SWITCHING THEORY AND LOGIC CIRCUITS


 Brett Campbell
 2 years ago
 Views:
Transcription
1 SWITCHING THEORY AND LOGIC CIRCUITS
2 COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra & postulates, kmap method and tabular method) to simplify the given Boolean function. 3. To understand the fundamentals of digital logic and to design various combinational and sequential circuits. 4. To understand the concepts of programmable logic devices(plds) 5. To understand formal procedure for the analysis and design of synchronous and asynchronous sequential logic
3 COURSE OUTCOMES After completion of the course the student will be able to. Understand the concepts and techniques of number systems and codes in representing numerical values in various number systems and perform number conversions between different number systems and codes. 2. Apply the simplification methods to simplify the given Boolean function (Boolean algebra, kmap and Tabular method). 3. Implement given Boolean function using logic gates, MSI circuits and/ or PLD s.
4 COURSE OUTCOMES After completion of the course the student will be able to 4. Design and decoders, analyze encoders, various combinational multiplexers, and circuits like demultiplexers, arithmetic circuits (half adder, full adder, multiplier etc). 5. Design and analyze various sequential circuits like flipflops, registers, counters etc. 6. Analyze and Design synchronous and asynchronous sequential circuits.
5 UNITI Introductory Concepts (Number systems, Base conversions)
6 Digital Systems Digital systems consider discrete amounts of data Examples 26 letters in the alphabet decimal digits Larger quantities can be built from discrete values: Words made of letters Numbers made of decimal digits (e.g ) Computers operate on binary values ( and ) Easy to represent binary values electrically Voltages and currents Can be implemented using circuits Create the building blocks of modern computers
7 Understanding Decimal Numbers Decimal numbers are made of decimal digits: (,,2,3,4,5,6,7,8,9) Base = How many items does decimal number 8653 represents? 8653 Weight = 8 x3 + 6 x2 + 5 x + 3 x Number = d3 x B3 + d2 x B2 + d x B + d x B = Value What about fractions? = 9x4 + 7x3 + 6x2 + 5x + 4x + 3x + 5x2 In formal notation ( )
8 Understanding Octal Numbers Octal numbers are made of octal digits: (,,2,3,4,5,6,7) How many items does an octal number represent? = Weights (4536)8 = 4x83 + 5x82 + 3x8 + 6x8 = (2398) What about fractions? (465.27)8 = 4x82 + 6x8 + 5x8 + 2x8 + 7x82 Octal numbers don t use digits 8 or 9
9 Understanding Hexadecimal Numbers Hexadecimal numbers are made of 6 digits: (,,2,3,4,5,6,7,8,9,A, B, C, D, E, F) How many items does a hex number represent? = Weights (3A9F)6 = 3x63 + x62 + 9x6 + 5x6 = 4999 What about fractions? (2D3.5)6 = 2x62 + 3x6 + 3x6 + 5x6 = Note that each hexadecimal digit can be represented with four bits ()2 = (E)6 Groups of four bits are called a nibble ()2
10 Understanding Binary Numbers Binary numbers are made of binary digits (bits): and How many items does a binary number represent? = Weights ()2 = x23 + x22 + x2 + x2 = () What about fractions? (.)2 = x22 + x2 + x2 + x2 + x22 Groups of eight bits are called a byte ()2 Groups of four bits are called a nibble ()2
11 Putting It All Together Binary, octal, and hexadecimal are similar Easy to build circuits to operate on these representations Possible to convert between the three formats
12 Why Use Binary Numbers? Easy to represent and using electrical values Possible to tolerate noise Easy to transmit data Easy to build binary circuits AND Gate
13 Conversion Between Number Bases Learn to convert between bases Already demonstrated how to convert from binary to decimal Octal (base 8) Decimal (base ) Binary (base 2) Hexadecimal (base 6)
14 Convert an Integer from Decimal to Another Base For each digit position:. Divide decimal number by the base (e.g. 2) 2. The remainder is the lowestorder digit 3. Repeat first two steps until no divisor remains Example for (3): Quotient Remainder 3/2 = 6/2 = 3/2 = /2 = Coefficient a = a = a2 = a3 = Answer (3) = (a3 a2 a a)2 = ()2 MSB LSB
15 Convert a Fraction from Decimal to Another Base For each digit position:. Multiply decimal number by the base (e.g. 2) 2. The integer is the highestorder digit 3. Repeat first two steps until fraction becomes zero Example for (.625): Integer.625 x 2 =.25 x 2 =.5 x 2 = Fraction Coefficient a = a2 = a3 = Answer (.625) = (.a a2 a3 )2 = (.)2 MSB LSB
16 The Growth of Binary Numbers n 2n n 2n 2= 8 28=256 2=2 9 29= =4 2= =8 2= =6 2 22= = =M Mega 6 26= =G Giga 7 27= =T Tera Kilo
17 Convert an Integer from Decimal to Octal For each digit position:. Divide decimal number by the base (8) 2. The remainder is the lowestorder digit 3. Repeat first two steps until no divisor remains Example for (75): Quotient 75/8 = 2/8 = 2/8 = Remainder Coefficient a = 7 a = 5 a2 = 2 Answer (75) = (a2 a a)8 = (257)8
18 Convert a Fraction from Decimal to Octal For each digit position:. Multiply decimal number by the base (e.g. 8) 2. The integer is the highestorder digit 3. Repeat first two steps until fraction becomes zero Example for (.325): Fraction Integer.325 x 8 =.5 x 8 = Coefficient.5. Answer (.325) = (.24)8 a = 2 a2 = 4
19 Conversion Between Base 6 and Base 2 Conversion is easy! Determine the 4bit binary value for each hex digit Note that there are 6 different values of four bits Easier to read and write in hexadecimal Representations are equivalent! 3A9F6 = 2 3 A 9 F
20 Conversion Between Base 6 and Base 8. Convert from Base 6 to Base 2 2. Regroup bits into groups of three starting from right 3. Ignore leading zeros 4. Each group of three bits forms an octal digit 3A9F6 = = A 9 F
21 Binary Addition Binary addition is very simple carries = 6 = 23 = 84
22 Binary Subtraction We can also perform subtraction (with borrows in place of carries) Let s subtract ()2 from ()2 borrows = 77 = = 54
23 Binary Multiplication Binary multiplication is much the same as decimal multiplication, except that the multiplication operations are much simpler X
24 Summary Binary numbers are made of binary digits (bits) Binary and octal number systems Conversion between number systems Addition, subtraction, and multiplication in binary
25 Introductory Concepts (Complements)
