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![Complements Radix Complement Example: Base-10 Example: Base-2The r's complement of an n-digit number N in base r is defined asrn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r 1) 'scomplement, we note that the r's complement is obtained by adding 1to the (r 1) 's complement, since rn – N = [(rn 1) – N] + 1.The 10's complement of 012398 is 987602The 10's complement of 246700 is 753300The 2's complement of 1101100 is 0010100The 2's complement of 0110111 is 1001001](/image.pl?url=https%3a%2f%2fimage.slidesharecdn.com%2fchapter1digitalsystemsandbinarynumbers-151021072016-lva1-app6891%2f75%2fChapter-1-digital-systems-and-binary-numbers-26-2048.jpg&f=jpg&w=240)
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Chapter 1 digital systems and binary numbers
Digital systems represent information using discrete binary values of 0 and 1 rather than continuous analog values. Binary numbers use a base-2 numbering system with place values that are powers of 2. There are various number systems like decimal, binary, octal and hexadecimal that use different number bases and represent the same number in different ways. Complements are used in binary arithmetic to perform subtraction by adding the 1's or 2's complement of a number. The 1's complement is obtained by inverting all bits, while the 2's complement is obtained by inverting all bits and adding 1.
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Introduction to digital systems, binary numbers, applications, and types such as digital computers and discrete processing systems.
Discussion on analog vs digital signals, binary signal representation, and different number systems: Decimal, Octal, Binary, and Hexadecimal.
Basic operations on number systems including Binary Addition, Subtraction, and Multiplication, emphasizing the column method.
Methods for converting between Decimal, Binary, Octal, and Hexadecimal systems, illustrating both integer and fractional conversions.
Description of complements, including 1's and 2's complements, their uses in subtraction, and signed binary numbers representation.
Presentation of various binary codes, including BCD, Gray Code, and ASCII, along with examples of usage and applications.
Explanation of ASCII character codes, their properties, and parity bits for error detection in data communication.
Insight into binary cells and registers, their functions in digital systems, and information transfer mechanisms.
Fundamentals of binary logic, definition, basic logical operations, and the role of logic gates in digital circuits.
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Chapter 1 digital systems and binary numbers
- 1.Chapter 1Digital Systemsand Binary NumbersDigital Logic Design
- 2.CSE-221 Lectures: 3hours/weekDigital Logic Design Credits: 3Number systems & codes, Digital logic: Boolean algebra, De-Morgan'sTheorems, logic gates and their truth tables, canonical forms,combinational logic circuits, minimization technique, Arithmetic anddata handling logic circuits, decoders and encoders, multiplexes and de-multiplexers, Combinational circuit design, Flip-flops, race aroundproblems; Counters: asynchronous counters, synchronous counters andtheir applications; PLA design; Synchronous and asynchronous logicdesign; State diagram, Mealy and Moore machines; Stateminimization’s and assignments; Pulse mode logic; Fundamental modedesign.CSE-222 Digital logic Design (Sessional) Contact hour: 3 hours/weekSessional based on CSE-221 Credits: 1.5
- 3.Outline of Chapter1 1.1 Digital Systems 1.2 Binary Numbers 1.3 Number-base Conversions 1.4 Octal and Hexadecimal Numbers 1.5 Complements 1.6 Signed Binary Numbers 1.7 Binary Codes 1.8 Binary Storage and Registers 1.9 Binary Logic
