01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2006

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1 Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 26

2 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom Fountain 2 course notes (Stanford EE82) Michael Wahl 2 lecture notes (U. of Siegen CS 3339) Ben Dugan 2 lecture notes (UW-CSE 378) Professor Scott Hauck lecture notes (UW EE 47) Mark L. Chang lecture notes for Digital Logic (NWU B) 2

3 Example: Car Electronics Door ajar light (driver door, passenger door): High-beam indicator (lights, high beam selected): 3

4 Seat Belt Light (driver belt in): Example: Car Electronics (cont.) Seat Belt Light (driver belt in, passenger belt in, passenger present): 4

5 Basic Logic Gates AND: If A and B are True, then Out is True A B Out OR: If A or B is True, or both, then Out is True A B Out Inverter (NOT): If A is False, then Out is True A Out 5

6 Digital vs. Analog V T ime V T ime 5 5 Digital: only assumes discrete values Analog: values vary over a broad range continuously 6

7 Analog systems: Advantages of Digital Circuits slight error in input yields large error in output Digital systems: more accurate and reliable readily available as self-contained, easy to cascade building blocks Computers use digital circuits internally Interface circuits (i.e., sensors & actuators) often analog 7

8 Binary/Boolean Logic Two discrete values: yes, on, 5 volts, TRUE, "" no, off, volts, FALSE, "" Advantage of binary systems: rigorous mathematical foundation based on logic IF IF the the garage garage door door is is open open AND AND the the car car is is running THEN THEN the the car car can can be be backed backed out out of of the the garage garage both the door must be open and the car running before I can back out IF IF passenger is is in in the the car car AND AND passenger belt belt is is in in AND AND driver driver belt belt is is in in THEN THEN we we can can turn turn off off the the fasten fasten seat seat belt belt light light the three preconditions must be true to imply the conclusion 8

9 Combinational vs. Sequential Logic Sequential logic X X X n Logic Network Z Z 2 Z m Network implemented from logic gates. The presence of feedback distinguishes between sequential and combinational networks. Combinational logic X X X 2 n Logic Network Z Z 2 Z m No feedback among inputs and outputs. Outputs are a function of the inputs only. 9

10 Black Box (Majority) Given a design problem, first determine the function Consider the unknown combination circuit a black box Truth Table Out A B C

11 Black Box Design & Truth Tables Given an idea of a desired circuit, implement it Example: Odd parity - inputs: A, B, C, output: Out

12 Truth Tables Algebra: variables, values, operations In Boolean algebra, the values are the symbols and If a logic statement is false, it has value If a logic statement is true, it has value Operations: AND, OR, NOT X Y X AND Y X NOT X X Y X OR Y 2

13 Boolean Algebra values:, variables: A, B, C,..., X, Y, Z operations: NOT, AND, OR,... Boolean Equations NOT X is written as X X AND Y is written as X & Y, or sometimes X Y X OR Y is written as X + Y Deriving Boolean equations from truth tables: A B Sum Carry Sum = A B + A B OR'd together product terms for each truth table row where the function is if input variable is, it appears in complemented form; if, it appears uncomplemented Carry = A B 3

14 4 Boolean Algebra Another example: A B Cin Sum Cout Sum = A B Cin + A B Cin + A B Cin + A B Cin Cout = A B Cin + A B Cin + A B Cin + A B Cin

15 Boolean Algebra Reducing the complexity of Boolean equations Laws of Boolean algebra can be applied to full adder's carry out function to derive the following simplified expression: B C in A C in A B A B C in C out Cout = A Cin + B Cin + A B Verify equivalence with the original Carry Out truth table: place a in each truth table row where the product term is true each product term in the above equation covers exactly two rows in the truth table; several rows are "covered" by more than one term 5

16 Representations of Boolean Functions Boolean Function: F = X + YZ Truth Table: X Y Z F Circuit Diagram: 6

17 Why Boolean Algebra/Logic Minimization? Logic Minimization: reduce complexity of the gate level implementation reduce number of literals (gate inputs) reduce number of gates reduce number of levels of gates fewer inputs implies faster gates in some technologies fan-ins (number of gate inputs) are limited in some technologies fewer levels of gates implies reduced signal propagation delays number of gates (or gate packages) influences manufacturing costs 7

18 Basic Boolean Identities: X + = X * = X + = X * = X + X = X * X = X + X = X * X = X = 8

19 Basic Laws Commutative Law: X + Y = Y + X XY = YX Associative Law: X+(Y+Z) = (X+Y)+Z X(YZ)=(XY)Z Distributive Law: X(Y+Z) = XY + XZ X+YZ = (X+Y)(X+Z) 9

20 Boolean Manipulations Boolean Function: F = XYZ + XY + XYZ Truth Table: X Y Z F Reduce Function: 2

21 Advanced Laws X+XY = XY + XY = X+XY = X(X+Y) = (X+Y)(X+Y) = X(X+Y) = 2

22 Boolean Function: F = XYZ + XZ Boolean Manipulations (cont.) Truth Table: X Y Z F Reduce Function: 22

23 Boolean Manipulations (cont.) Boolean Function: F = (X+Y+XY)(XY+XZ+YZ) Truth Table: X Y Z F Reduce Function: 23

24 DeMorgan s Law (X + Y) = X * Y X Y X Y X + Y X Y (X * Y) = X + Y X Y X Y X Y X + Y DeMorgan's Law Law can can be be used used to to convert AND/OR expressions to to OR/AND expressions Example: Z = A B C + A B C + A B C + A B C Z = (A + B + C) * (A + B + C) * (A + B + C) * (A + B + C) 24

25 DeMorgan s Law example If F = (XY+Z)(Y+XZ)(XY+Z), F = 25

26 26 NAND and NOR Gates NAND Gate: NOT(AND(A, B)) NOR Gate: NOT(OR(A, B)) X Y X NAND Y X Y X NOR Y

27 NAND and NOR gates are universal NAND and NOR Gates can implement all the basic gates (AND, OR, NOT) NAND NOR NOT AND OR 27

28 Bubble Manipulation Bubble Matching DeMorgan s Law 28

29 XOR and XNOR Gates XOR Gate: Z= if X is different from Y X Y Z XNOR Gate: Z= if X is the same as Y X Y Z X Y Z X Y Z 29

30 Boolean Equations to Circuit Diagrams F = XYZ + XY + XYZ F = XY + X(WZ + WZ) 3

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008

01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2008 Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 28 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom