Chapter 1

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Analysis vs. Design

• Analysis: From circuit to description

• Design: From description to circuit

QUIZ: Analysis vs. Design

The Dread Pirate Roberts tells his engineer:

Build me a ship-carried catapult that can throw a 100 lb rock over a distance of 100 yards!

Analysis or design?

QUIZ: Analysis vs. Design

The King tells his wise counselor:

This guy showed up the other day with a new game, played on a board with 64 black and white squares. He said he would teach it to me if I gave him one grain of rice for the first square, two for the second, four for the third, and so on, doubling the number for each square …

QUIZ: Analysis vs. Design

… Sounds like a pretty good deal to me (I am very bored). What do you advise, wise counselor?

Analysis or design?

The main conceptual tools used in

computer analysis and design

• ABSTRACTION • BLACK BOX • LEVELS OF DESCRIPTION

1.2 Boolean Algebra and Venn Diagrams

One set:

1.2 Boolean Algebra and Venn Diagrams

Two sets:

1.2 Boolean Algebra and Venn Diagrams

Two sets:

QUIZ: Boolean Algebra and Venn Diagrams

Shade the areas corresponding to

the following Boolean expressions:

What to do for next time?

• Review the concepts and quizzes covered during today’s lecture; read your notes and the text.

• Solve the quizzes again, without peeking at the solution! (Quizzes provide about 75% of the exam questions.)

• Individual work: Solve end-of-chapter problems 1 through 15 in notebook.

This is not homework; do not submit to instructor for grading!

Notebook will be graded at exam time!

QUIZ: Boolean Algebra and Venn Diagrams

1. Draw a Venn diagram for 3 sets: A, B, C

2. Shade the areas corresponding to

the following Boolean expression: AB’ + BC’

Basic logic symbols, a.k.a. gates

Truth Tables

Truth tables are showing all possible input combinations

and the corresponding output in condensed form

Truth Tables

Draw the truth table for: • AND with 3 inputs • OR with 4 inputs

QUIZ: How many lines are there in a truth

table with

4 inputs?

5 inputs?

k inputs?

Postulates (axioms) of Boolean algebra

Note that each postulate has a dual!

Postulates (axioms) of Boolean algebra

Rules for finding the dual:

• Switch all 0 and 1 values

• Switch all + and ∙ operators

• Parenthesize to ensure order of operations stays the same!

Precedence of Boolean operators

• NOT (Highest)

• AND

• OR (Lowest)

Note: A + B∙C means (A + B∙C )’

Practice taking the dual

Transform:

• P3a into P3b

• P3b into P3a

Dual pitfall

Based on the complement, we may think that finding the dual

also involves doing something about the NOT operators, but …

NOT operators stay the same!

More advanced Boolean identities (theorems)

Every Boolean identity translates into equivalent circuits

Image source: http://www.wisc-online.com/objects/ViewObject.aspx?ID=DIG2703

Example: associativity

Use the appropriate postulate or theorem to find a

circuit equivalent to the one given

????

Find Boolean expression first

Use the appropriate postulate or theorem to find a

circuit equivalent to the one given

Simplification!

How do we prove properties and theorems?

Another proof example

QUIZ: Use perfect induction to prove DeMorgan’s theorem

QUIZ: Use manipulation to prove DeMorgan’s theorem

How many minterms in Venn diagrams?

• One set

• Two sets

• Three sets

• Four sets ?

• Five sets?

• …

• N sets?

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QUIZ: Use perfect induction to prove the dual DeMorgan’s

theorem

Conclusion: We have just covered four representations of digital circuits:

– Venn diagrams

– Gate diagrams

– Boolean functions

– Truth tables

Given any of them, we should be able to derive the other 3 …

For circuit design, the most useful is:

1.3 Deriving boolean functions from truth tables

Example:

For circuit design, the most useful is:

1.3 Deriving boolean functions from truth tables

Example:

Hint: Use the ones in the

output column!

