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Slide 1

Logic Design

A Review

Binary numbers

Binary numbers to decimalBinary 2 decimal

Decimal2 binary

Boolean Algebra

Two Values: zero and one

Three Basic Functions: And, Or, Not Any Boolean Function Can beConstructed from These Three

And

0

1

Or

0

1

Not

0

0

0

0

0

1

0

1

1

0

1

1

1

1

1

0

Algebraic Laws

Classification

Law

Identity

a1=1a=a

a+0=0+a=a

Dominance

a0=0a=0

1+a=a+1=1

Commutativity

a+b=b+a

ab=ba

Associativity

a(bc)=(ab)c

a+(b+c)=(a+b)+c

Distributive

a(b+c)=ab+ac

a+bc=(a+b)(a+c)

Demorgans Laws

(a + b) = ab

(ab) = a + b

Boolean Expressions

Multiplication represents AND

Not is represented by a prime a or anoverbar a

Examples:

s = abc + abc + abc + abc q = ab + bc + ac + abc

Superfluous Terms

The following Two Equations RepresentThe Same Function.

q = ab + bc + ac + abcq = ab + bc + ac

abcq

0000

0010

0100

0111

1000

1011

1101

1111

Prime Implicants

A Prime Implicant is a Product of

Variables or Their Complements, eg.abcd

If a Prime Implicant has the Value 1,then the Function has the Value 1

A Minimal Equation is a Sum of PrimeImplicants

Minimization and Minterms

Minimization Reduces the Size andNumber of Prime Implicants

A MinTerm is a Prime Implicant withthe Maximum Number of Variables

For a 3-input Function abc is aMinTerm, while ab is not.

Prime Implicants can be Combined toEliminate Variables, abc+abc = ab

Minimization with Maps

A Karnaugh Map

C

B

00

01

11

10

0

1

0

0

1

A

^

110

00

Procedure

Select Regions Containing All 1s

Regions should be as Large as Possible Regions must contain 2k cells Regions should overlap as little aspossible

The complete set of regions mustcontain all 1s in the map

Procedure 2

Top and Bottom of Map are Contiguous Left and Right of Map are Contiguous Regions represent Prime Implicants Use Variable name guides to constructequation

Completely inside the region of a variablemeans prime implicant contains variable

Completely outside the region of a variablemeans prime implicant contains negation

Applied to Previous Map

C

B

00

01

11

10

0

1

0

0

1

A

^

110

00

q=cb+ca

A 4-Variable Karnaugh Map

D

C

00

01

11

10

00

0

0

0

0

B

^

010

110

111

101

A

100

111

First Minimization

D

C

00

01

11

10

00

0

0

0

0

B

^

010

110

111

101

A

100

111

Second Minimization

D

C

00

01

11

10

00

0

0

0

0

B

^

010

110

111

101

A

100

111

Minimal Forms for PreviousSlides:

abd + bcd + abc + acdacd + abd + bcd + abc

Moral: A Boolean Function May HaveSeveral Different Minimal Forms

Karnaugh Maps are Ineffective for

Functions with More than Six Inputs.

Basic Logic Symbols

And

Or

Not

The Exclusive Or Function

Xor

0

1

0

0

1

1

1

0

A Simple Logic Diagram

Signal Flow

Nand Nor Buffer Xnor

DecoderA binary decoder has a n-bit binary code input code and a one activates output out of 2n outputs is called a binary decoder.A binary decoder is used when it is necessary to activate exactly one of 2n outputs based on an n-bit input value. Notice Enable signals:

Truth table of a Decoder

2x4 decoder3x8 decoder

Encoder

A digital circuit that performs the inverse operation of a decoder is called an encoder.An encoder has 2n input lines and n output linesIn encoder the output lines generate binary code corresponding to the input value.

Truth table of 8x3 Encoder

MultiplexerThemultiplexeris a circuit whose output is one of several inputs, depending on the value given by some otherselectioninputs. In a circuit, a multiplexer is drawn as a trapezoid as follows. Normally, there are 2n input lines and n selection lines whose bit combinations determine which input is selected.

Circuit of a Multiplexer Truth table

Circuit

Symbol

Tristate Elements

Three States:

Zero (Output is grounded)

One (Output connected to Power Terminal) High-Impedance (Output Not Connected toEither Power Or Ground)

Can be Used to Construct CheapMultiplexors

CMOS Tri-state Buffers

Non-Inverting

Inverting

Tri-State Buffer Issues

The Gate Amplifies its Signal

May be Inverting or Non-Inverting

Often used to Construct MultiplexorsUsing Wired-Or Connections

More Tri-State Issues

In a Wired-Or Connection, Only OneBuffer can be in Non-Tristate State

Violating This Rule Can Destroy TheCircuit Due a Power/Ground Short

1

1

1

0

DANGER!