- 4.Digital Systems andBinary Numbers Digital age and information age Digital computers General purposes Many scientific, industrial and commercial applications Digital systems Telephone switching exchanges Digital camera Electronic calculators, Digital TV Discrete information-processing systems Manipulate discrete elements of information For example, {1, 2, 3, …} and {A, B, C, …}…
- 5.Analog and DigitalSignal Analog system The physical quantities or signals may vary continuously over a specifiedrange. Digital system The physical quantities or signals can assume only discrete values. Greater accuracytX(t)tX(t)Analog signal Digital signal
- 6.Binary Digital SignalAn 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 valuesor ranges of values of physical quantities.tV(t)Binary digital signalLogic 1Logic 0undefine
- 7.Decimal Number SystemBase (also called radix) = 10 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } Digit Position Integer & fraction Digit Weight Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation1 0 -12 -25 1 2 7 410 1 0.1100 0.01500 10 2 0.7 0.04d2*B2+d1*B1+d0*B0+d-1*B-1+d-2*B-2(512.74)10
- 8.Octal Number SystemBase = 8 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 } Weights Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation1 0 -12 -28 1 1/864 1/645 1 2 7 45 *82+1 *81+2 *80+7 *8-1+4 *8-2=(330.9375)10(512.74)8
- 9.Binary Number SystemBase = 2 2 digits { 0, 1 }, called binary digits or “bits” Weights Weight = (Base) Position Magnitude Sum of “Bit x Weight” Formal Notation Groups of bits 4 bits = Nibble8 bits = Byte1 0 -12 -22 1 1/24 1/41 0 1 0 11 *22+0 *21+1 *20+0 *2-1+1 *2-2=(5.25)10(101.01)21 0 1 11 1 0 0 0 1 0 1
- 10.Hexadecimal Number SystemBase = 16 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F } Weights Weight = (Base) Position Magnitude Sum of “Digit x Weight” Formal Notation1 0 -12 -216 1 1/16256 1/2561 E 5 7 A1 *162+14 *161+5 *160+7 *16-1+10 *16-2=(485.4765625)10(1E5.7A)16
- 11.The Power of2n 2n0 20=11 21=22 22=43 23=84 24=165 25=326 26=647 27=128n 2n8 28=2569 29=51210 210=102411 211=204812 212=409620 220=1M30 230=1G40 240=1TMegaGigaTeraKilo
- 12.Addition Decimal Addition5555+011= Ten ≥ Base Subtract a Base11 Carry
- 13.Binary Addition ColumnAddition1 0 11111111 0+0000 1 11≥ (2)10111111= 61= 23= 84
- 14.Binary Subtraction Borrowa “Base” when needed0 0 11101111 0−0101 1 10= (10)2222 210001= 77= 23= 54
- 15.Binary Multiplication Bitby bit01 1 1 101 1 000 0 0 001 1 1 101 1 1 10 0 0000110111 0x
- 16.Number Base ConversionsDecimal(Base10)Octal(Base 8)Binary(Base 2)Hexadecimal(Base 16)EvaluateMagnitudeEvaluateMagnitudeEvaluateMagnitude
- 17.Decimal (Integer) toBinary Conversion Divide the number by the ‘Base’ (=2) Take the remainder (either 0 or 1) as a coefficient Take the quotient and repeat the divisionExample: (13)10Quotient Remainder CoefficientAnswer: (13)10 = (a3 a2 a1 a0)2 = (1101)2MSB LSB13/ 2 = 6 1 a0 = 16 / 2 = 3 0 a1 = 03 / 2 = 1 1 a2 = 11 / 2 = 0 1 a3 = 1
- 18.Decimal (Fraction) toBinary Conversion Multiply the number by the ‘Base’ (=2) Take the integer (either 0 or 1) as a coefficient Take the resultant fraction and repeat the divisionExample: (0.625)10Integer Fraction CoefficientAnswer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2MSB LSB0.625 * 2 = 1 . 250.25 * 2 = 0 . 5 a-2 = 00.5 * 2 = 1 . 0 a-3 = 1a-1 = 1
- 19.Decimal to OctalConversionExample: (175)10Quotient Remainder CoefficientAnswer: (175)10 = (a2 a1 a0)8 = (257)8175 / 8 = 21 7 a0 = 721 / 8 = 2 5 a1 = 52 / 8 = 0 2 a2 = 2Example: (0.3125)10Integer Fraction CoefficientAnswer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)80.3125 * 8 = 2 . 50.5 * 8 = 4 . 0 a-2 = 4a-1 = 2
