Lecture 2: Digital Logic Basis Xufeng Kou School of Information Science and Technology ShanghaiTech University 1
Outline Truth Table Basic Logic Operation and Gates Logic Circuits NOR Gates and NAND Gates Boolean Theorems Demorgan s Theorems 2
Analog vs. Digital 3
Analog vs. Digital Analog Digital Audio System CD Player 4
Analog vs. Digital Signal CONTINUOUS (Sine Waves) DISCRETE (Square Waves) Applications Audio and Video trans. Computing Bandwidth Can be done in real time Less bandwidth More bandwidth to carry out the same information Memory Form of wave signal Form of binary bit Accuracy High Low Response to Noise Subjected to deterioration Noise-Immune w/o deterioration during trans. Power Large Small 5
Boolean Constants and Variables Boolean algebra allows only two values 0 and 1 Three basic logic operations: OR AND NOT 6
Truth Table Describes the relationship between the input and output of a logic circuit. # of entries corresponds to # of inputs. A 2-input table: 2 2 = 4 entries A 3-input table: 2 3 = 8 entries 7
Truth Table Example Examples of truth tables with 2, 3, and 4 inputs.
Simplest Logic Function - NOT The Boolean expression for the NOT operation: X = A The overbar represents the NOT operation. Read as: X equals NOT A X equals the inverse of A X equals the complement of A NOT Truth Table This NOT Gate always has only a single input, and the output logic level is always opposite to the logic level of this input.
Review NMOS & PMOS NMOS V G = HIGH -> ON V G = LOW -> OFF PMOS V G = LOW -> ON V G = HIGH -> OFF
NOT Gate Circuit Implementation Inverter V out Input V in Output Whenever the input = 0, output = 1, and vice versa.
Application of NOT Gate This circuit provides an expression that is true when the button is not pressed.
OR Operation The Boolean expression for the OR operation is: X = A + B Read as X equals A OR B The + sign does not stand for ordinary addition it stands for the OR operation The OR operation is similar to addition, but when A = 1 and B = 1, the OR operation produces: 1 + 1 = 1 not 1 + 1 = 2 In the Boolean expression x = 1 + 1 + 1 = 1 x is true (1) when A is true (1) OR B is true (1) OR C is true (1)
OR Gate An OR gate is a circuit with two or more inputs, whose output is equal to the OR combination of the inputs. 2-Input OR gate Symbol Truth Table Waveform
3-Input OR Gate Truth table/circuit symbol for a 3 input OR gate. The OR symbol means the output will go HIGH when any input is HIGH.
Application of OR Gate Question: What function does this system implement?
AND Operation The Boolean expression for the AND operation is: X = A B C Read as X equals A AND B AND C 2-Input AND gate The sign does not stand for ordinary multiplication it stands for the AND operation. Symbol Truth Table Waveform
3-Input AND Gate Truth table/circuit symbol for a 3 input AND gate. The AND symbol on a logic-circuit diagram tells you output will go HIGH only when all inputs are HIGH.
Application of AND Gate Seat Belt Alarm System Question: How does this system work?
Boolean Operations Summarized rules for OR, AND and NOT These three basic Boolean operations can describe any logic circuit.
Logic Circuit Algebra If an expression contains both AND and OR gates, the AND operation will be performed first. Unless there is a parenthesis in the expression.
Logic Circuit Algebra (Cont.) Whenever an INVERTER is present, output is equivalent to input, with a bar over it. Input A through an inverter equals A.
Exercise Question: What is the output state?
Evaluate Logic Circuit Outputs Rules for evaluating a Boolean expression: Perform all inversions of single terms. Perform all operations within parenthesis. Perform AND operation before an OR operation unless parenthesis indicate otherwise. If an expression has a bar over it, perform operations inside the expression, and then invert the result.
Evaluate Logic Circuit Outputs The best way to analyze a circuit made up of multiple logic gates is to use a truth table. It allows you to analyze one gate or logic combination at a time. It allows you to easily double-check your work. When you are done, you have a table of tremendous benefit in troubleshooting the logic circuit.
