Digital Logic Design Lecture Notes (PDF)

NET 104: Digital Logic Design Lecture 2+3 Digital Logic Design 10/19/2024 1  Describing Circuit Functionality: Inverter Basic logic functions have symbols The same functionality can be represented with a truth table Truth table completely specif...

NET 104: Digital Logic Design Lecture 2+3 Digital Logic Design 10/19/2024 1  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 0 is inverted to a 1 A Y An input of 1 is inverted to a 0 0 1 1 0 A Y Input Output Symbol Digital Logic Design 10/19/2024 2 10/19/2024 Digital Logic Design 2  The AND Gate This is an AND gate Truth Table If the two input signals A B Y are asserted (i.e. high) the 0 0 0 output will also be asserted. 0 1 0 Otherwise, the output will 1 0 0 be deasserted (i.e. low) 1 1 1 Y=A.B A Y A B B Digital Logic Design 10/19/2024 3 10/19/2024 Digital Logic Design 3  The OR Gate This is an OR gate A B Y If either of the two 0 0 0 input signals is 0 1 1 asserted, or both of 1 0 1 them are, the output 1 1 1 will be asserted A Y=A+B A Y B B Digital Logic Design 10/19/2024 4 10/19/2024 Digital Logic Design 4  The NAND Gate The NAND gate is a combination of an AND gate followed by an inverter NAND(A, B) → (A AND B)’ A B Y A Y 0 0 1 B 0 1 1 1 0 1 Y=A.B 1 1 0 Digital Logic Design 10/19/2024 5 10/19/2024 Digital Logic Design 5 The Universal Property of NAND  You can implement any function using: NOT, AND, and OR. They represent a logically complete set.  You can use only NAND gates to implement the above three gates. Therefore, NAND alone is a logically complete set. NOT X X X X.Y AND Y X OR X+Y Y Digital Logic Design 10/19/2024 6 10/19/2024 Digital Logic Design 6  The NOR Gate A NOR gate is a combination of an OR gate followed by an inverter NOR(A , B) → (A+B)’ A B Y 0 0 1 A Y 0 1 0 B 1 0 0 Y=A+B 1 1 0 Digital Logic Design 10/19/2024 7 10/19/2024 Digital Logic Design 7 The Universal Property of NOR  Similarly, you can use only NOR gates to implement NOT, AND, and OR. Therefore, NOR alone is a logically complete set. NOT X X X AND X.Y Y OR X X+Y Y Digital Logic Design 10/19/2024 8 10/19/2024 Digital Logic Design 8 Exclusive-OR Circuits Exclusive-OR (XOR) produces a HIGH output whenever the two inputs are at opposite levels Digital Logic Design 10/19/2024 9 10/19/2024 Digital Logic Design 9 Exclusive-NOR Circuits Exclusive-NOR (XNOR) produces a HIGH output whenever the two inputs are at the same level Digital Logic Design 10/19/2024 10 10/19/2024 Digital Logic Design 10 XOR Function XOR function can also be implemented with AND/OR gates (also NANDs) Digital Logic Design 10/19/2024 11 10/19/2024 Digital Logic Design 11  Describing Circuit Functionality: Waveforms Waveforms provide another approach for representing functionality Values are either high (logic 1) or low (logic 0) Can you create a truth table from the waveforms? AND Gate x y f 0 0 0 0 1 0 1 0 0 1 1 1 Digital Logic Design 10/19/2024 12 10/19/2024 Digital Logic Design 12 Consider Three-input Gates 3 Input OR Gate Digital Logic Design 10/19/2024 13 10/19/2024 Digital Logic Design 13  Boolean Algebra Useful for identifying and minimizing circuit functionality Identity elements  a+0=a  a 1=a 0 is the identity element for the + operation 1 is the identity element for the operation The Complement: for every element ‘a’, there exists a unique element called a’ (or ā) (complement of a) such that :  a + a’ = 1  a a’ = 0 Digital Logic Design 10/19/2024 14 10/19/2024 Digital Logic Design 14 George Boole (1815 - 1864)  Father of Boolean algebra  Boole’s system (detailed in his 'An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities', 1854) was based on a binary approach, processing only two objects - the yes-no, true-false, on-off, zero-one approach.  Surprisingly, given his standing in the academic community, Boole's idea was either criticized or completely ignored by the majority of his peers.  Eventually, one bright student, Claude Shannon (1916-2001), picked up the idea and ran with it. Digital Logic Design 10/19/2024 15 10/19/2024 Digital Logic Design 15  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Laws of Boolean Algebra  Commutative Law of ORing: A+B=B+A  Commutative Law of ANDing: A.B=B.A Digital Logic Design 10/19/2024 16 10/19/2024 Digital Logic Design 16  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Laws of Boolean Algebra (Cont’d)  Associative Law of ORing: A + (B + C) = (A + B) + C  Associative Law of ANDing: A. (B. C) = (A. B). C Digital Logic Design 10/19/2024 17 10/19/2024 Digital Logic Design 17  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Laws of Boolean Algebra (Cont’d)  Distributive Law: Note: 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) = AB + AC Digital Logic Design 10/19/2024 18 10/19/2024 Digital Logic Design 18  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Digital Logic Design 10/19/2024 19 10/19/2024 Digital Logic Design 19  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 1 OR Truth Table Rule 2 OR Truth Table Digital Logic Design 10/19/2024 20 10/19/2024 Digital Logic Design 20  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 3 AND Truth Table Rule 4 AND Truth Table Digital Logic Design 10/19/2024 21 10/19/2024 Digital Logic Design 21  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 5 OR Truth Table Rule 6 OR Truth Table Digital Logic Design 10/19/2024 22 10/19/2024 Digital Logic Design 22  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 7 AND Truth Table Rule 8 AND Truth Table Digital Logic Design 10/19/2024 23 10/19/2024 Digital Logic Design 23  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 9 Rule 10 (the absorption rule): A + AB = A Digital Logic Design 10/19/2024 24 10/19/2024 Digital Logic Design 24 Digital Logic Design 10/19/2024 25  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Rules of Boolean Algebra Cont’d  Rule 11: A  AB  A  B Rule 12: (A + B)(A + C) = A + BC Digital Logic Design 10/19/2024 26 10/19/2024 Digital Logic Design 26 Extra Digital Logic Design 10/19/2024 27 Digital Logic Design 10/19/2024 28 De Morgan’s Theorems (by Augustus De Morgan (1806 - 1871), an English mathematician and logician) A.B  A  B A  B  A.B General: A.B. C.D  A  B  C  D A  B  C  D  A.B. C.D Digital Logic Design 10/19/2024 29 10/19/2024 Digital Logic Design 29 De Morgan’s Theorems Example (A.B  C) (A  B.C)  (A.B  C)  (A  B.C)  ( A.B. C )  ( A.B.C )  ( A  B ). C  A.(B  C )  A.C  B.C  A.B  A.C  A.C  B.C  A.B Digital Logic Design 10/19/2024 30 10/19/2024 Digital Logic Design 30  Copyright © The McGraw-Hill Companies, Inc. Converting AND to OR  Using De Morgan’s Theorems, AND can be converted to OR (with some help from NOT)  Consider the following gate: To convert AND to OR A B A B A B A B (or vice versa), 0 0 1 1 1 0 invert inputs and output. 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 Same as A+B Digital Logic Design 10/19/2024 31 10/19/2024 Digital Logic Design 31  © 2006 Pearson Education, Inc., “Digital Fundamentals”, 9/e by Floyd Standard Forms of Boolean Expressions  The sum-of-product (SOP) form Example: X = AB + CD + EF  The product of sum (POS) form Example: X = (A + B)(C + D)(E + F) Digital Logic Design 10/19/2024 32 10/19/2024 Digital Logic Design 32 Sum-of-Products Expression  A literal is a variable or the complement of a variable. Examples: X , Y , X , Y  A product term is a single literal or a logical product of two or more literals. Examples: Z , W.X.Y , X.Y.Z , W.Y.Z  A sum-of-products (SOP) expression is a logical sum of product terms. Example: Z  W.X.Y  X.Y.Z  W.Y.Z 1 Z  Every SOP expression W can be realized by a X Y two-level circuit containing AND X Y gates followed by Z an OR gate W Y Digital Logic Design Z 10/19/2024 33 10/19/2024 Digital Logic Design 33  Function Representation Conversion Need to transit between a Boolean expression, a truth table, and a circuit (symbols) All three formats are equivalent Circuit Boolean Expression Truth Table Digital Logic Design 10/19/2024 34 10/19/2024 Digital Logic Design 34 Minterm  For each truth-table row a product term can be defined that evaluates to 1 only when the inputs have the values listed in that row.  If the product term contains each input variable exactly once it is called a minterm Row A B C Minterm Maxterm 0 0 0 0 A.B.C A + B + C 1 0 0 1 A.B.C A + B + C 2 0 1 0 A.B.C A + B + C 3 0 1 1 A.B.C A + B + C 4 1 0 0 A.B.C A + B + C 5 1 0 1 A.B.C A + B + C 6 1 1 0 A.B.C A + B + C 7 1 1 1 A.B.C A + B + C Digital Logic Design 10/19/2024 35 10/19/2024 Digital Logic Design 35 From Truth Table to Boolean Expression  Any logic function can be expressed algebraically by taking the OR of all those minterms corresponding to the truth-table rows for which the function produces a 1 output.  Example: Majority detector. Row A B C Minterm R 0 0 0 0 A.B.C 0 1 0 0 1 A.B.C 0 R=A.B.C+ A.B.C+ A.B.C+ A.B.C 2 0 1 0 A.B.C 0 3 0 1 1 A.B.C 1 4 1 0 0 A.B.C 0 5 1 0 1 A.B.C 1 6 1 1 0 A.B.C 1 7 1 1 1 A.B.C 1 Alternate forms: R = m3 + m5 + m6 + m7 R = ∑ (3, 5, 6, 7) Digital Logic Design 10/19/2024 36 10/19/2024 Digital Logic Design 36 Converting to a Circuit Number of 1’s in truth table output column equals AND terms for Sum-of-Products (SOP) x y z G 0 0 0 0 0 0 1 0 0 1 0 0 G 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 x y z G = xyz + xyz’ + x’yz Digital Logic Design 10/19/2024 37 10/19/2024 Digital Logic Design 37  Thank You Digital Logic Design 10/19/2024 38