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/7.jpg)
Applications of Digital Systems
• Communication
![Page 8: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/8.jpg)
Applications of Digital Systems
• Business Transactions
![Page 9: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/9.jpg)
Applications of Digital Systems
• Traffic Control
![Page 10: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/10.jpg)
Applications of Digital Systems
• Space Guidance• Weather
Monitoring
![Page 11: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/11.jpg)
Applications of Digital Systems
• Medicine
![Page 12: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/12.jpg)
Applications of Digital Systems
• Internet
![Page 13: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/13.jpg)
Applications of Digital Systems
• Commercial
![Page 14: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/14.jpg)
Applications of Digital Systems
• Commercial
![Page 15: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/15.jpg)
Applications of Digital Systems
• Commercial
![Page 16: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/16.jpg)
Applications of Digital Systems
Commercial
![Page 17: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/17.jpg)
Applications of Digital Systems
Industry
![Page 18: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/18.jpg)
Applications of Digital Systems
• Scientific Enterprises
![Page 19: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/19.jpg)
Applications of Digital Systems
• Military
![Page 20: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/20.jpg)
Advantages of Digital Systems
• Easier to Design• Information storage is easy• Accuracy and Precision
through out the system• Operations can be
programmed• Digital Circuits are less prone
to noise
![Page 21: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/21.jpg)
Disadvantages of Digital Systems
• Real world is analog• Digitization of information is a
time consuming process.
![Page 22: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/22.jpg)
Digital Computer• It can follow a sequence of instructions
called a program, that operates on given data.
• Program and data can be varied according to the user’s needs.
• Hence, it can perform various information processing tasks that fulfill several applications.
![Page 23: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/23.jpg)
Digital Computer
• One important characteristic of digital computer is the ability to manipulate discrete elements of information.
• Discrete information must contain finite number of elements.
• Eg: 10 Decimal digits, 26 alphabets, 52 playing cards, 64 chess squares
![Page 24: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/24.jpg)
Digital Computer• The name Digital Computer emerged
from an application.• Early computers are used for numeric
computations in which discrete elements are digits.
• Physical quantities used to represent discrete information are signals.
• Some of the signals are voltage and current signals (Implemented by transistors)
![Page 25: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/25.jpg)
Digital Computer• But signals used in digital systems have 2
discrete values 0 & 1 ( binary)• Binary Digit- Bit 0 or 1• Group of bits – Binary Codes• Hence using various techniques, groups of bits
can represent discrete symbols.• These symbols are again used to develop
system in digital format.• So, we can say that digital system manipulates
discrete elements of information which is again represented internally in binary form
![Page 26: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/26.jpg)
Design of Digital SystemsSystem Design
Logic Design
Circuit Design
![Page 27: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/27.jpg)
System Design •Breaking overall system into subsystems & specifying chcs. of each sub-systems
Logic Design •Determines how to interconnect and control these sub-systems
Circuit Design •Specifying the interconnection of specific components like resistors, diodes and transistors to form gates, flip flops, or other logic building blocks
![Page 28: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/28.jpg)
Switching Circuits
• Many of sub systems of digital systems take form of a switching circuit.
• It consists of one or more inputs and outputs having discrete values.
Switching Circuit
x₁
x₂..
xm
Z₁
Z₂..
Zm
![Page 29: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/29.jpg)
Classification of Switching Circuits
Combinational Circuits Sequential Circuits
Output depends on present on input only
Output depends on both past and present inputs
Building blocks are logic gates
Building blocks are logic gates and flip flops
No memory is required as no storage is necessary
Memory is required as past inputs are to be stored
Eg: Multiplexers, Decoders, Encoders, PLDs, PLAs, PALs, CPLDs, FPGAs etc.
Eg: Ring Counter, Synchronous Counter, Ripple Counter
![Page 30: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/30.jpg)
Why binary in Digital Systems
• In general, switching devices used in digital systems are generally two-state devices.