26 How To Represent Signed Numbers Plus and minus signs are used for decimal numbers: 25 (or +25), 6, etc In computers, everything is represented as bits Three types of signed binary number representations: signed magnitude s complement 2 s complement In each case: leftmost bit indicates the sign: for positive and for negative
27 Signed Magnitude Representation The left most bit is designated as the sign bit while the remaining bits form the magnitude The sign bit should not be included in addition / subtraction operations 2 = 2 Sign bit Magnitude 2 = 2 Sign bit Magnitude
28 One s Complement Representation The one s complement of a binary number is done by complementing (i.e. inverting) all bits s comp of is s comp of is For a nbit number N the s complement is (2n ) N Called diminished radix complement by Mano To find the negative of a s complement number take its s complement 2 = 2 Sign bit Magnitude 2 = 2 Sign bit Code
29 One s Complement Representation 7 4 bits 6 6 combinations
30 Two s Complement Representation The two s complement of a binary number is done by complementing (inverting) all bits then adding 2 s comp of is 2 s comp of is For an nbit number N the 2 s complement is (2n ) N + Called radix complement by Mano To find the negative of a 2 s complement number take its 2 s complement 2 = 2 Sign bit Magnitude 2 = 2 Sign bit Code
31 Two s Complement Shortcuts Algorithm : Complement each bit then add to the result N = [N] = + + Algorithm 2: Starting with the least significant bit, copy all of the bits up to and including the first bit, then complement the remaining bits N [N] = =
32 Two s Complement Representation 7 4 bits 6 6 combinations
33 FinitePrecision Number Representation Machines that use 2 s complement arithmetic can represent integers in the range 2n N 2n n is the number of bits used for representing N Note that 2n = (..)2 and 2n = (..)2 2 s complement code has more negative numbers than positive s complement code has 2 representations for zero For a nbit number in base (i.e. radix) z there are zn different unsigned values (combinations) (,, zn)
34 s Complement Subtraction Using s complement representation, subtracting numbers is also easy Step : Take s complement of 2nd operand Step 2: Add binary numbers Step 3: Add carry as a low order bit  For example: (+2) () s comp (+2) = +()2 + = 2 Add ( ) = ()2 Add carry = 2 in s comp Final Result
35 2 s Complement Subtraction Using 2 s complement representation, subtracting numbers is also easy Step : Take 2 s complement of 2nd operand Step 2: Add binary numbers Step 3: Ignore the resulting carry bit For example: (+2) () (+2) = +()2 = 2 ( ) = ()2 = 2 in 2 s comp.  2 s comp + Add Final Result Ignore Carry
36 2 s Complement Subtraction Example 2: (3) (5) (3) = +()2 = ()2 ( 5) = ()2 = ()2 Adding these two 5bit codes: + Carry Discarding the carry bit, the sign bit is seen to be zero, indicating a positive result Indeed: ()2 = +(8)
37 2 s Complement Subtraction Example 3: (5) (2) (5) = +()2 = ()2 ( 2) = ()2 = ()2 Adding these two 5bit codes: + Carry Here, there is no carry bit and the sign bit is. This indicates a negative result, which is what we expect: ()2 = (7)
38 Summary Binary numbers can also be represented in octal and hexadecimal Easy to convert between binary, octal, and hexadecimal Signed numbers are represented in 3 codes: signed magnitude, s complement, or 2 s complement 2 s complement code is most important (only representation for zero) Important to understand the treatment of the sign bit for s and 2 s complement codes
39 Introductory Concepts (Codes)
40 Binary Coded Decimal Binary Coded Decimal (BCD) represents each decimal digit with four bits Ex. = This is NOT the same as 2 Why do this? Because people think in decimal Digit BCD Code Digit BCD Code
41 Putting It All Together BCD is not very efficient Used in early computers (94s, 95s) Used to encode numbers for sevensegment displays Easier to read?
42 Binary Gray Code Only one bit changes from one decimal digit to the next 6 7 Useful for reducing errors in communication Digit Gray Code Gray code is not a number system It is an alternate way to represent four bit data Can be scaled to larger numbers
43 ASCII Code American Standard Code for Information Interchange ASCII is a 7bit code, frequently used with a 8th bit for error detection (more about that later) Character ASCII (bin) ASCII (hex) Decimal Octal A 4 65 B C Z a
44 ASCII Codes and Data Transmission ASCII Codes A Z (26 codes), a z (26 codes) 9 ( codes), others Transmission susceptible to noise Typical transmission rates (5 Kbps, 56.6 Kbps) How to keep data transmission accurate?
45 Parity Codes Parity codes are formed by concatenating a parity bit, P to each code word C In an evenparity code, the parity bit is specified so that the total number of ones is even In an oddparity code, the parity bit is specified so that the total number of ones is odd P Information Bits Added even parity bit Added odd parity bit
46 Parity Code Example Concatenate a parity bit to the ASCII code for the characters, X, and = to produce both oddparity and evenparity codes Character ASCII OddParity ASCII EvenParity ASCII X =
47 Binary Data Storage Binary cells store individual bits of data Multiple cells form a register Data in registers can indicate different values Hex (binary) BCD ASCII Binary Cell
48 Register Transfer Data can move from a register to a register Digital logic used to process data Register A Register B Digital Logic Circuits Register C
49 Transfer of Information Data input at keyboard Shifted into place Stored in memory NOTE: Data input in ASCII
50 Building a Computer We need processing We need storage We need communication You will learn to use and design these components
51 Summary Although 2 s complement is most important, other number codes exist ASCII code is used to represent characters (such as those on the keyboard) Registers store binary data
52 UnitII Boolean Algebra and Logic gates
53 Digital Systems Analysis problem: Inputs.. Logic Circuit.. Outputs Determine the binary output for each input combination Design problem: given a task, develop a circuit that accomplishes that task Many possible implementations Best circuit: based on some criterion (size, power, performance, etc.)
