CpE358/CS381. Switching Theory and Logical Design. Class 2

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1 CpE358/CS38 Switching Theor and Logical Design Class 2 CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -45

2 Toda s Material Fundamental concepts of digital sstems (Mano Chapter ) Binar codes, number sstems, and arithmetic (Ch ) Boolean algebra (Ch 2) Simplification of switching equations (Ch 3) Digital device characteristics (e.g., TTL, CMOS)/design considerations (Ch ) Combinatoric logical design including LSI implementation (Chapter 4) Hazards, Races, and time related issues in digital design (Ch 9) Flip-flops and state memor elements (Ch 5) Sequential logic analsis and design (Ch 5) Snchronous vs. asnchronous design (Ch 9) Counters, shift register circuits (Ch 6) Memor and Programmable logic (Ch 7) Minimization of sequential sstems Introduction to Finite Automata CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -46

3 Basic Concepts in Set Theor t S S t S S = {, } Set R Set S + Operators i = [if S and if S c = i S] CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -47

4 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and +, + is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -48

5 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and +, + is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -49

6 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and +, + is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -5

7 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and +, + is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -5

8 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and +, + is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -52

9 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and *, * is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -53

10 . Closure Properties S is closed with respect to operator if 2. Associative A binar operator on set S is associative if 3. Commutative A binar operator on set S is commutative if 4. Identit A set S has an identit element e with respect to operator if 5. Inverse A set S with identit element e with respect to operator has an inverse if 6. Distributive ab, S, ab i S abc,, S, ( ab i ) ic= ai( bc i ) ab, S, ab i = ba i S, e S: ie = ei = S, S: i = e For a set S with operators and *, * is distributive over if, z, S, ( z i ) = ( ) i( z) S CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -54

11 Mathematical Sstems Z Boolean Algebra Special W + Operators Closure Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -55

12 Boolean Algebra Z Boolean Algebra Special W + Operators There are two operators: AND ( ) and OR (+) Closure Associative Commutative Inverse Distributive Properties Complement NOTE: I am using smbols that look similar and act similarl to +(plus) and (times) would act for normal arithmetic. The are not PLUS and TIMES!!! CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -56

13 Boolean Algebra There are at least two elements ( and ) Z Boolean Algebra Special W + Operators There are two operators: AND ( ) and OR (+) Closure Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -57

14 Boolean Algebra There are at least two elements ( and ) Z Boolean Algebra + Special W + Operators There are two operators: AND ( ) and OR (+) Closure with respect to AND ( ), {,}, i {,} Closure with respect to OR (+), {,}, + {,} Closure Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -58

15 Boolean Algebra There are at least two elements ( and ) There are two identit elements: Z= is the identit with respect to OR W= is the identit element with respect to AND + {,}, i = Special Z W + Operators There are two operators: AND ( ) and OR (+) Closure Boolean Algebra Associative Commutative Inverse Distributive Properties Complement {,}, + = CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -59

16 Boolean Algebra There are at least two elements ( and ) Z Boolean Algebra + Special W + Operators There are two operators: AND ( ) and OR (+) Commutative with respect to AND ( ), {,}, i = i Commutative with respect to OR (+), {,}, + = + Closure Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -6

17 Boolean Algebra There are at least two elements ( and ) + Special Z W + Operators There are two operators: AND ( ) and OR (+) AND ( ) is distributive over OR(+), z, {,}, i( + z) = ( i ) + ( z i ) OR (+) is distributive over AND( ), z, {,}, + ( z i ) = ( + ) i( + z) Closure Boolean Algebra Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -6

18 Boolean Algebra There are at least two elements ( and ) Z Boolean Algebra + Special W + Operators There are two operators: AND ( ) and OR (+) There is a complement element with respect to AND and OR {,}, ' {,) : + ' = i ' = Closure Associative Commutative Inverse Distributive Properties Complement Alternate was to epress complement: ' = = CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -62

19 Boolean Algebra There are at least two elements ( and ) + Special Z W + Operators There are two operators: AND ( ) and OR (+) AND ( ) is associative, z, {,}, i( z i ) = ( i ) iz OR (+) is associative, z, {,}, + ( + z) = ( + ) + z Closure Boolean Algebra Associative Commutative Inverse Distributive Properties Complement CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -63

20 Boolean Algebra α β There are at least two elements α β α α β β α β Special Z W + Operators There are two operators: AND ( ) and OR (+) Closure Boolean Algebra Associative Commutative Inverse Distributive Properties Complement + α β α α α α β β β β It is possible to define a Boolean Algebra with more than two elements, e.g., {,α,β,}. All the properties defined above can be shown to be valid. CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -64

21 Proof b Truth Tables Show that the Boolean Algebra defined above is distributive: a b c b+c a (b+c) a b a c (a b)+(a c) +, z, {,}, i( + z) = ( i ) + ( z i ), z, {,}, + ( z i ) = ( + ) i( + z) Addition is also distributive Multiplication is not CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -65

22 Theorems and Postulates of Boolean Algebra + = i = + ' = i ' = + = i = + = i = ( ')' = + = + i = i + ( + z) = ( + ) + z i( z i ) = ( i ) iz i( + z) = i + i z + ( z i ) = ( + ) i( + z) ( + )' = ' i ' ( i)' = ' + ' + i = + i = DeMorgan s Law Absorption CpE358/CS38 Switching Theor and Logical Design Summer- 24 Dualit: Interchange identit Interchange operator Copright 24 Stevens Institute of Technolog -66