• So, output can assume only two discrete values.
![Page 31: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/31.jpg)
Switching devices
Relay
On
Diode Transistor
![Page 32: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/32.jpg)
Number Systems
• Decimal Number System (10)• Binary Number System ( 2 )• Octal Number System ( 8 )• Hexadecimal Number System (16)
![Page 33: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/33.jpg)
Decimal Number System
Representation
=5*10^2 + 0*10^1 + 1*10^0 + 6*10^-1 + 8*10^-2
(501.68)10
![Page 34: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/34.jpg)
Decimal Number System
• Numbers have positional importance• 349.2510
In the binary system, positional importance follows powers of 2
3 x 102 = 3 x 100 = 300
4 x 101 = 4 x 10 = 40
9 x 100 = 9 x 1 = 9 2 x 10-1 = 2/10
5 x 10-2 = 5/100
![Page 35: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/35.jpg)
Conversion from Binary to Decimal
• (11001.11)₂
= 1*2^4 + 1*2^3 + 0*2^2 + 0* 2^1 + 1*2^0 + 1*2^-1 + 1*2^-2 = 16 + 8 + 0 + 0 + 1 + 0.5 + 0.25
= ( 25.75 )10
![Page 36: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/36.jpg)
Conversion from Octal to Decimal
( 347.205)₈
= 3 * 8^2 + 4 * 8^1 + 7 * 8^0 + 2 * 8^-1 + 0 * 8^-2 + 5 * 8^-3
= 192 + 32 + 7 + 0.25 + 0 + 0.01
= ( 231.26)10
![Page 37: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/37.jpg)
Conversion from Hexadecimal to Decimal
• ( 23A4.EC)16
= 2 * 16^3 + 3 * 16^2 + A * 16^1 + 4 * 16^0 + E * 16^-1 + C * 16^-2= 8192 + 768 + 160 + 4 + 0.875 + 0.0468
= (9214.9218 )10
![Page 38: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/38.jpg)
Conversion from Decimal to Binary
( 61 )10
![Page 39: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/39.jpg)
Conversion from Decimal to Binary
( 61 )10
Decimal value Integer Fraction Coefficient
0001
![Page 40: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/40.jpg)
Conversion from Decimal to Octal
(247.6875)10
= (367.54)₈
![Page 41: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/41.jpg)
Octal Number SystemOctal Numbers Binary Equivalents
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
![Page 42: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/42.jpg)
Hexadecimal Number SystemDecimal Values Hexadecimal
RepresentationBinary Equivalents
0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
10 A 1010
11 B 1011
12 C 1100
13 D 1101
14 E 1110
15 F 1111
![Page 43: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/43.jpg)
Binary Arithmetic
• Binary Addition• Binary Subtraction• Binary Multiplication• Binary Division
![Page 44: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/44.jpg)
Binary Addition
1 0 1 1 0 1 0 1 + 1 1 0 0 + 1 1 1 1 _ _ _ _ _ _ _ _ _ _ 1 0 1 1 1 1 0 1 0 0
![Page 45: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/45.jpg)
Addition of two binary numbers
1410 = 011102
+2510 = 1100121
0
1
0
1
0
0
1
0
1
1
0
= 32 + 7 = 39
![Page 46: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/46.jpg)
1 0 1 1 0 1
+ 0 1 1 1 0 1
1
0
0
1
1
0
1
1
1
0
1
0
1
Carry
Sum
Check your work
45
+ 29
= 74
![Page 47: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/47.jpg)
Binary Subtraction
1 0 1 1 Minuend11101
- 0 1 1 0 Subtrahend - 10011
_ _ _ _ _ _ _ _ _ _ _ _ 0 1 0 1 Difference
01010
![Page 48: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/48.jpg)
Binary Multiplication
1001 Mul t ip l icand *1101 Mul t ip l ier _____
1001 0000 1001
1001 _________
1110101
Partial Products
Final Product
![Page 49: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/49.jpg)
Binary Division 1101 1001 1110101 1001 - - - - - - - - - 1011 1001 - - - - - - - - - - - 1001 1001
- - - - - - - - - - - 0000
![Page 50: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/50.jpg)
Complements
Used in digital computers • for simplifying the subtraction
operation and• for logical manipulation.
