# CS 226: Digital Logic Design

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1 CS 226: Digital Logic Design I S S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29

2 Objectives In this lecture we will introduce: 1. Logic functions and circuits, 2. Boolean algebra for manipulating with logic functions, 3. Logic gates, and 4. Synthesis of simple logic circuits. 2 of 29

3 Objectives Logic functions and circuits Boolean Algebra Synthesis of Simple Circuits Introduction to CAD tools 3 of 29

4 Variables and Functions Logic circuits form the foundation of digital systems Binary logic circuits perform operations of binary signals 4 of 29

5 Variables and Functions Logic circuits form the foundation of digital systems Binary logic circuits perform operations of binary signals Let s consider the simplest element logic circuit: switch. A switch can be in two states: open and close. Graphical symbol for a switch A simple application using a switch : lightbulb 4 of 29

6 Variables and Functions Logic circuits form the foundation of digital systems Binary logic circuits perform operations of binary signals Let s consider the simplest element logic circuit: switch. A switch can be in two states: open and close. Graphical symbol for a switch A simple application using a switch : lightbulb State of the switch can be given as a binary variable x s.t.: x = 0 when switch is open, and x = 1 when switch is closed. The function of a switch: F = 0 when x = 0 and F = 1 when x = 1. In other words F(x) = x. 4 of 29

7 Variables and Functions Logic circuits form the foundation of digital systems Binary logic circuits perform operations of binary signals Let s consider the simplest element logic circuit: switch. A switch can be in two states: open and close. Graphical symbol for a switch A simple application using a switch : lightbulb State of the switch can be given as a binary variable x s.t.: x = 0 when switch is open, and x = 1 when switch is closed. The function of a switch: F = 0 when x = 0 and F = 1 when x = 1. In other words F(x) = x. Keywords: Logic expression defining output, logic function, and input variable. 4 of 29

8 Variables and Functions Logic circuits form the foundation of digital systems Binary logic circuits perform operations of binary signals Let s consider the simplest element logic circuit: switch. A switch can be in two states: open and close. Graphical symbol for a switch A simple application using a switch : lightbulb State of the switch can be given as a binary variable x s.t.: x = 0 when switch is open, and x = 1 when switch is closed. The function of a switch: F = 0 when x = 0 and F = 1 when x = 1. In other words F(x) = x. Keywords: Logic expression defining output, logic function, and input variable. Let s use two switches to control the lightbulb by connecting them in series and parallel. 4 of 29

9 Introducing AND nd OR Logical AND operation (series connection) We write the expression L(x, y) = x y to say L(x, y) = x AND y when { 1 if x = 1 and y = 1. L = 0 otherwise. 5 of 29

10 Introducing AND nd OR Logical AND operation (series connection) We write the expression L(x, y) = x y to say L(x, y) = x AND y when { 1 if x = 1 and y = 1. L = 0 otherwise. Logical OR operation (parallel connection) We write the expression L(x, y) = x + y to say L(x, y) = x OR y when { 0 if x = 0 and y = 0. L = 0 otherwise. 5 of 29

11 Introducing AND nd OR Logical AND operation (series connection) We write the expression L(x, y) = x y to say L(x, y) = x AND y when { 1 if x = 1 and y = 1. L = 0 otherwise. Logical OR operation (parallel connection) We write the expression L(x, y) = x + y to say L(x, y) = x OR y when { 0 if x = 0 and y = 0. L = 0 otherwise. A series-parallel connection 5 of 29

12 Inversion Can we use a switch to implement an inverted switch? We would like to implement the following function: L(x) = x to say L(x) = NOT x or complement of x when { 0 if x = 1. L = 1 if x = 0 6 of 29

13 Inversion Can we use a switch to implement an inverted switch? We would like to implement the following function: L(x) = x to say L(x) = NOT x or complement of x when { 0 if x = 1. L = 1 if x = 0 The inverting circuit 6 of 29

14 Inversion Can we use a switch to implement an inverted switch? We would like to implement the following function: L(x) = x to say L(x) = NOT x or complement of x when { 0 if x = 1. L = 1 if x = 0 The inverting circuit We can complement not only variables but also expressions, s.t. f (x, y) = x + y. 6 of 29

15 Specification of a logical function: Truth tables The specification of a logical function can be enumerate as a truth-table Since input variables are finite, there are only finitely many possible combinations for a finite set of inputs. Example of truth table for x y, x + y and x. Truth-table for n-input AND, OR, or NOT operations 7 of 29

16 Logic Gates A switch can be implemented using a transistor. Also other Boolean operations can be implemented using various combinations of switches. AND gate, OR gate, NOT gate represent encapsulation of transistor circuits implementing Boolean function x y x y x y x + y x x 8 of 29