54 Toll Booth Controller Consider the design of a toll booth controller Inputs: quarter, car sensor Outputs: gatelift signal, gateclose signal $.25 Car? Logic Circuit Raise gate Close gate If driver pitches in quarter, raise gate When car has cleared gate, close gate
55 Describing Circuit Functionality: Inverter Basic logic functions have symbols The same functionality can be represented with a truth table Truth table completely specifies outputs for all input combinations This is an inverter Truth Table An input of is inverted to a A Y An input of is inverted to a A Y Symbol Input Output
56 The AND Gate This is an AND gate Truth Table If the two input signals A B Y are asserted (i.e. high) the output will also be asserted. Otherwise, the output will be deasserted (i.e. low) A B Y A B
57 The OR Gate This is an OR gate If either of the two input signals is asserted, or both of them are, the output A B Y will be asserted A A B Y B
58 Describing Circuit Functionality: Waveforms Waveforms provide another approach for representing functionality Values are either high (logic ) or low (logic ) Can you create a truth table from the waveforms? AND Gate x y f
59 Consider threeinput gates 3 Input OR Gate
60 Ordering Boolean Functions How to interpret A B + C? Is it A B ORed with C? Is it A ANDed with B + C? Order of precedence for Boolean algebra: AND before OR Note that parentheses are needed here:
61 Boolean Algebra A Boolean algebra is defined as a closed algebraic system containing a set K of two or more elements and the two operators, and + Useful for identifying and minimizing circuit functionality Identity elements a+=a a =a is the identity element for the + operation is the identity element for the operation
62 Commutativity and Associativity of the Operators Commutative Property: For every a and b in K, a+b=b+a a b=b a Associative Property: For every a, b, and c in K, a + (b + c) = (a + b) + c a (b c) = (a b) c
63 Distributivity of the Operators and Complements Distributive Property: For every a, b, and c in K, a+(b c)=(a+b) (a+c) a (b+c)=(a b)+(a c) The Existence of the Complement: For every a in K there exists a unique element called a (or ā) (complement of a) such that, a + a = a a = To simplify notation, the operator is frequently omitted. When two elements are written next to each other, the AND ( ) operator is implied a+b c=(a+b) (a+c) a + bc = ( a + b )( a + c )
64 Duality The principle of duality is an important concept: If an expression is valid in Boolean algebra, the dual of that expression is also valid To form the dual of an expression, replace all + operators with operators, all operators with + operators, all ones with zeros, and all zeros with ones Form the dual of the equation: a + (bc) = (a + b)(a + c) Following the replacement rules: a(b + c) = ab + ac Take care not to alter the location of the parentheses if they are present
65 Involution This theorem states: a = a a=a Remember that: aa = aa= a+a = a+a= Therefore, a is the complement of a and a is also the complement of a Taking the double inverse of a value produces the initial value
66 Absorption This theorem states: a + ab = a a(a+b) = a To prove the first half of this theorem: a + ab = a + ab = a ( + b) = a (b + ) = a () a + ab = a
67 DeMorgan s Theorem A key theorem in simplifying Boolean algebra expressions is DeMorgan s Theorem. It states: (a + b) = a b (ab) = a + b a+b =a b a b =a +b Example: Complement and simplify the expression a(b + z(x + a )) a (b + z ( x + a )) = a + (b + z (x + a )) = a + b (z (x + a )) = a + b (z + (x + a )) = a + b (z + x a) = a + b (z + x a)
68 Summary Basic logic functions can be made from AND, OR, and NOT (invert) functions The behavior of digital circuits can be represented with waveforms, truth tables, or symbols Primitive gates can be combined to form larger circuits Boolean algebra defines how binary variables can be combined Rules for associativity, commutativity, and distribution are similar to algebra DeMorgan s rules are important Will allow us to reduce circuit sizes
69 UNITII Boolean Algebra and Logic gates
70 Boolean Functions Boolean algebra deals with binary variables and logic operations Function results in binary or x y z xy yz G x xy y G = xy +yz z yz How to transit between an equation, a circuit, and a truth table?
71 Representation Conversion Need to transit between a Boolean expression, a truth table, and a circuit (symbols) Conversion between truth table and expression is easy Conversion between expression and circuit is easy Conversion to truth table is more difficult Boolean Expression Circuit Truth Table
72 Truth Table to Expression Converting a truth table to an expression Each row with an output of becomes a product term Sum the product terms together x y z G Any Boolean Expression can be represented in sum of products form! xyz + xyz + x yz
73 Equivalent Representations of Circuits All three formats are equivalent Number of s in truth table output column equals AND terms for SumofProducts (SOP) x y z G G = xyz + xyz + x yz x G y z
74 Reducing Boolean Expressions Is this the smallest possible implementation of this expression? No! G = xyz + xyz + x yz Use Boolean Algebra rules to reduce complexity while preserving functionality Step : Use Theorem (a + a = a) xyz + xyz + x yz = xyz + xyz + xyz + x yz Step 2: Use distributive rule a(b + c) = ab + ac xyz + xyz + xyz + x yz = xy(z + z ) + yz(x + x ) Step 3: Use Postulate 3 (a + a = ) xy(z + z ) + yz(x + x ) = xy. + yz. Step 4: Use Postulate 2 (a. = a) xy. + yz. = xy + yz = xyz + xyz + x yz
75 Reduced Hardware Implementation Reduced equation requires less hardware! Same function is implemented! x y z G x G = xyz + xyz + x yz = xy + yz y G z
76 Minterms and Maxterms Each variable in a Boolean expression is a literal Boolean variables can appear in normal (x) or complemented form (x ) Each AND combination of terms is a minterm Each OR combination of terms is a maxterm For example: x For example: y z Minterm x y z m x y z m xy z m4 xyz m7 x y z Maxterm x+y+z M x+y+z M x +y+z x +y +z M7 M4
77 Representing Functions with Minterms Minterm number is same as row position in truth table (starting with at the top) Shorthand way to represent functions x y z G G = xyz + xyz + x yz G = m7 + m6 + m3 = Σ(3, 6, 7)
78 Complementing Functions Minterm number is same as row position in truth table (starting with at the top) Shorthand way to represent functions x y z G G G = xyz + xyz + x yz G = (xyz + xyz + x yz) =? Can we find a simpler representation?
79 Complementing Functions Step : assign temporary names b+ c z (a + z) = G G = a + b+ c G = (a + b + c) Step 2: Use DeMorgans Law (a + z) = a z Step 3: Resubstitute (b+c) for z a z = a (b + c) Step 4: Use DeMorgans Law a (b + c) = a (b c ) Step 5: Associative rule a (b c ) = a b c G = a + b+ c G = a b c = a b c
80 Complementation Example Find complement of F = x z + yz F = (x z + yz) DeMorgan s F = (x z) (yz) DeMorgan s F = (x +z ) (y +z ) Reduction eliminate double negation on x F = (x+z ) (y +z ) This format is called product of sums
81 Conversion Between Canonical Forms Easy to convert between minterm and maxterm representations For maxterm representation, select rows with s x y z G G = xyz + xyz + x yz G = m7 + m6 + m3 = Σ(3, 6, 7) G = MMM2M4M5 = π(,,2,4,5) G = (x+y+z)(x+y+z )(x+y +z)(x +y+z)(x +y+z )
82 Representation of Circuits Any logic expression can be represented in a 2level circuit Circuits can be reduced to minimal 2level representations Sum of products representation is most common in industry
83 Summary Truth table, circuit, and Boolean expression formats are equivalent Easy to translate a truth table to SOP and POS representations Boolean algebra rules can be used to reduce circuit size while maintaining functionality All logic functions can be made from AND, OR, and NOT Easiest way to understand: Do examples!
