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B01: Introduction to Digital Logic Design

Instructor: Katherine ComptonTech L470 (moving to L458 around 7/1)<kati@ece.nwu.edu>, (847)491-2083Office Hours: by appointment

TA: Probably not.

Book: Randy H. Katz’s Contemporary Logic Design

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Grading

20% - Homeworks 25% - Tutorials & Labs 25% - Midterm Exam 30% - Final Exam Homework is due at the beginning of class on the

specified date. Late penalties:

<24 hours: -15% <48 hours: -30% <72 hours: -60% >72 hours: not accepted

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Joint Work Policy

All work done for the class must be a student’s own, original work.

Everyone must do their own work separately. You cannot work together on the specifics of

homeworks or labs. You also cannot use answers from previous years

or answer keys.

Any students caught using someone else’s work will receive a 0 for that assignment/lab/exam.

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Class & Lab Meetings

Lectures: TTh 6:30-9:30PM Labs : Must sign up for a weekly time slot

Sign-up sheet will go up on door of L470 on Wednesday

Sign up by next class time!

Will have open labs (help sessions), time to be announced later.

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Motivation

Electronics an increasing part of our lives Computers & the Internet Car electronics Robots Electrical Appliances Telephones

Class covers digital logic design & implementation

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Example: Car Electronics

Brake Light (foot brake, parking brake):

High-beam indicator (lights, high beam selected):

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Example: Car Electronics (cont.)

Seat Belt Light (driver belt in):

Seat Belt Light (driver belt in, passenger belt in, passenger present):

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Basic Logic Gates AND: If A and B are True, then Out is True

OR: If A or B is True, or both, then Out is True

Inverter (NOT): If A is False, then Out is True

AB Out

Out

Out

AB

A

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TTL Logic (Transistor-transistor Logic)

Vdd (TRUE)

GND (False)

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Digital vs. Analog

Analog: values vary over a broad range continuously

Digital: only assumes discrete values

+5

V

–5

T ime

+5

V

–5

1 0 1

T ime

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Analog systems: 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

Advantages of Digital Circuits

This course is about logic design, not system design (processor architecture), not circuit design (transistor level)

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Binary/Boolean Logic

• Two discrete values (TTL): yes, on, 5 volts, TRUE, "1" no, off, 0 volts, FALSE, "0"

• Advantage of binary systems: rigorous mathematical foundation based on logic

the three preconditions must be true to imply the conclusion

IF the garage door is openAND the car is runningTHEN the car can be backed out of the garage

IF the garage door is openAND the car is runningTHEN the car can be backed out of the garage

both the door must be open and the carrunning before I canback out

IF passenger is in the carAND passenger belt is inAND driver belt is inTHEN we can turn off the fasten seat belt light

IF passenger is in the carAND passenger belt is inAND driver belt is inTHEN we can turn off the fasten seat belt light

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Combinational vs. Sequential Logic

No feedback among inputs and outputs. Outputs are a function of the inputs only.

Network implemented from logic gates.The presence of feedback distinguishes between sequentialand combinational networks.

---

X1X2

Xn

Logic

Network

Z1Z2

Zm

---

Sequential logic

---

X1X2

Xn

Logic

Network

Z1Z2

Zm

---

Combinational logic

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Black Box (Majority)

Given a design problem, first determine the function

Consider the unknown combination circuit a “black box”

A B C

Out

Truth Table

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“Black Box” Design & Truth Tables

Given an idea of a desired circuit, implement it Example: Odd parity - inputs: A, B, C, output: Out

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Truth Tables

Algebra: variables, values, operations

In Boolean algebra, the values are the symbols 0 and 1 If a logic statement is false, it has value 0 If a logic statement is true, it has value 1

Operations: AND, OR, NOT

0 0 1 1

X Y X AND Y

0 1 0 1

0 0 0 1

X Y X OR Y

0 0 1 1

0 1 0 1

0 1 1 1

X NOT X

0 1

1 0

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Boolean EquationsBoolean Algebra

values: 0, 1variables: A, B, C, . . ., X, Y, Zoperations: NOT, AND, OR, . . .

NOT X is written as XX AND Y is written as X & Y, or sometimes X YX OR Y is written as X + Y

A

0011

B

0101

Sum

0110

Carry

0001

Sum = A B + A B

Carry = A B

OR'd together product terms for each truth table

row where the function is 1

if input variable is 0, it appears in complemented form;

if 1, it appears uncomplemented

Deriving Boolean equations from truth tables:

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Boolean Algebra

A

00001111

B

00110011

Cin

01010101

Sum

01101001

Cout

00010111

Another example:

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

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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:

Cout = A Cin + B Cin + A B

Verify equivalence with the original Carry Out truth table:

place a 1 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

A 0 0 0 0 1 1 1 1

C in 0 1 0 1 0 1 0 1

B 0 0 1 1 0 0 1 1

C out 0 0 0 1 0 1 1 1

B C in

A C in

A B

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Representations of Boolean Functions

Boolean Function: F = X + YZ

Truth Table:

X Y Z F0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

Circuit Diagram:

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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

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Basic Boolean Identities:

X + 0 = X * 1 =

X + 1 = X * 0 =

X + X = X * X =

X + X = X * X =

X =

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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)

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Boolean Manipulations

Boolean Function: F = XYZ + XY + XYZ

Truth Table:

X Y Z F0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

Reduce Function:

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Advanced Laws (Absorbtion)

X+XY = XY + XY = X+XY = X(X+Y) = (X+Y)(X+Y) = X(X+Y) =

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Boolean Manipulations (cont.)

Boolean Function: F = XYZ + XZ

Truth Table:

X Y Z F0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

Reduce Function:

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Boolean Manipulations (cont.)

Boolean Function: F = (X+Y+XY)(XY+XZ+YZ)

Truth Table:

X Y Z F0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

Reduce Function:

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DeMorgan’s Law

(X + Y) = X * Y

(X * Y) = X + Y

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)

DeMorgan's Law can be used to convert AND/OR expressionsto OR/AND expressions

DeMorgan's Law can be used to convert AND/OR expressionsto OR/AND expressions

X 0 0 1 1

Y 0 1 0 1

X 1 1 0 0

Y 1 0 1 0

X + Y X•Y

X 0 0 1 1

Y 0 1 0 1

X 1 1 0 0

Y 1 0 1 0

X + Y X•Y

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DeMorgan’s Law example

If F = (XY+Z)(Y+XZ)(XY+Z),

F =