17 Logic Gates A switch can be implemented using a transistor. Also other Boolean operations can be implemented using various combinations of switches. AND gate, OR gate, NOT gate represent encapsulation of transistor circuits implementing Boolean function x y x y x y x + y x x x y z x y z x y y x + y 8 of 29

18 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis 9 of 29

19 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis Let s analyze the logic network f (x 1, x 2, x 3 ) = x 1 + x 2 x 3. Construct its Truth-Table 9 of 29

20 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis Let s analyze the logic network f (x 1, x 2, x 3 ) = x 1 + x 2 x 3. Construct its Truth-Table Timing diagram of a circuit Functionally equivalent circuits: consider the network that implement Boolean function g(x 1, x 2 ) = x 1 + x 2. and compare with the function f (x 1, x 2 ). 9 of 29

21 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis Let s analyze the logic network f (x 1, x 2, x 3 ) = x 1 + x 2 x 3. Construct its Truth-Table Timing diagram of a circuit Functionally equivalent circuits: consider the network that implement Boolean function g(x 1, x 2 ) = x 1 + x 2. and compare with the function f (x 1, x 2 ). How do we know if two functions are logically equivalent? 9 of 29

22 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis Let s analyze the logic network f (x 1, x 2, x 3 ) = x 1 + x 2 x 3. Construct its Truth-Table Timing diagram of a circuit Functionally equivalent circuits: consider the network that implement Boolean function g(x 1, x 2 ) = x 1 + x 2. and compare with the function f (x 1, x 2 ). How do we know if two functions are logically equivalent? How many different logically equivalent functions over n-variables? 9 of 29

23 Analysis and Synthesis of a Logic Network Tasks of a designer of a digital system: analysis and synthesis Let s analyze the logic network f (x 1, x 2, x 3 ) = x 1 + x 2 x 3. Construct its Truth-Table Timing diagram of a circuit Functionally equivalent circuits: consider the network that implement Boolean function g(x 1, x 2 ) = x 1 + x 2. and compare with the function f (x 1, x 2 ). How do we know if two functions are logically equivalent? How many different logically equivalent functions over n-variables? How to minimize circuit complexity (and cost)? 9 of 29

24 Objectives Logic functions and circuits Boolean Algebra Synthesis of Simple Circuits Introduction to CAD tools 10 of 29

25 Boolean Algebra In 1849 George Boole introduced algebra for manipulating processes involving logical thoughts and reasoning This scheme, with some refinements, is not know as Boolean algebra. Claude Shannon in 1930 showed that Boolean algebra is awesome for describing circuits with switches! It is, then, of course awesome to describe logical circuits. Let s see how it is a powerful tool to design and analyze logical circuits. 11 of 29

26 Boolean Algebra: Axioms = 0 (1) = 1 (2) = 1 (3) = 0 (4) = 1 0 = 0 (5) = = 1 (6) 4. If x = 0 then x = 1 (7) If x = 1 then x = 0 (8) 12 of 29

27 Boolean Algebra: Single variable Theorems 1. x 0 = 0 (9) x + 1 = 1 (10) 2. x 1 = x (11) x + 0 = x (12) 3. x x = x (13) x + x = x (14) 4. x x = 0 (15) x + x = 1 (16) 5. x = x (17) 13 of 29

28 Boolean Algebra: 2- and 3- Variable Properties 1. Commutativity: x y = y x (18) x + y = y + x (19) 2. Associativity: x (y z) = (x y) z (20) x + (y + z) = (x + y) + z (21) 3. Distributivity: x (y z) = (x y) z (22) x + (y + z) = (x + y) + z (23) 4. Absorption: x + x y = x (24) x (x + y) = x (25) 14 of 29

29 Boolean Algebra: 2- and 3- Variable Properties 1. Combining: x y + x y = x (26) (x + y) (x + y) = x (27) 2. Consensus: x y + y z + x z = x y + x z (28) (x + y) (y + z) (x + z) = (x + y) (x + z) (29) 15 of 29

30 Boolean Algebra: DeMorgan s theorem Augustus De Morgan (27 June March 1871) 16 of 29

31 Boolean Algebra: DeMorgan s theorem Augustus De Morgan (27 June March 1871) Proof by Perfect Induction (Truth-tables) x y = x + y (30) x + y = x y (31) 16 of 29

32 Other Remarks Venn diagram are quite useful proving theorems in Boolean algebra 17 of 29

33 Other Remarks Venn diagram are quite useful proving theorems in Boolean algebra Common symbols to represent OR: x y and x + y Common symbols to represent AND: x y and x y Common symbols to represent NOT: x and x 17 of 29

34 Other Remarks Venn diagram are quite useful proving theorems in Boolean algebra Common symbols to represent OR: x y and x + y Common symbols to represent AND: x y and x y Common symbols to represent NOT: x and x Precedence of Operations: NOT > AND > OR Example x 1 x 2 + x 1 x 2 (x 1 x 2) + ((x 1 ) (x 2)) x 1 (x 2 + x 1 ) x 2 17 of 29