![Page 51: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/51.jpg)
Diminished Radix Complement
• Also called (R-1)’s complement• (R-1)’s complement of any number system
can be defined as ( Rⁿ-1 )-N• R = Base or Radix of a given number system• N = given number• n = no. of digits present in the given number• For Decimal Number System, R-1’s
complement is (10ⁿ-1) –N• For Binary Number System, R-1’s
complement is (2ⁿ-1) –N
![Page 52: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/52.jpg)
For Decimal Number System
• R = 10 ==> (R-1) = 9• ( R-1 )’s complement = 9’s complement = (10ⁿ-1) –N (R-1)’s complement of 546700 is 9’s complement of 546700 = (10⁶-1)-
(546700)
= 99999-546700
= 453299
![Page 53: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/53.jpg)
For Binary Number System
• R = 2 ==> (R-1) = 1• ( R-1 )’s complement = 9’s complement = (2ⁿ-1) –N (R-1)’s complement of 1011000 is 1’s complement of 1011000 =(2⁷-1)-
(1011000)
= 1111111-1011000
= ( 0100111 )₂
![Page 54: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/54.jpg)
Radix Complement
• Also called R’s complement• R ’s complement of any number system can
be defined as [ ( Rⁿ-1 )-N ]+ 1• R = Base or Radix of a given number system• N = given number• n = no. of digits present in the given number• For Decimal Number System, R-1’s
complement is [ (10ⁿ-1) –N ] + 1• For Binary Number System, R ’s complement
is [ (2ⁿ-1) –N ] + 1
![Page 55: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/55.jpg)
For Decimal Number System
• R = 10 R’s complement = 9’s complement
= [ (10ⁿ-1) –N ] + 1 R’s complement of 546700 is 10’s complement of 546700 = 9’s
complement + 1
9’s complement = (10⁶-1)- (546700) = 999999-546700
= 45329910’s complement = 453299 + 1 = 453300
![Page 56: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/56.jpg)
For Binary Number System
• R = 2 ==> R’s complement = 2’s complement
= (2ⁿ-1) –N R’s complement of 1011000 is 1’s
complement + 1
1’s complement of 1011000 =(2⁷-1)- (1011000)
= 1111111-1011000
= ( 0100111 )₂
2’s complement = 0100111 + 1 = (01001000)₂
![Page 57: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/57.jpg)
Subtraction with Complements(Unsigned Numbers)
• During subtraction of two n- digit unsigned numbers M & N of same base R, there occurs two cases
• Case (i): M>N• Case (ii): M<N• This operation can be applied for
any number system
![Page 58: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/58.jpg)
For Decimal Number System
• M= minuend = 72532
• N = subtrahend = 3250• Perform M-N• Case (i) M>N is to be applied
M= 72532 M= 72532N = 03250 10’s complement N = 96750
--_______ +_______
69282 169282
![Page 59: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/59.jpg)
For Decimal Number System
So case (ii) : M<NM = 03250 M = 03250N = 72532 N = 27468
-______ +_____
-69282 30718
![Page 60: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/60.jpg)
For Binary Number System
• M= minuend = 1010100• N = subtrahend = 1000011• Perform M-N• Case (i) M>N is to be applied
M= 1010100 M= 1010100N = 1000011 2’s complement N = 0111101
-- _________ +_________ 0010001 10010001
![Page 61: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/61.jpg)
For Binary Number System
So case (ii) : M<NM = 1000011 M = 1000011N = 1010100 N = 0101100
-_______ +________
- 0010001 1 1 0 1 1 1 1
Final Answer = - ( 2’s complement of 1101111) = - ( 0010001)₂
![Page 62: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/62.jpg)
Signed- Binary Numbers
• “+ve” sign indicates a positive number• “-ve” sign indicates a negative number• Digital Circuits can understand only two
numbers 0 & 1• Hence to indicate the sign, an additional
bit is placed as the most significant bit.• 0 represents a +ve number• 1 represents a –ve number
• “+ve” sign indicates a positive number• “-ve” sign indicates a negative number• Digital Circuits can understand only two
numbers 0 & 1• Hence to indicate the sign, an additional
bit is placed as the most significant bit.• 0 represents a +ve number• 1 represents a –ve number