This solution is called

The Canonical Sum of Products (SOP)

or

The Sum of Minterms

Note: The text uses the acronym CSOP for Canonical SOP. B/c of the possible confusion between Canonical and Compact, we prefer to write Canonical SOP; In the end-of-chapter problems, CSOP means Canonical SOP.

QUIZ: Write the canonical SOP for the function in this truth

table

QUIZ

Write the canonical SOP, and then the gate diagram for the “majority voter”:

Individual work:

• Read pp. 4-11 of text (and take notes!)

• Solve end-of-chapter problems (in notebook):

• 16, 17, 18

• 35

QUIZ: End-of-chapter 1.26

Describe how to obtain the dual of a Boolean

expression.

Find the dual of X + YZW + A

Swap zeroes and ones Swap AND and OR (∙ and +), while preserving the order of operations (parentheses!!)

Short notation: the compact sum of

minterms …

But, since there are more 0s than ones, isn’t it a better idea

to design based on the 0s? … But how? … Use DeMorgan!

Write the dual of this statement:

Consider all the lines where the function is 1, write the

minterm for each, add up the minterms.

Q: What is the dual of a minterm? A: Use duality again → maxterm!

Conclusion: The entire design process has a

dual! • Zeroes in the truth table

• Maxterms

• Product of sums

• Bar the variables that are equal to 1

Someone asked: Why don’t we complement the

variables? Isn’t this what we do in maxterms?

A: Yes, we do complement variables in maxterms, but

not b/c duality says anything about complementing (it

doesn’t!).

We do it b/c duality says to swap 0 and 1; so we take

the definition of the minterm: “variable has no bar if value is 1 and bar if value is 0”

and we write its dual for the maxterm: “variable has no bar if value is 0 and bar if value is 1”

Short notation: the compact sum of

minterms …

EoL

… and its dual: the compact product of

maxterms

EoL

QUIZ

1.37: Find truth tables for these compact sums of minterms:

• F(X,Y) = Sm(3)

• F(Y) = Sm(0)

• F(A,B) = Sm(1,2,3)

QUIZ

1.42: Find truth tables for these compact products of maxterms:

• F(X,Y) = PM(0,1,2)

• F(Y) = PM(0)

• F(A,B) = Sm(1,2)

1.4 VHDL for simple gate functions

Another representation for circuits:

Hardware Description Languages (HDLs)

1.4 VHDL for simple gate functions

WAVEFORMS ARE EQUIVALENT TO TRUTH TABLES!

And yet another representation:

Waveforms (a.k.a. timing diagrams)

EoL 4

QUIZ: Timing diagrams

Write the truth table for the function represented in this

diagram:

QUIZ: Timing diagrams

Write the truth table for the function represented in this

diagram:

Can you explain in words what this function does?

QUIZ: Timing diagrams

Write the truth table for the function represented in this

diagram:

QUIZ … AND gate

EOL

QUIZ: For the function represented in this truth table, derive:

• canonical SOP

• compact sum of minterms

• canonical POS

• compact product of maxterms

Derive this from the truth table!

1.5 More about logic gates

Active-low inputs and outputs

Logically, these symbols for an inverter are

equivalent

‘B’ is for Bubble

Gate-level DeMorgan manipulations

Functionally complete gates

NAND is functionally complete!

???

???

Functionally complete gates

NAND is functionally complete!

NOR is also functionally complete!

???

NOR is also functionally complete!

QUIZ: Functional completeness

Implement the circuit below with only NAND or

only NOR gates

When we have AND-OR circuits, it’s easy

to bring them to NAND-NAND form

When we have OR-AND circuits, it’s easy

to bring them to NOR-NORform

QUIZ: How? Show the steps.

QUIZ: From OR-AND to NOR-NOR

QUIZ: For the function represented in this truth table, derive:

• gate diagram AND-OR

• gate diagram NAND-NAND

• gate diagram OR-AND

• gate diagram NOR-NOR

SKIP IEEE symbols

(we’re not going to use them)

Homework for Ch.1, due Fri, Jan.30

27,29, 35, 37, 41, 42, 51 through 55

The homework assignments and slides will also available on the course webpage.

EoL