23 Operator Precedence. Parenthesis 2. NOT 3. AND 4. OR i + z = ( i) + z i( + z) z ( )+z (+z) (+z) CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -67

24 Boolean Functions Fz (,, ) = i + ' iz z F(,,z) F(,,z) z CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -68

25 Boolean Functions Epressed as Timing Diagrams z F(,,z) This is an idealized timing diagram: There are no delas through gates, all events occur at instants of time. CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -69

26 Simplifing Boolean Functions F (,, ) ' ' ' ' a z = i iz+ iiz+ i ' i' iz ' iz i F(,,z) z i ' CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -7

27 Simplifing Boolean Functions F (,, z) = ' i' iz+ ' iiz+ i' a = ' izi( ' + ) + i' = ' iz+ ' i' iz i' ' iiz F(,,z) z i ' CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -7

28 Simplifing Boolean Functions F (,, ) ' ' b z = iz+ i ' iz F(,,z) z i ' CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -72

29 Verifing Simplification F (,, ) ' ' ' ' a z = i iz+ iiz+ i F (,, ) ' ' b z = iz+ i z z z z F a (,,z) F b (,,z) CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -73

30 Gate Compleit of Boolean Epression Gate requirements can be estimated directl from epression for Boolean function Fz (,, ) = i + ' iz+ z i One 3-input OR operation Three 2-input AND operations Epression can be simplified algebraicall: i + ' iz+ iz = i + ' iz+ iz( + ') = i + ' iz+ z i i + ' iz i = i i( + z) + ' izi( + ) Two 2-input AND operations = i + ' iz Fz (,, ) = i + ' iz One 2-input OR operation CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -74

31 More Logic Functions AND and OR function are not often seen as such in real logic designs. NOT-AND and NOT-OR are easier to implement in hardware and are generall faster NOT-AND NAND = NOT-OR NOR = The bubble can be placed on an lead to indicate inversion. Two other common logic functions are: CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -75

32 More on DeMorgan s Law ( i)' = ' + ' NAND NAND ( + )' = ' i ' NOR NOR CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -76

33 Functions of Two Inputs There are 6 possible functions of two inputs F F F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F F F 2 F 3 F 4 F 5 Name of F n Z e r o A N D i n h i b i n h i b X O R O R N O R = -> -> N A N D O n e CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -77

34 Generic Boolean Functions A function can be epressed in terms of min-terms or ma-terms z Minterm Name Materm Name z m + +z M z m + +z M z m 2 ++z M 2 z m 3 ++z M 3 z m 4 + +z M 4 z m 5 + +z M 5 z m 6 ++z M 6 z m 7 ++z M 7 An alternative description of a function is to onl specif the minterms Fz (,, ) ' ' z ' z' z m m m (, 4, 7) = i i + i i + i i = = CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -78

35 Sum of Products vs. Product of Sums z z z z Σ(,4,7) z z z Π(,2,3,5,6) z z z z CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -79

36 Conversion between Σ(Π) and Π(Σ) m = { m, m, m,, m } 2 7 Fz (,, ) = ( abcd,,, ) Fz (,, ) = m+ m + m+ m F'(,, z) = ( m { m, m, m, m }) = ( m, m, m, m ) = m + m + m + m Fz (,, ) = ( m+ m+ m + m)' = m' im' im' im' = MiMiMiM a b c d Fz (,, ) = ( efgh,,, ) a b c d e f g h e f g h e f g h e f g h e f g h CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -8

37 Design Considerations These two designs are logicall equivalent: z z z z z z z z Σ(,4,7) z z z z + z z z + z Σ(,4,7) But this one incurs two more gate delas CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -8

38 Other Basic Logic Gates Eclusive OR (XOR) = ' i + i ' = Eclusive NOR (equivalence) ( ) = ' i' + i CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -82

39 Digital Logic Families RTL Resistor Transistor Logic Original logic famil used in ICs VERY slow and power hungr b recent standards Mostl replaced b TTL in 96s-7s TTL Transistor-Transistor Logic 74 series - workhorse of logic designs Mostl replaced b CMOS with 74C equivalents ECL Emitter Coupled Logic Popular in 97s Ver high speed High power consumption MOS Metal Oide Silicon High densit, introduced for memor applications CMOS Complementar Metal Oide Silicon Etremel low quiescent power consumption Wide range of speed/power tradeoffs Industr standard toda CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -83

40 Summar Fundamental concepts of digital sstems (Mano Chapter ) Binar codes, number sstems, and arithmetic (Ch ) Boolean algebra (Ch 2) Simplification of switching equations (Ch 3) Digital device characteristics (e.g., TTL, CMOS)/design considerations (Ch ) Combinatoric logical design including LSI implementation (Chapter 4) Hazards, Races, and time related issues in digital design (Ch 9) Flip-flops and state memor elements (Ch 5) Sequential logic analsis and design (Ch 5) Snchronous vs. asnchronous design (Ch 9) Counters, shift register circuits (Ch 6) Memor and Programmable logic (Ch 7) Minimization of sequential sstems Introduction to Finite Automata CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -84

41 Homework 2 due in Class 4 Problems 2-, 2-5, 2-5, 2-9. Show all work CpE358/CS38 Switching Theor and Logical Design Summer- 24 Copright 24 Stevens Institute of Technolog -85

DIGITAL LOGIC CIRCUITS

DIGITAL LOGIC CIRCUITS Introduction Logic Gates Boolean Algebra Map Specification Combinational Circuits Flip-Flops Sequential Circuits Memor Components Integrated Circuits BASIC LOGIC BLOCK - GATE - Logic