84 UNITII Boolean Algebra and Logic Gates
85 Boolean Functions Boolean algebra deals with binary variables and logic operations Function results in binary or x y z F x y z z y+z F = x(y+z ) F = x(y+z )
86 Logic functions of N variables Each truth table represents one possible function (AND, OR etc) If there are N inputs, there are 2 2N For example, if N is 2 then there are 6 possible truth tables So far, we have defined 2 of these functions 4 more are possible Why consider new functions? Cheaper hardware, more flexibility x y G
87 The NAND Gate The NAND gate is a combination of an AND gate followed by an inverter NAND gates have several interesting properties NAND(a,a) (aa) = a NOT(a) NAND (a,b) (ab) = ab AND(a,b) NAND(a,b ) (a b ) = a+b OR(a,b) A B Y Y=AB A B Y
88 The NAND Gate Those three properties show that: a NAND gate with both of its inputs driven by the same signal is equivalent to a NOT gate a NAND gate whose output is complemented is equivalent to an AND gate a NAND gate with complemented inputs acts as an OR gate Hence, we can use a NAND gate to implement all three of the elementary operators (AND, OR, NOT) Therefore, ANY switching function can be constructed using only NAND gates. Such a gate is said to be primitive or functionally complete (Universal Gate)
89 NAND Gates into Other Gates What are these circuits? A Y NOT Gate A B Y AND Gate A Y B OR Gate
90 The NOR Gate A NOR gate is a combination of an OR gate followed by an inverter NOR gates also have several interesting properties NOR(a,a) (a+a) = a NOT(a) NOR (a,b) (a+b) = a+b OR(a,b) NOR(a,b ) (a +b ) = ab AND(a,b) A B Y Y=A+B A B Y
91 Functionally Complete Gates Just like the NAND gate, the NOR gate is functionally complete any logic function can be implemented using just NOR gates Both NAND and NOR gates are very valuable as any design can be realized using either one It is easier to build an IC chip using all NAND or NOR gates than to combine AND, OR, and NOT gates NAND/NOR gates are typically faster in switching and cheaper to produce
92 NOR Gates into Other Gates What are these circuits? A Y NOT Gate A B Y OR Gate A Y B AND Gate
93 The XOR Gate (ExclusiveOR) This is a XOR gate XOR gates assert their output when exactly one of the inputs is asserted, hence the name The switching algebra symbol for this operation is : A B Y = and = Y=A B A B Y
94 The XNOR Gate This is a XNOR gate This functions as an exclusivenor gate, or simply the complement of the XOR gate The switching algebra symbol A B Y for this operation is : = and = Y=A B A B Y
95 NOR Gate Equivalence NOR Symbol, Equivalent Circuit, Truth Table
96 DeMorgan s Theorem A key theorem in simplifying Boolean algebra expression is DeMorgan s Theorem. It states: (a + b) = a b (ab) = a + b a+b =a b a b =a +b Example: Complement and simplify the expression a(b + z(x + a )) a (b + z ( x + a )) = a + (b + z (x + a )) = a + b (z (x + a )) = a + b (z + (x + a )) = a + b (z + x a) = a + b (z + x a)
97 Example Determine the output expression for the following circuit and simplify it using DeMorgan s Theorem
98 Universality of NAND gate
99 Universality of NOR gate
100 Example
101 Interpretation of the two NAND gate symbols DeMorgan s Theorem
102 Interpretation of the two OR gate symbols DeMorgan s Theorem
103 Summary Basic logic functions can be made from NAND, and NOR functions The behavior of digital circuits can be represented with waveforms, truth tables, or Boolean expressions Primitive gates can be combined to form larger circuits Boolean algebra defines how binary variables can be combined with NAND, NOR DeMorgan s rules are important Allow conversion to NAND/NOR representations
104 KMAP
105 Karnaugh maps Alternate way of representing Boolean functions A Karnaugh map is a graphical tool for assisting in the general simplification procedure Each row in the truth table is represented by a square Each square represents a minterm x y x y x y x y x xy xy y x y F F = Σ(m,m) = x y + x y
106 Karnaugh Maps Two variable maps B A F=AB+AB B A F=AB +AB +AB Three variable maps BC A + A B F=ABC +ABC +ABC + ABC + ABC + ABC C F
107 Karnaugh maps Numbering scheme is based on Gray code BC e.g.,,, Only a single bit changes in code for adjacent map cells Observe the variable transitions B A A B 3 2 A A G(A,B,C) = B BC C BC C B A A C F(A,B,C) = m(,2,6,7) = A C +AB
108 Karnaugh Maps Two variable maps B A B A F=AB+AB F=AB +AB +AB F=A+B Three variable maps BC A F=A +BC +BC F=ABC +ABC +ABC + ABC + ABC + ABC
109 More Karnaugh Map Examples b b Examples a f=b bc a cout = ac+bc + ab a f = a' bc a f=b. Circle the largest groups possible 2. Group dimensions must be a power of 2 3. Remember what circling means!
110 Application of Karnaugh Maps: The Onebit Adder Cin A Adder S B + A B Cin S Cout How to use a Karnaugh Map instead of the Algebraic simplification? Cout S = A B Cin + A BCin + AB Cin + ABCin Cout = A BCin + A B Cin + ABCin + ABCin = A BCin + ABCin + AB Cin + ABCin + ABCin + ABCin = (A + A)BCin + (B + B)ACin + (Cin + Cin)AB = BCin + ACin + AB = BCin + ACin + AB
111 Application of Karnaugh Maps: The Onebit Adder Cin A Adder S B Cout BC A A + B Cin S Cout B A C Karnaugh Map for Cout Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. Cout = BCin + AB + ACin
112 Application of Karnaugh Maps: The Onebit Adder Cin A Adder S B Cout BC A A + A B Cin S Cout B C Karnaugh Map for S Now we have to cover all the s in the Karnaugh Map using the largest rectangles and as few rectangles as we can. S = A B C in + A B Cin + A B Cin+A BC in No Possible Reduction!