35 Objectives Logic functions and circuits Boolean Algebra Synthesis of Simple Circuits Introduction to CAD tools 18 of 29

36 Synthesis of simple circuits Definition (Synthesis) Given a description of the desired functional behavior, the synthesis is the process to generate a circuit that realizes this behavior. Commonly Used logic Gates: x y x y x y x + y x x x y x y x y x + y x y x y 19 of 29

37 Synthesis of simple circuits Theorem Any Boolean function can be synthesized using only AND, OR, and NOT gates. x y x y x y x + y x x 20 of 29

38 Synthesis of simple circuits Theorem Any Boolean function can be synthesized using only AND, OR, and NOT gates. x y x y x y x + y x x Theorem Any Boolean function can be synthesized using only NAND gates. x y x y 20 of 29

39 Synthesis of simple circuits Theorem Any Boolean function can be synthesized using only AND, OR, and NOT gates. x y x y x y x + y x x Theorem Any Boolean function can be synthesized using only NAND gates. x y x y Theorem Any Boolean function can be synthesized using only NOR gates. x y x + y 20 of 29

40 Synthesis using AND, OR, and NOT gates Given a Boolean function f given in the form of a truth table, the expression that realizes f can be obtained: (SUM-OF-PRODUCTS) by considering rows for which f = 1, or (PRODUCTS-OF-SUMS) by considering rows for which f = of 29

41 Synthesis using AND, OR, and NOT gates Given a Boolean function f given in the form of a truth table, the expression that realizes f can be obtained: (SUM-OF-PRODUCTS) by considering rows for which f = 1, or (PRODUCTS-OF-SUMS) by considering rows for which f = 0. Example: x 1 x 2 f (x 1, x 2 ) f (x 1, x 2 ) = x 1 x 2 + x 1 x 2 = m 1 + m 3 f (x 1, x 2 ) = (x 1 x 2 + x 1 x 2 ) = (x 1 x 2 ) (x 1 x 2 ) = (x 1 + x 2 ) (x 1 + x 2 ) = M 0 M 2 21 of 29

42 Sum-of-Product Canonical Form For a function of n variables a minterm is a product term in which each of the variable occur exactly once. For example: x 0 x 1 x 2 and x 0 x 1 x 2. A function f can be represented by an expression that is sum of minterms. A logical expression that consists of product terms that are summed (ORed) together is called a sum-of-product form. If each of the product term is a minterm, then the expression is called canonical sum-of-product expression. Notice that two logically equivalent functions will have the same canonical representation. For a truth-table of n variables, we represent the minterm corresponding to the i-th row as m i where i {0, 2 n 1}. A unique representation of a function can be given as an explicit sum of minterms for rows for which function is true. For example f (x 1, x 2 ) = x 1 x 2 + x 1 x 2 = m 1 + m 3 = (m 1, m 3 ) = m(1, 3). 22 of 29

43 Sum-of-Product Canonical Form For a function of n variables a maxterm is a sum term in which each of the variable occur exactly once. For example (x 0 + x 1 + x 2 ) and (x 0 + x 1 + x 2 ). A function f can be represented by an expression that is product of maxterms. A logical expression that consists of sum terms that are factors of logical product (AND) is called a product-of-sum form. If each of the sum term is a maxterm, then the expression is called canonical product-of-sum expression. Notice that two logically equivalent functions will have the same canonical product-of-sum representation. For a truth-table of n variables, we represent the maxterm corresponding to complement of the minterm of the i-th row as M i where i {0, 2 n 1}. A unique representation of a function can be given as an explicit product of maxterms for rows for which function is false. For example f (x 1, x 2 ) = (x 1 x 2 + x 1 x 2 ) = (M 0, M 2 ) = M(0, 2). 23 of 29

44 Questions Give SOP form of the function f (x 1, x 2, x 3 ) = m(2, 3, 4, 6, 7) and simplify it. Give POS for of the function f (x 1, x 2, x 3 ) = M(0, 1, 5) and simplify it. 24 of 29

45 NAND and NOR logic networks De Morgan s theorem in terms of logic gates Using NAND gates to implement a sum-of-products Using NOR gates to implement a product-of-sums 25 of 29

46 Examples Design the logic circuits for the following problems: Three-way light control Multiplexer circuit 26 of 29

47 Objectives Logic functions and circuits Boolean Algebra Synthesis of Simple Circuits Introduction to CAD tools 27 of 29

48 Steps in Design process 1. Design Entry Schematic capture Hardware description language 2. Synthesis (or translating/ compiling) 3. Functional Simulation 4. Physical Design 5. Timing Simulation 6. Chip configuration (programming) 28 of 29

49 Introduction to VHDL 29 of 29

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