- 20.Binary − OctalConversion 8 = 23 Each group of 3 bits represents an octaldigitOctal Binary0 0 0 01 0 0 12 0 1 03 0 1 14 1 0 05 1 0 16 1 1 07 1 1 1Example:( 1 0 1 1 0 . 0 1 )2( 2 6 . 2 )8Assume ZerosWorks both ways (Binary to Octal & Octal to Binary)
- 21.Binary − HexadecimalConversion 16 = 24 Each group of 4 bits represents ahexadecimal digitHex Binary0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1A 1 0 1 0B 1 0 1 1C 1 1 0 0D 1 1 0 1E 1 1 1 0F 1 1 1 1Example:( 1 0 1 1 0 . 0 1 )2( 1 6 . 4 )16Assume ZerosWorks both ways (Binary to Hex & Hex to Binary)
- 22.Octal − HexadecimalConversion Convert to Binary as an intermediate stepExample:( 0 1 0 1 1 0 . 0 1 0 )2( 1 6 . 4 )16Assume ZerosWorks both ways (Octal to Hex & Hex to Octal)( 2 6 . 2 )8Assume Zeros
- 23.Decimal, Binary, Octaland HexadecimalDecimal Binary Octal Hex00 0000 00 001 0001 01 102 0010 02 203 0011 03 304 0100 04 405 0101 05 506 0110 06 607 0111 07 708 1000 10 809 1001 11 910 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F
- 24.1.5 Complements Thereare two types of complements for each base-r system: the radix complement anddiminished radix complement. Diminished Radix Complement - (r-1)’s Complement Given a number N in base r having n digits, the (r–1)’s complement of N isdefined as:(rn –1) – N Example for 6-digit decimal numbers: 9’s complement is (rn – 1)–N = (106–1)–N = 999999–N 9’s complement of 546700 is 999999–546700 = 453299 Example for 7-digit binary numbers: 1’s complement is (rn – 1) – N = (27–1)–N = 1111111–N 1’s complement of 1011000 is 1111111–1011000 = 0100111 Observation: Subtraction from (rn – 1) will never require a borrow Diminished radix complement can be computed digit-by-digit For binary: 1 – 0 = 1 and 1 – 1 = 0
- 25.Complements 1’s Complement(Diminished Radix Complement) All ‘0’s become ‘1’s All ‘1’s become ‘0’sExample (10110000)2 (01001111)2If you add a number and its 1’s complement …1 0 1 1 0 0 0 0+ 0 1 0 0 1 1 1 11 1 1 1 1 1 1 1
- 26.Complements Radix ComplementExample: Base-10 Example: Base-2The r's complement of an n-digit number N in base r is defined asrn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r 1) 'scomplement, we note that the r's complement is obtained by adding 1to the (r 1) 's complement, since rn – N = [(rn 1) – N] + 1.The 10's complement of 012398 is 987602The 10's complement of 246700 is 753300The 2's complement of 1101100 is 0010100The 2's complement of 0110111 is 1001001
- 27.Complements 2’s Complement(Radix Complement) Take 1’s complement then add 1 Toggle all bits to the left of the first ‘1’ from the rightExample:Number:1’s Comp.:0 1 0 1 0 0 0 01 0 1 1 0 0 0 00 1 0 0 1 1 1 1+ 1OR1 0 1 1 0 0 0 000001010
- 28.Complements Subtraction withComplements The subtraction of two n-digit unsigned numbers M – N in base r can bedone as follows:
- 29.Complements Example 1.5Using 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.
- 30.Complements Example 1.7Given the two binary numbers X = 1010100 and Y = 1000011, perform thesubtraction (a) X – Y ; and (b) Y X, by using 2's complement.There is no end carry.Therefore, the answer isY – X = (2's complementof 1101111) = 0010001.
- 31.Complements Subtraction ofunsigned numbers can also be done by means of the (r 1)'scomplement. Remember that the (r 1) 's complement is one less then the r'scomplement. 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 of1101110) = 0010001.
- 32.1.6 Signed BinaryNumbersTo represent negative integers, we need a notation for negativevalues.It is customary to represent the sign with a bit placed in theleftmost position of the number since binary digits.The convention is to make the sign bit 0 for positive and 1 fornegative.Example:Table 1.3 lists all possible four-bit signed binary numbers in thethree representations.