Step 1 List all input combinations. Create a column in the truth table for each node. Fill the values for u.
Step 2 Fill in the values for column v. v =AB Node v should be HIGH when A (node u) is HIGH AND B is HIGH
Step 3 Predict the values at node w which is the logical product of BC. This column is HIGH whenever B is HIGH AND C is HIGH
Step 4 Logically combine columns v and w to predict the output x. Since x = v + w, the x output will be HIGH when v OR w is HIGH
Draw the Circuit from Boolean Expression A circuit with output y = AC + BC + ABC contains three terms which are ORed together. Backward
Draw the Circuit from Boolean Expression Each OR gate input is an AND product term, An AND gate with appropriate inputs can be used to generate each of these terms.
Exercise Question: Draw the circuit to implement x = (A + B) (B + C)
NOR and NAND Gates Combine basic AND, OR, and NOT operations. Simplifying the writing of Boolean expressions Output of NAND and NOR gates may be found by determining the output of an AND or OR gate, and inverting it. The truth tables for NOR and NAND gates show the complement of truth tables for OR and AND gates.
NOR Gate The NOR gate is an inverted OR gate. x = A + B Only if both inputs are 0, the output will be 1
NAND Gate The NAND gate is an inverted AND gate. x = AB
Logic Circuit using NOR/NAND Gates Logic circuit with the expression x = AB (C + D) using only NOR and NAND gates.
Universality of NOR and NAND Gates NAND or NOR gates can be used to create the three basic logic expressions. OR, AND, and INVERT. Provides flexibility very useful in logic circuit design.
Universality of NAND Gate
Universality of NOR Gate
74 Series IC Packages
Logic Circuit using 74 Series x = AB + CD Possible Implementations # 1
Logic Circuit using 74 Series x = AB + CD Possible Implementations # 2
Boolean Theorems Th1: if any variable is ANDed with 0, the result must be 0. Th2: if any variable is ANDed with 1, the result must be itself Logic AND is just like ordinary multiplication!
Boolean Theorems Th3: A variable ANDed with itself is always equal to itself Th4: A variable ANDed with its complement is always 0
Boolean Theorems Th5: A variable ORed with 0 is always equal to itelf Th6: A variable ORed with 1 is always 1
Boolean Theorems Th7: A variable ORed with itself is always equal to itself Th8: A variable ORed with its complement is always 1
Commutative laws Multivariable Theorems Associative laws Distributive law
Multivariable Theorems Theorems (14) and (15) do not have counterparts in ordinary algebra. Each can be proved by - Trying all possible cases for x and y. Analysis table & factoring for Theorem (14)
DeMorgan s Theorems Each of DeMorgan s theorems can readily be proven by checking for all possible combinations of x and y.
DeMorgan s Theorems Equivalent circuits implied by Theorem (16) The alternative symbol for the NOR function.
DeMorgan s Theorems Equivalent circuits implied by Theorem (17) The alternative symbol for the NOR function.
References David M. Harris and Sarah L. Harris, Digital Design and Computer Architecture Thomas L. Floyd, Digital Fundamentals,11 th edition Ronald J. Tocci and Neal Widmer, Digital Systems Principles and Applications,11 th edition
Advanced Topic: CMOS Logic
NMOS Transistors in Series/Parallel Transistor switch controlled by its gate signal - NMOS switch closes when switch control input is high AND Y = X if A AND B OR Y = X if A OR B
PMOS Transistors in Series/Parallel PMOS switch closes when switch control input is low NOR Y = X if A AND B (A+B) NAND Y = X if A OR B (AB) 55
Complementary CMOS Logic Style Apply DeMorgan s Theorems A+B = A B AB = A + B The complementary gate is inverting AND = NAND + INV 56
Example: NAND Gate PDN: G = AB -> Conduction to GND PUN: F = A + B = AB -> Conduction to V DD G(In 1, In 2, In 3, ) = F (In 1, In 2, In 3, ) 57
Example: NOR Gate 58
Complex CMOS Gate 59
Constructing a Complex Gate F = D +A(B+C) Pull-down network Deriving the pull-up network hierarchically by identifying sub-nets Complete gate 60