# SWITCHING THEORY AND LOGIC CIRCUITS

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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, k-map 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, k-map 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 de-multiplexers, arithmetic circuits (half adder, full adder, multiplier etc). 5. Design and analyze various sequential circuits like flip-flops, registers, counters etc. 6. Analyze and Design synchronous and asynchronous sequential circuits.

5 UNIT-I 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- + 5x-2 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- + 7x8-2 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- + x2-2 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 lowest-order 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 highest-order 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- = a-2 = a-3 = Answer (.625) = (.a- a-2 a-3 )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 lowest-order 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 highest-order 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 a-2 = 4

19 Conversion Between Base 6 and Base 2 Conversion is easy! Determine the 4-bit 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: left-most 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 n-bit 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 n-bit 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 Finite-Precision 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 n-bit 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 5-bit 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 5-bit 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 seven-segment 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 7-bit 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 even-parity code, the parity bit is specified so that the total number of ones is even In an odd-parity 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 even-parity codes Character ASCII Odd-Parity ASCII Even-Parity 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 Unit-II 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: gate-lift signal, gate-close 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 three-input 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 UNIT-II 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 Sum-of-Products (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 2-level circuit Circuits can be reduced to minimal 2-level 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 UNIT-II 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 (Exclusive-OR) 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 exclusive-nor 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 K-MAP

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 One-bit 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 One-bit 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 One-bit 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 K-MAP

115 Karnaugh Maps for 4 Input Functions Represent functions of 4 inputs with 6 minterms Use same rules developed for 3-input 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 K-map for LT F = A C + A B D + B CD

118 Design Examples K-map for EQ F = A B C D + A BC D + ABCD + AB CD

119 Design Examples K-map for GT F = AC + BC D + ABD

120 Physical Implementation Step : Truth table Step 2: K-map Step 3: Minimized sum-of-products Step 4: Physical implementation with gates

121 Physical Implementation A B C K-map 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 sum-of-products 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 K-maps 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 sum-of-products Result contains prime implicants Essential prime implicants are required in the implementation

135 NAND-NAND & NOR-NOR 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 NAND-NAND Networks Mapping from AND/OR to NAND/NAND

137 Implementations of 2-Level Logic Sum-of-products AND gates to form product terms (minterms) OR gate to form sum Product-of-sums OR gates to form sum terms (maxterms) AND gates to form product

138 Two-level Logic using NAND Gates Replace minterm AND gates with NAND gates Place compensating inversion at inputs of OR gate

139 Two-level 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 Two-level NAND-NAND 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 Multi-level Logic x = (A + B + C) (D + E) F + G Factored form not written as two-level S-o-P x 3-input OR gate, 2 x 2-input OR gates, x 3-input AND gate wires (7 literals plus 3 internal wires) A B C D E F G X

142 Conversion of Multi-level Logic to NAND Gates F = A (B + C D) + B C'

143 Exclusive-OR Circuits Exclusive-OR (XOR) produces a HIGH output whenever the two inputs are at opposite levels

144 Exclusive-NOR Circuits Exclusive-NOR (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 K-maps 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 K-maps

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