- 33.
- 34.Signed Binary NumbersArithmetic addition The addition of two numbers in the signed-magnitude system follows the rules ofordinary 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. 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:
- 35.Signed Binary NumbersArithmetic Subtraction In 2’s-complement form: Example: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 BA B A B ( 6) ( 13) (11111010 11110011)(11111010 + 00001101)00000111 (+ 7)
- 36.1.7 Binary CodesBCD Code A number with k decimal digits willrequire 4k bits in BCD. Decimal 396 is represented in BCDwith 12bits as 0011 1001 0110, witheach group of 4 bits representing onedecimal digit. A decimal number in BCD is thesame as its equivalent binary numberonly when the number is between 0and 9. The binary combinations 1010through 1111 are not used and haveno meaning in BCD.
- 37.Binary Code Example:Consider decimal 185 and its corresponding value in BCD and binary: BCD addition
- 38.Binary Code Example:Consider the addition of 184 + 576 = 760 in BCD: Decimal Arithmetic: (+375) + (-240) = +135Hint 6: using 10’s of BCD
- 39.Binary Codes OtherDecimal Codes
- 40.Binary Codes) GrayCode The advantage is that only bit in thecode group changes in going fromone number to the next.» Error detection.» Representation of analog data.» Low power design.000 001010100110 1111010111-1 and onto!!
- 41.Binary Codes AmericanStandard Code for Information Interchange (ASCII) Character Code
- 42.Binary Codes ASCIICharacter Code
- 43.ASCII Character CodesAmerican Standard Code for Information Interchange (Refer toTable 1.7) 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 andflow control (e.g. STX and ETX start and end text areas).
- 44.ASCII Properties ASCIIhas 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.
- 45.Binary Codes Error-DetectingCode To detect errors in data communication and processing, an eighth bit issometimes added to the ASCII character to indicate its parity. A parity bit is an extra bit included with a message to make the totalnumber of 1's either even or odd. Example: Consider the following two characters and their even and odd parity:
- 46.Binary Codes Error-DetectingCode Redundancy (e.g. extra information), in the form of extra bits, can beincorporated into binary code words to detect and correct errors. A simple form of redundancy is parity, an extra bit appended onto the codeword 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. Example:100010011000100110 (odd parity)Message B:Message A: (even parity)
- 47.1.8 Binary Storageand Registers Registers A binary cell is a device that possesses two stable states and is capable of storingone of the two states. A register is a group of binary cells. A register with n cells can store any discretequantity of information that contains n bits. 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 in next slides.n cells 2n possible states
- 48.A Digital ComputerExampleSynchronous orAsynchronous?Inputs: Keyboard,mouse, modem,microphoneOutputs: CRT,LCD, modem,speakersMemoryControlunit DatapathInput/OutputCPU
- 49.Transfer of informationFigure1.1 Transfer of information among register
- 50.Transfer of informationThe other major componentof a digital system Circuit elements tomanipulate individual bits ofinformation Load-store machineLD R1;LD R2;ADD R2, R1;SD R3;Figure 1.2 Example of binary information processing
- 51.1.9 Binary LogicDefinition 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, Three basic logical operations: AND, OR, and NOT.
- 52.Binary Logic TruthTables, Boolean Expressions, and Logic Gatesx y z0 0 00 1 01 0 01 1 1x y z0 0 00 1 11 0 11 1 1x z0 11 0AND OR NOTxy z xy zz = x • y = x y z = x + y z = x = x’x z
- 53.
- 54.Binary Logic Logicgates Example of binary signals0123Logic 1Logic 0Un-defineFigure 1.3 Example of binary signals
- 55.Binary Logic Logicgates Graphic Symbols and Input-Output Signals for Logic gates:Fig. 1.4 Symbols for digital logic circuitsFig. 1.5 Input-Output signals for gates
- 56.Binary Logic Logicgates Graphic Symbols and Input-Output Signals for Logic gates:Fig. 1.6 Gates with multiple inputs