![Page 63: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/63.jpg)
Signed- Binary Numbers
Consider an 8 bit number (01000100)₂
MSB of this no. is 0 which represents a +ve sign i.e., a positive number
Its equivalent is (01000100)₂ = (+68)10
Consider another number (11000100)₂
MSB of this no. is 1 which represents –ve sign i.e., a negative number
Its equivalent is (11000100)₂ = (-68)10
![Page 64: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/64.jpg)
Signed- Binary Numbers
(101100)₂ =
MSB = 1 -ve number Magnitude =
(01100) = (12)₂ 10
Answer = Sign & Magnitude = (-12)10
![Page 65: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/65.jpg)
Signed- Binary Numbers
(0111)₂ =
MSB = 0 +ve number Magnitude =
(111) = ( 7 )₂ 10
Answer = Sign & Magnitude = ( +7 )10
![Page 66: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/66.jpg)
Signed- Binary Numbers1’s Complement representation
Also called Signed Complement representation
i.e., Sign + Complement
( 0101 )₂ = (+5)10
( 1010 )₂ = ( -5 )10 in 1’s complement form
( 01000 )₂ = (+8)10
( 10111 )₂ = ( -8 )10 in 1’s complement form
![Page 67: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/67.jpg)
Signed- Binary Numbers2’s Complement representation
Also called Signed 2’s Complement representation
i.e., Sign + 2’s Complement of magnitude
( 0101 )₂ = (+5)10
( 1011 )₂ = ( -5 )10 in 2’s complement form
( 01000 )₂ = (+8)10
( 11000)₂ = ( -8 )10 in 2’s complement form
![Page 68: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/68.jpg)
Signed Decimal Numbers
Signed Magnitude Signed 1’s complement
Signed 2’s complement
(+7) 0111 0111 0111
(+6) 0110 0110 0110
(+5) 0101 0101 0101
(+4) 0100 0100 0100
(+3) 0011 0011 0011
(+2) 0010 0010 0010
(+1) 0001 0001 0001
(+0) 0000 0000 0000
(-0) 1000 1111 --
(-1) 1001 1110 1111
(-2) 1010 1101 1110
(-3) 1011 1100 1101
(-4) 1100 1011 1100
(-5) 1101 1010 1011
(-6) 1110 1001 1010
(-7) 1111 1000 1001
(-8) -- -- 1000
![Page 69: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/69.jpg)
Addition of 2 signed numbers
+ 6 00000110 +13 00001101 - - - - - -
- - - - - - - - - - - - - - - - - - - +19 00010011
- 6 11111010 +13 00001101 ------ ------------------- +19 00000111
- 6 11111010 -13 11110011 ------ ------------------- +19 11101101
+ 6 00000110 -13 11110011 ------ ------------------- +19 11111001
![Page 70: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/70.jpg)
Subtraction of 2 signed numbers
• (+ A) – (+B) = (+ A) + (-B)• (+ A) – (-B) = (+ A) + (+B)
![Page 71: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/71.jpg)
Overflow, Signed Integers
• As has been shown, when numbers are treated as signed integers, a “carry” of 1 from the addition of the most significant bits DOES NOT indicate an overflow, 3 00011+ (-3) +11101= 0 = 00000, with a carry of “1”
• For signed integers, overflow occurs when:
• The addition of two positive numbers results in a negative number, orThe addition of two negative numbers results in a positive number
![Page 72: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/72.jpg)
Overflow Examples• In a 6-bit register
+ 17 = 010001+ 16 = +010000
=100001100001 = - (011110 + 1) = - 011111 = -31
• In an 8-bit register- 100 = - (0110 0100) = 1001 1011 +1 = 1001 1100- 50 = - (0011 0010) = 1100 1101 +1 = 1100 1110
= 0110 10100110 1010 = 6A16=6*16 + 10 = +106
![Page 73: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/73.jpg)
Range of a numberOverflow during addition
• A fixed-length register can only hold a Range of numbers
• For a 4-bit device, the range of positive integers is 0 - 15
• For an 8-bit device the range of positive integers is 0 - 255
• When adding positive integers, Overflow occurs when the sum falls outside the range of the register
![Page 74: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/74.jpg)
Overflow Summary
• For positive integers, overflow occurs when the carry from addition of the leftmost bits is a “1”
• For signed integers, overflow occurs when either
The addition of two negative numbers gives a positive number, or
The addition of two positive numbers gives a negative number.