113 Summary Karnaugh map allows us to represent functions with new notation Representation allows for logic reduction Implement same function with less logic Each square represents one minterm Each circle leads to one product term Not all functions can be reduced
114 KMAP
115 Karnaugh Maps for 4 Input Functions Represent functions of 4 inputs with 6 minterms Use same rules developed for 3input functions
116 F(A,B,C,D) = m(, 2, 3, 5, 6, 7, 8,,, 4, 5) F = C + A BD + B D
117 Design Examples Kmap for LT F = A C + A B D + B CD
118 Design Examples Kmap for EQ F = A B C D + A BC D + ABCD + AB CD
119 Design Examples Kmap for GT F = AC + BC D + ABD
120 Physical Implementation Step : Truth table Step 2: Kmap Step 3: Minimized sumofproducts Step 4: Physical implementation with gates
121 Physical Implementation A B C Kmap for EQ D EQ EQ F = A B C D + A BC D + ABCD + AB CD
122 Karnaugh Maps Four variable maps CD AB F=A BC +A CD +ABC +AB C D +ABC +AB C F=BC + AC +CD + AD Need to make sure all s are covered Try to minimize total product terms Design could be implemented using NANDs and NORs
123 Karnaugh Maps: Don t Cares In some cases, outputs are undefined We don t care if the circuit produces a or a This knowledge can be used to simplify functions A AB CD X X X D C B  Treat X s like either s or s  Very useful  OK to leave some X s uncovered
124 Karnaugh Maps: Don t Cares F(A,B,C,D) = (,3,5,7,9) + d(6,2,3) AB C CD X X X B A D F=A D + B C D Without don t cares f = A'D + B'C'D without don't cares F=A D + C D With don t cares + A + B C D F X X X
125 Don t Care Conditions In some situations, we don t care about the value of a function for certain combinations of the variables these combinations may be impossible in certain contexts or the value of the function may not matter when the combinations occur In such situations we say the function is incompletely specified and there are multiple (completely specified) logic functions that can be used in the design so we can select a function that gives the simplest circuit When constructing the terms in the simplification procedure, we can choose to either cover or not cover the don t care conditions
126 Map Simplification with Don t Cares CD AB x x x x x F=A C D+B +AC CD AB x x x x x F= A B C D +BC +AC+ABC Alternative covering:
127 Karnaugh Maps: Product of Sums F(A,B,C,D) = (2,3,9,,3) + d(6,4) CD AB x x F = AC D + AB D + A B C
128 Karnaugh Maps: Product of Sums G(A,B,C,D) = (,,4,5,7,8,,2,5) + d(6,4) CD AB x x G = AD + A C + BC
129 Karnaugh Maps: Product of Sums F(A,B,C,D) = (2,3,9,,3) + d(6,4) CD AB x x F = AC D + A B C+ A B C AB D (A+C)(A +D) F = (B +C ) (A+C)
130 Prime Implicants Any single or group of s in the Karnaugh map of a function F is an implicant of F. A product term is called a prime implicant of F if it cannot be combined with another term to eliminate a variable. CD AB (a) A B C (b) BD (c) A B C D (d) A C (e) A B D Implicants: (a),(c),(d),(e) Prime Implicants: (d),(e)
131 Essential Prime Implicants A product term is an essential prime implicant if there is a minterm that is only covered by that prime implicant The minimal sumofproducts form of F must include all the essential prime implicants of F
132 Examples to Illustrate Terms C CD AB A X A D, AC, A BC, CD, BC'D' B essential minimum cover: AC + A D + BC'D' D
133 Examples to Illustrate Terms CD AB A C 5 prime implicants: BD, ABC, ABC AC'D, A'BC' A'BC, A'CD B D minimum cover: 4 essential implicants
134 Summary Kmaps of four literals were considered Larger examples exist Don t care conditions help minimize functions Output for don t cares are originally undefined Result of minimization is a minimal sumofproducts Result contains prime implicants Essential prime implicants are required in the implementation
135 NANDNAND & NORNOR Networks DeMorgan s Law: (a + b) = a b (a b) = a + b a+b = ab ab =a +b a + b = (a b ) a+b = a b (a b) = (a + b ) ab = a +b push bubbles or introduce in pairs or remove pairs
136 NANDNAND Networks Mapping from AND/OR to NAND/NAND
137 Implementations of 2Level Logic Sumofproducts AND gates to form product terms (minterms) OR gate to form sum Productofsums OR gates to form sum terms (maxterms) AND gates to form product
138 Twolevel Logic using NAND Gates Replace minterm AND gates with NAND gates Place compensating inversion at inputs of OR gate
139 Twolevel Logic using NAND Gates (cont d) OR gate with inverted inputs is a NAND gate DeMorgan's: A' + B' = (A B)' A+B=A B Twolevel NANDNAND network Inverted inputs are not counted In a typical circuit, inversion is done once and signal is then distributed
140 Conversion Between Forms (cont d) Example: verify equivalence of two forms A A B B NAND Z NAND C C D D NAND Z = [ (A B)' (C D)' ]' = [ (A' + B') (C' + D') ]' = [ (A' + B')' + (C' + D')' ] = (A B) + (C D) Z
141 Multilevel Logic x = (A + B + C) (D + E) F + G Factored form not written as twolevel SoP x 3input OR gate, 2 x 2input OR gates, x 3input AND gate wires (7 literals plus 3 internal wires) A B C D E F G X
142 Conversion of Multilevel Logic to NAND Gates F = A (B + C D) + B C'
143 ExclusiveOR Circuits ExclusiveOR (XOR) produces a HIGH output whenever the two inputs are at opposite levels
144 ExclusiveNOR Circuits ExclusiveNOR (XNOR) produces a HIGH output whenever the two inputs are at the same level
145 XOR Function XOR function can also be implemented with AND/OR gates (also NANDs)
146 XOR Function Even function even number of inputs are Odd function odd number of inputs are
147 Parity Generation and Checking
148 Summary Follow rules to convert representation and symbols between AND/OR Conversions are based on DeMorgan s Law NOR gate implementations are also possible XORs provide straightforward implementation for some functions Used for parity generation and checking XOR circuits AND/ORs can also be implemented using
149 The Problem How can we convert from a circuit drawing to an equation or truth table? Two approaches Create intermediate equations Create intermediate truth tables A B C Out A B C
150 Label Gate Outputs. Label all gate outputs that are functions of input variables 2. Label gates that are functions of input variables and previously labeled gates 3. Repeat process until all outputs are labeled A B C A B C R S T Out
151 Approach : Create Intermediate Equations Step : Create an equation for each gate output based on its inputs R = ABC S=A+B T = C S Out = R + T A B C A B C R S T Out
152 Approach : Substitute in subexpressions Step 2: Form a relationship based on input variables R = ABC S=A+B T = C S = C (A + B) Out = R+T = ABC + C (A+B) A B C A B C R S T Out
153 Approach : Substitute in subexpressions Step 3: Expand equation to SOP Out = ABC + C (A+B) = ABC + AC + BC A B C Out A C B C
154 Approach 2: Truth Table Step : Determine outputs for functions of input variables A B C A B C R S T A B C Out R S
155 Approach 2: Truth Table Step 2: Determine outputs for A B functions of intermediate variables. T = S C A R C C R B C A B C S T Out S T
156 Approach 2: Truth Table Step 3: Determine outputs for function. A Out = R + T A B C A B C R S T B C R S Out T Out
157 More Difficult Example Note labels on interior nodes
158 More Difficult Example: Truth Table Remember to determine intermediate variables starting from the inputs When all inputs are determined for a gate, determine its output The truth table can be reduced using Kmaps A B C F2 F 2 T T2 T3 F
159 Summary Important to be able to convert circuits into truth table and equation form WHY? Leads to minimized sum of products representation Two approaches illustrated Approach : Create an equation with circuit outputs dependent on circuit inputs Approach 2: Create a truth table which shows relationship between circuit inputs and circuit outputs Both results can then be minimized using Kmaps
ENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 2 Intro to Electrical and Computer Engineering Lecture 5 Boolean Algebra Overview Logic functions with s and s Building digital circuitry Truth tables Logic symbols and waveforms Boolean algebra
More informationGet Free notes at ModuleI One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)
More informationIT 201 Digital System Design Module II Notes
IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationDIGITAL SYSTEM DESIGN
DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital SystemsIntroduction to number systems, Binary numbers, Number
More informationGate Level Minimization Map Method
Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically
More informationLecture 2: Number Systems
Lecture 2: Number Systems Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Contents
More informationCombinational Logic & Circuits
WeekI Combinational Logic & Circuits Spring' 232  Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other
More informationChapter 2 Boolean algebra and Logic Gates
Chapter 2 Boolean algebra and Logic Gates 2. Introduction In working with logic relations in digital form, we need a set of rules for symbolic manipulation which will enable us to simplify complex expressions
More informationSpecifying logic functions
CSE4: Components and Design Techniques for Digital Systems Specifying logic functions Instructor: Mohsen Imani Slides from: Prof.Tajana Simunic and Dr.Pietro Mercati We have seen various concepts: Last
More informationChapter 3. GateLevel Minimization. Outlines
Chapter 3 GateLevel Minimization Introduction The Map Method FourVariable Map FiveVariable Map Outlines Product of Sums Simplification Don tcare Conditions NAND and NOR Implementation Other TwoLevel
More informationChapter 2. Boolean Expressions:
Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean
More informationCombinational Logic Circuits
Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 21 Binary Logic and Gates 22 Boolean Algebra 23 Standard Forms 24 TwoLevel Circuit Optimization
More informationGate Level Minimization
Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science
More informationUNIT4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable.
UNIT4 BOOLEAN LOGIC Boolean algebra is an algebra that deals with Boolean values((true and FALSE). Everyday we have to make logic decisions: Should I carry the book or not?, Should I watch TV or not?
More informationCS8803: Advanced Digital Design for Embedded Hardware
CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter
More informationExperiment 3: Logic Simplification
Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed ElSaied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions
More informationX Y Z F=X+Y+Z
This circuit is used to obtain the compliment of a value. If X = 0, then X = 1. The truth table for NOT gate is : X X 0 1 1 0 2. OR gate : The OR gate has two or more input signals but only one output
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter
More informationGateLevel Minimization
GateLevel Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method
More informationCode No: 07A3EC03 Set No. 1
Code No: 07A3EC03 Set No. 1 II B.Tech I Semester Regular Examinations, November 2008 SWITCHING THEORY AND LOGIC DESIGN ( Common to Electrical & Electronic Engineering, Electronics & Instrumentation Engineering,
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2006 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems
More informationB.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN
B.Tech II Year I Semester () Regular Examinations December 2014 (Common to IT and CSE) (a) If 1010 2 + 10 2 = X 10, then X is  Write the first 9 decimal digits in base 3. (c) What is meant by don
More informationGateLevel Minimization
MEC520 디지털공학 GateLevel Minimization JeeHwan Ryu School of Mechanical Engineering GateLevel MinimizationThe Map Method Truth table is unique Many different algebraic expression Boolean expressions may
More informationUnitIV Boolean Algebra
UnitIV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of
More information2.6 BOOLEAN FUNCTIONS
2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses
More informationDKT 122/3 DIGITAL SYSTEM 1
Company LOGO DKT 122/3 DIGITAL SYSTEM 1 BOOLEAN ALGEBRA (PART 2) Boolean Algebra Contents Boolean Operations & Expression Laws & Rules of Boolean algebra DeMorgan s Theorems Boolean analysis of logic circuits
More informationR.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai
L T P C R.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai 601206 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC8392 UNIT  I 3 0 0 3 OBJECTIVES: To present the Digital fundamentals, Boolean
More informationGateLevel Minimization. BME208 Logic Circuits Yalçın İŞLER
GateLevel Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates Binary logic is used in all of today's digital computers and devices Cost of the circuits is an important factor Finding simpler and cheaper but equivalent circuits can
More informationAssignment (36) Boolean Algebra and Logic Simplification  General Questions
Assignment (36) Boolean Algebra and Logic Simplification  General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems
More information1. Mark the correct statement(s)
1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another
More informationCombinational Circuits
Combinational Circuits Combinational circuit consists of an interconnection of logic gates They react to their inputs and produce their outputs by transforming binary information n input binary variables
More informationInjntu.com Injntu.com Injntu.com R16
1. a) What are the three methods of obtaining the 2 s complement of a given binary (3M) number? b) What do you mean by Kmap? Name it advantages and disadvantages. (3M) c) Distinguish between a halfadder
More informationCombinational Logic Circuits
Chapter 3 Combinational Logic Circuits 12 Hours 24 Marks 3.1 Standard representation for logical functions Boolean expressions / logic expressions / logical functions are expressed in terms of logical
More informationGateLevel Minimization
GateLevel Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method
More informationCHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey
CHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input
More informationUniversity of Technology
University of Technology Lecturer: Dr. Sinan Majid Course Title: microprocessors 4 th year Lecture 5 & 6 Minimization with Karnaugh Maps Karnaugh maps lternate way of representing oolean function ll rows
More informationNH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN
NH 67, Karur Trichy Highways, Puliyur C.F, 639 114 Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN UNIT 1 BOOLEAN ALGEBRA AND LOGIC GATES Review of binary
More informationComputer Science. Unit4: Introduction to Boolean Algebra
Unit4: Introduction to Boolean Algebra Learning Objective At the end of the chapter students will: Learn Fundamental concepts and basic laws of Boolean algebra. Learn about Boolean expression and will
More informationwww.vidyarthiplus.com Question Paper Code : 31298 B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2013. Third Semester Computer Science and Engineering CS 2202/CS 34/EC 1206 A/10144 CS 303/080230012DIGITAL
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : STLD(16EC402) Year & Sem: IIB.Tech & ISem Course & Branch: B.Tech
More informationIncompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap Representation and Methods Examples
Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap Representation and Methods Examples Incompletely specified functions
More information2.1 Binary Logic and Gates
1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 22 121115 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Binary Number Representation Binary Arithmetic Combinatorial Logic
More informationSIR C.R.REDDY COLLEGE OF ENGINEERING, ELURU DEPARTMENT OF INFORMATION TECHNOLOGY LESSON PLAN
SIR C.R.REDDY COLLEGE OF ENGINEERING, ELURU DEPARTMENT OF INFORMATION TECHNOLOGY LESSON PLAN SUBJECT: CSE 2.1.6 DIGITAL LOGIC DESIGN CLASS: 2/4 B.Tech., I SEMESTER, A.Y.201718 INSTRUCTOR: Sri A.M.K.KANNA
More informationVALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur603 203 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS YEAR / SEMESTER: II / III ACADEMIC YEAR: 20152016 (ODD
More informationD I G I T A L C I R C U I T S E E
D I G I T A L C I R C U I T S E E Digital Circuits Basic Scope and Introduction This book covers theory solved examples and previous year gate question for following topics: Number system, Boolean algebra,
More informationClass Subject Code Subject Prepared By Lesson Plan for Time: Lesson. No 1.CONTENT LIST: Introduction to UnitI 2. SKILLS ADDRESSED: Listening I year, 02 sem CS6201 Digital Principles & System Design S.Seedhanadevi
More informationENDTERM EXAMINATION
(Please Write your Exam Roll No. immediately) ENDTERM EXAMINATION DECEMBER 2006 Exam. Roll No... Exam Series code: 100919DEC06200963 Paper Code: MCA103 Subject: Digital Electronics Time: 3 Hours Maximum
More information2008 The McGrawHill Companies, Inc. All rights reserved.
