# Chapter 4 - PowerPoint PPT Presentation

1 / 29

Chapter 4. Optimized Implementation of Logic Functions. Chapter Objectives. Synthesis of logic functions Analysis of logic circuits Techniques for deriving minimum-cost implementations of logic functions Graphical representation of logic functions in the form of Karnaugh maps

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Chapter 4

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## Chapter 4

Optimized Implementation of Logic Functions

### Chapter Objectives

• Synthesis of logic functions

• Analysis of logic circuits

• Techniques for deriving minimum-cost implementations of logic functions

• Graphical representation of logic functions in the form of Karnaugh maps

• Cubical representation of logic functions

• Use of CAD tools and VHDL to implement logic functions

### Karnaugh Map Simplification

• Truth table representation of a function is unique.

• Algebraic representation of a function is not unique.

• Sometimes it is difficult to simplify an algebraic function.

• A Karnaugh map is a pictorial arrangement of the truth table which allows an easy interpretation for choosing the minimum number of terms needed to express the function algebraically.

### Karnaugh Map Simplification

• Each combination of variables in a truth table is called a minterm.

• A truth table of n variables has 2n minterms.

• A Boolean function is equal to 1 for some minterms and 0 for others.

• Another representation of the function is to list the decimal equivalent of those minterms that produce a 1 for the function.

x

x

2

1

x

x

1

2

0

1

m

0

0

0

m

m

m

0

1

0

1

0

2

m

1

0

2

m

m

1

1

3

m

1

1

3

(a) Truth table

(b) Karnaugh map

1

### Examples of Three-Variable Karnaugh Maps

x

x

1

2

x

3

00

01

11

10

0

0

0

1

1

f

x

x

x

x

=

+

3

2

1

3

1

1

0

0

1

(a) The function of Figure 2.18

x

x

1

2

x

3

00

01

11

10

0

1

1

1

1

f

x

x

x

=

+

3

2

1

1

0

0

0

1

(b) The function of Figure 4.1

### A Four-Variable Karnaugh Map

A Literal is a Boolean variable appearing in either complemented or uncomplemented form.

Minterm is a product term of all the literals in a function.

Example: AB’CD, ABCD, ABC’D’

Maxterm is a sum term of all the literals in a function.

Example: A’+B+C’+D’, A’+B’+C’+D’, A’+B’+C+D

### Karnaugh Map Simplification Definitions

Implicant is any product term of the variables.

Example: AB’C, ABCD, and B’C are all implicants of a function of (A, B, C, D).

Prime-Implicant (PI): The minimal implicant (maximal group size).

Example: If an implicant ABC can be further simplified by adding more 1-cells (maximizing the group size) to get the implicant AB, then AB is the PI. ABC in this case is redundant because ABC is covered by AB.

### Karnaugh Map Simplification Definitions

Essential ‘1’ Cells are those ‘1’ cells which are covered by only one Prime Implicant.

Essential Prime Implicants are the Prime Implicants corresponding to essential 1-cells.

### Karnaugh Map Simplification Definitions

Inspect the map and identify all Prime Implicants.

Identify all essential 1-cells.

Pick all esential Prime Implicants (PIs covering all essential 1-cells).

Pick a minimum number of additional Pis to cover the remaining 1-cells (entire function) and form the Boolean function.

Repeat for at least 25 K-maps to become a master.

### A Three-Variable Function f(x1, x2, x3)

• What are the minterms of this function?

### A Three-Variable Function

• Cover: A valid collection of implicants covering the function.

• Cover 1 would be using all minterms.

• Cover 2 would be using all 1-cells only in pairs.

• Cover 3 (Best) would be using maximized groups together with all remaining essential groups.

• Cost: Number of gates plus the total number of inputs to all gates.

x

x

1

2

x

x

3

4

00

01

11

10

x

x

x

00

1

1

1

3

4

x

x

x

01

1

1

3

2

4

x

x

x

11

1

1

1

3

4

10

1

1

x

x

x

2

4

3

x

x

x

x

x

x

1

2

4

1

2

4

x

x

x

x

x

x

1

3

2

2

1

3

### Product of Sums Minimization

• f(x1, x2, x3) = ΠM(4, 5, 6)

• Applying DeMorgan’s Theorem

• f‘ = x1x3’ + x1x2’

• f = (f’)’ = (x1’ + x3)(x1’ + x2)

### Another Product of Sums Minimization

• What are the Maxterms?

x

x

1

2

x

x

3

4

00

01

11

10

(

)

x

+

x

00

0

0

0

0

3

4

01

0

1

1

0

(

)

x

+

x

2

3

11

1

1

0

1

10

1

1

1

1

(

)

x

+

x

+

x

+

x

1

2

3

4

### Incompletely Specified Functions

• These functions use Don’t Cares.

• For example: In two interlocked switches, ’11’ may not occur at all, both switches cannot be ‘on’ at the same time.

• Minterms that may produce either 0 or 1 for the function are don’t care conidtions.

• Mark don’t cares with an x or d in the K-map.

• When combining squares to simplify the expression, don’t cares can be taken to be either 1 or 0 – whichever gives the simplest expression.

x

x

1

2

x

x

3

4

00

01

11

10

00

0

1

d

0

x

x

x

x

3

2

1

2

01

0

1

d

0

x

x

3

4

00

01

11

10

(

)

x

+

x

11

0

0

d

0

2

3

00

0

1

d

0

x

x

10

1

1

d

1

4

3

01

0

1

d

0

(

)

x

+

x

11

0

0

d

0

(a) SOP implementation

3

4

10

1

1

d

1

(b) POS implementation

### Two Implementations of a Function

x

x

1

2

x

x

3

4

00

01

11

10

00

1

1

x

2

01

1

1

1

x

3

11

1

1

x

4

f

1

10

1

1

x

1

x

3

(a) Function

f

1

x

1

x

x

1

2

x

x

x

3

3

4

00

01

11

10

f

2

x

2

00

1

1

x

3

01

1

1

x

4

11

1

1

1

f

f

(c) Combined circuit for

and

1

2

10

1

1

(b) Function

f

2

### Optimal Multiple-Output Synthesis

x

x

x

x

1

2

1

2

x

x

x

x

3

4

3

4

00

01

11

10

00

01

11

10

00

00

01

1

1

1

01

1

1

1

11

1

1

1

11

1

1

1

10

1

10

1

(a) Optimal realization of

(b) Optimal realization of

f

f

3

4

x

x

x

x

1

2

1

2

x

x

x

x

3

4

3

4

00

01

11

10

00

01

11

10

00

00

01

1

1

1

01

1

1

1

11

1

1

1

11

1

1

1

10

1

10

1

(c) Optimal realization of

f

and

f

together

3

4

### Optimal Multiple-Output Synthesis

x

1

x

4

x

f

1

3

x

2

x

4

x

1

x

2

x

3

x

f

4

4

x

2

x

4

f

f

(d) Combined circuit for

and

3

4