![Page 75: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/75.jpg)
Binary Codes
Codes
Weighted
Non-weighted
![Page 76: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/76.jpg)
Weighted Codes
• BCD (8421) • (2421)• (5421)• (63-1-1)• (7421)
• N= w₃a₃ + w₂a₂ + w₁a₁
![Page 77: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/77.jpg)
BINARY CODES
• Most compatible system for a computer or a digital system is binary system
• Most of the users are accustomed to decimal number system
• To reduce this gap, decimal numbers are converted to binary, arithmetic calculations are performed in binary, and then converted back to decimal.
![Page 78: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/78.jpg)
BINARY CODES• Code is a symbolic representation of an
information transform
• During this process, we need to store decimal numbers in computer for performing conversion
• But computers accept only digits 0s & 1s
• So, we must represent these decimal digits by means of a code consisting of 0s & 1s.
• Arithmetic operations can be directly performed with decimal numbers when they are stored in computer in coded from.
![Page 79: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/79.jpg)
BCD Code
• In simplest form of binary code, each decimal digit is represented by its binary equivalent.
8 5 4 . 7 9 2
1000 0101 0100 0111 1001 0010
![Page 80: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/80.jpg)
Representation of BCD Code
( 3 4 5 )10 = ( 0011 0100 0101)BCD
= ( 101011001 )₂
( 1 5 7 )10 = ( 0001 0101 0111 )₂
BCD representation – 12 bits – denotes a decimal number
Binary value – 8 bits – denotes binary value itself
![Page 81: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/81.jpg)
BCD Addition
1 8 4 0001 1000 0100 + 5 7 6 0101 01 1 1 01 10 ---------- ---------------------- 7 6 0 0 111 10000 1010 0 11 0 0110
----------------------- 0 111 0110 0000
![Page 82: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/82.jpg)
Non-Weighted Codes
• Excess – 3 code• Gray code• 2 out of 5 code• Biquinary code
![Page 83: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/83.jpg)
Excess-3 code
![Page 84: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/84.jpg)
Gray Codes
The property of this code is that the successive decimal digits differ in exactly one bit.
![Page 85: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/85.jpg)
Conversion from Binary to Gray
![Page 86: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/86.jpg)
Conversion of Gray to Binary
![Page 87: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/87.jpg)
Alpha numeric codes
• ASCII code• EBCDIC code
![Page 88: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/88.jpg)
![Page 89: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/89.jpg)
![Page 90: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/90.jpg)
Binary Storage & Registers
![Page 91: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/91.jpg)
Register Transfer
![Page 92: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/92.jpg)
Logical Operation in Registers
![Page 93: Chapter 1 Ppt](https://reader033.fdocuments.net/reader033/viewer/2022061107/544c8e5ab1af9f36788b49d2/html5/thumbnails/93.jpg)
Binary Logic