28 The McGrawHill Companies, Inc. All rights reserved. 28 The McGrawHill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28
More informationLOGIC DESIGN. Dr. Mahmoud Abo_elfetouh
LOGIC DESIGN Dr. Mahmoud Abo_elfetouh Course objectives This course provides you with a basic understanding of what digital devices are, how they operate, and how they can be designed to perform useful
More informationDigital Techniques. Lecture 1. 1 st Class
Digital Techniques Lecture 1 1 st Class Digital Techniques Digital Computer and Digital System: Digital computer is a part of digital system, it based on binary system. A block diagram of digital computer
More informationDigital logic fundamentals. Question Bank. Unit I
Digital logic fundamentals Question Bank Subject Name : Digital Logic Fundamentals Subject code: CA102T Staff Name: R.Roseline Unit I 1. What is Number system? 2. Define binary logic. 3. Show how negative
More informationSUBJECT CODE: IT T35 DIGITAL SYSTEM DESIGN YEAR / SEM : 2 / 3
UNIT  I PART A (2 Marks) 1. Using Demorgan s theorem convert the following Boolean expression to an equivalent expression that has only OR and complement operations. Show the function can be implemented
More informationII/IV B.Tech (Regular/Supplementary) DEGREE EXAMINATION. Answer ONE question from each unit.
Hall Ticket Number: 14CS IT303 November, 2017 Third Semester Time: Three Hours Answer Question No.1 compulsorily. II/IV B.Tech (Regular/Supplementary) DEGREE EXAMINATION Common for CSE & IT Digital Logic
More informationUNIT II. Circuit minimization
UNIT II Circuit minimization The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented.
More informationBINARY SYSTEM. Binary system is used in digital systems because it is:
CHAPTER 2 CHAPTER CONTENTS 2.1 Binary System 2.2 Binary Arithmetic Operation 2.3 Signed & Unsigned Numbers 2.4 Arithmetic Operations of Signed Numbers 2.5 Hexadecimal Number System 2.6 Octal Number System
More informationLogic Design: Part 2
Orange Coast College Business Division Computer Science Department CS 6 Computer Architecture Logic Design: Part 2 Where are we? Number systems Decimal Binary (and related Octal and Hexadecimal) Binary
More informationSimplification of Boolean Functions
Simplification of Boolean Functions Contents: Why simplification? The Map Method Two, Three, Four and Five variable Maps. Simplification of two, three, four and five variable Boolean function by Map method.
More informationDHANALAKSHMI SRINIVASAN COLLEGE OF ENGINEERING AND TECHNOLOGY
DHANALAKSHMI SRINIVASAN COLLEGE OF ENGINEERING AND TECHNOLOGY Dept/Sem: II CSE/03 DEPARTMENT OF ECE CS8351 DIGITAL PRINCIPLES AND SYSTEM DESIGN UNIT I BOOLEAN ALGEBRA AND LOGIC GATES PART A 1. How many
More informationMGUBCA205 Second Sem Core VI Fundamentals of Digital Systems MCQ s. 2. Why the decimal number system is also called as positional number system?
MGUBCA205 Second Sem Core VI Fundamentals of Digital Systems MCQ s Unit1 Number Systems 1. What does a decimal number represents? A. Quality B. Quantity C. Position D. None of the above 2. Why the
More informationGateLevel Minimization. section instructor: Ufuk Çelikcan
GateLevel Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to
More informationBOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.
COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless
More informationQuestion No: 1 ( Marks: 1 )  Please choose one A SOP expression is equal to 1
ASSALAM O ALAIKUM all fellows ALL IN ONE Mega File CS302 Midterm PAPERS, MCQz & subjective Created BY Farhan& Ali BS (cs) 3rd sem Hackers Group Mandi Bahauddin Remember us in your prayers Mindhacker124@gmail.com
More informationPoints Addressed in this Lecture. Standard form of Boolean Expressions. Lecture 4: Logic Simplication & Karnaugh Map
Points Addressed in this Lecture Lecture 4: Logic Simplication & Karnaugh Map Professor Peter Cheung Department of EEE, Imperial College London Standard form of Boolean Expressions SumofProducts (SOP),
More informationCOPYRIGHTED MATERIAL INDEX
INDEX Absorption law, 31, 38 Acyclic graph, 35 tree, 36 Addition operators, in VHDL (VHSIC hardware description language), 192 Algebraic division, 105 AND gate, 48 49 Antisymmetric, 34 Applicable input
More informationMicrocomputers. Outline. Number Systems and Digital Logic Review
Microcomputers Number Systems and Digital Logic Review Lecture 11 Outline Number systems and formats Common number systems Base Conversion Integer representation Signed integer representation Binary coded
More informationR10. II B. Tech I Semester, Supplementary Examinations, May
SET  1 1. a) Convert the following decimal numbers into an equivalent binary numbers. i) 53.625 ii) 4097.188 iii) 167 iv) 0.4475 b) Add the following numbers using 2 s complement method. i) 48 and +31
More informationCOMBINATIONAL LOGIC CIRCUITS
COMBINATIONAL LOGIC CIRCUITS 4.1 INTRODUCTION The digital system consists of two types of circuits, namely: (i) Combinational circuits and (ii) Sequential circuits A combinational circuit consists of logic
More informationQUESTION BANK FOR TEST
CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad  500043 Course Name : DIGITAL LOGIC DESISN Course Code : AEC020 Class : B Tech III Semester Branch : CSE Academic Year : 2018 2019
More informationComputer Logical Organization Tutorial
Computer Logical Organization Tutorial COMPUTER LOGICAL ORGANIZATION TUTORIAL Simply Easy Learning by tutorialspoint.com tutorialspoint.com i ABOUT THE TUTORIAL Computer Logical Organization Tutorial Computer
More informationModule 7. Karnaugh Maps
1 Module 7 Karnaugh Maps 1. Introduction 2. Canonical and Standard forms 2.1 Minterms 2.2 Maxterms 2.3 Canonical Sum of Product or SumofMinterms (SOM) 2.4 Canonical product of sum or ProductofMaxterms(POM)
More informationUNIT V COMBINATIONAL LOGIC DESIGN
UNIT V COMBINATIONAL LOGIC DESIGN NOTE: This is UNITV in JNTUK and UNITIII and HALF PART OF UNITIV in JNTUA SYLLABUS (JNTUK)UNITV: Combinational Logic Design: Adders & Subtractors, Ripple Adder, Look
More informationExperiment 4 Boolean Functions Implementation
Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.
More informationDigital Systems and Binary Numbers
Digital Systems and Binary Numbers Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) 1 / 51 Overview 1 Course Summary 2 Binary Numbers 3 NumberBase
More informationCh. 5 : Boolean Algebra &
Ch. 5 : Boolean Algebra & Reduction elektronik@fisika.ui.ac.id Objectives Should able to: Write Boolean equations for combinational logic applications. Utilize Boolean algebra laws and rules for simplifying
More informationVALLIAMMAI ENGINEERING COLLEGE
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203 DEPARTMENT OF INFORMATION TECHNOLOGY & COMPUTER SCIENCE AND ENGINEERING QUESTION BANK II SEMESTER CS6201 DIGITAL PRINCIPLE AND SYSTEM DESIGN
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad  500 043 COMPUTER SCIENCE AND ENGINEERING TUTORIAL QUESTION BANK Name : DIGITAL LOGIC DESISN Code : AEC020 Class : B Tech III Semester
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard
More informationENGINEERS ACADEMY. 7. Given Boolean theorem. (a) A B A C B C A B A C. (b) AB AC BC AB BC. (c) AB AC BC A B A C B C.
Digital Electronics Boolean Function QUESTION BANK. The Boolean equation Y = C + C + C can be simplified to (a) (c) A (B + C) (b) AC (d) C. The Boolean equation Y = (A + B) (A + B) can be simplified to
More informationIntroduction to Computer Architecture
Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of
More information數位系統 Digital Systems 朝陽科技大學資工系. Speaker: FuwYi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷
數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: FuwYi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,
More informationCDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012
CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6311 and Excess3? Basic Operations of Boolean Algebra Examples
More informationR07
www..com www..com SET  1 II B. Tech I Semester Supplementary Examinations May 2013 SWITCHING THEORY AND LOGIC DESIGN (Com. to EEE, EIE, BME, ECC) Time: 3 hours Max. Marks: 80 Answer any FIVE Questions
More informationBinary. Hexadecimal BINARY CODED DECIMAL
Logical operators Common arithmetic operators, like plus, minus, multiply and divide, works in any number base but the binary number system provides some further operators, called logical operators. Meaning
More informationCMPE223/CMSE222 Digital Logic
CMPE223/CMSE222 Digital Logic Optimized Implementation of Logic Functions: Strategy for Minimization, Minimum ProductofSums Forms, Incompletely Specified Functions Terminology For a given term, each
More informationSimplification of Boolean Functions
COM111 Introduction to Computer Engineering (Fall 20062007) NOTES 5  page 1 of 5 Introduction Simplification of Boolean Functions You already know one method for simplifying Boolean expressions: Boolean
More informationDIGITAL ARITHMETIC: OPERATIONS AND CIRCUITS
C H A P T E R 6 DIGITAL ARITHMETIC: OPERATIONS AND CIRCUITS OUTLINE 6 Binary Addition 62 Representing Signed Numbers 63 Addition in the 2 s Complement System 64 Subtraction in the 2 s Complement
More informationDigital Logic Design (CEN120) (3+1)
Digital Logic Design (CEN120) (3+1) ASSISTANT PROFESSOR Engr. Syed Rizwan Ali, MS(CAAD)UK, PDG(CS)UK, PGD(PM)IR, BS(CE)PK HEC Certified Master Trainer (MTFPDP) PEC Certified Professional Engineer (COM/2531)
More informationNODIA AND COMPANY. GATE SOLVED PAPER Computer Science Engineering Digital Logic. Copyright By NODIA & COMPANY
No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Computer
More informationSummary. Boolean Addition
Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse
More informationDIGITAL ELECTRONICS. Vayu Education of India
DIGITAL ELECTRONICS ARUN RANA Assistant Professor Department of Electronics & Communication Engineering Doon Valley Institute of Engineering & Technology Karnal, Haryana (An ISO 9001:2008 ) Vayu Education
More informationChap.3 3. Chap reduces the complexity required to represent the schematic diagram of a circuit Library
3.1 Combinational Circuits 2 Chap 3. logic circuits for digital systems: combinational vs sequential Combinational Logic Design Combinational Circuit (Chap 3) outputs are determined by the present applied
More informationVALLIAMMAI ENGINEERING COLLEGE
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203 DEPARTMENT OF INFORMATION TECHNOLOGY QUESTION BANK Academic Year 2018 19 III SEMESTER CS8351DIGITAL PRINCIPLES AND SYSTEM DESIGN Regulation
More information1. Draw general diagram of computer showing different logical components (3)
Tutorial 1 1. Draw general diagram of computer showing different logical components (3) 2. List at least three input devices (1.5) 3. List any three output devices (1.5) 4. Fill the blank cells of the
More informationDIGITAL ELECTRONIC CIRCUITS
DIGITAL ELECTRONIC CIRCUITS SUBJECT CODE: PEI4I103 B.Tech, Fourth Semester Prepared By Dr. Kanhu Charan Bhuyan Asst. Professor Instrumentation and Electronics Engineering COLLEGE OF ENGINEERING AND TECHNOLOGY
More information