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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

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Chapter 4

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Chapter 4

Chapter 4

Optimized Implementation of Logic Functions


Chapter objectives

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

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 simplification1

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.


The function f x 1 x 2 x 3 m 0 2 4 5 6

The Function f(x1, x2, x3) = Σm(0, 2, 4, 5, 6)


Location of two variable minterms

x

x

2

1

Location of Two-Variable Minterms

x

x

1

2

0

1

m

0

0

0

m

m

m

0

1

0

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0

2

m

1

0

2

m

m

1

1

3

m

1

1

3

(a) Truth table

(b) Karnaugh map


The function f x 1 x 2 x 2 x 1

1

The Function f(x1, x2) = x2 + x1’


Location of three variable minterms

Location of Three-Variable Minterms


Examples of three variable karnaugh maps

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

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1

1

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f

x

x

x

=

+

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0

1

(b) The function of Figure 4.1


A four variable karnaugh map

A Four-Variable Karnaugh Map


Karnaugh map simplification definitions

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


Karnaugh map simplification definitions1

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


Karnaugh map simplification definitions2

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


Karnaugh map minimization procedure

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.

Karnaugh Map Minimization Procedure


Examples of four variable karnaugh maps

Examples of Four-Variable Karnaugh Maps


More examples of four variable karnaugh maps

More Examples of Four-Variable Karnaugh Maps


A five variable karnaugh map

A Five-Variable Karnaugh Map


A three variable function f x 1 x 2 x 3

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

  • What are the minterms of this function?


A three variable 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.


A four variable function f x 1 x 2 x 3 x 4

A Four-Variable Function f(x1, x2, x3, x4)


Another four variable function f x 1 x 2 x 3 x 4

Another Four-Variable Function f(x1, x2, x3, x4)


Another four variable function f x 1 x 2 x 3 x 41

Another Four-Variable Function f(x1, x2, x3, x4)

x

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1

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3


Product of sums minimization

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

Another Product of Sums Minimization

  • What are the Maxterms?

x

x

1

2

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4

00

01

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10

(

)

x

+

x

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(

)

x

+

x

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(

)

x

+

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+

x

+

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4


Incompletely specified functions

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.


Two implementations of a function

x

x

1

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00

01

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(

)

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+

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01

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(

)

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


Optimal multiple output synthesis

x

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01

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(a) Function

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f

(c) Combined circuit for

and

1

2

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(b) Function

f

2

Optimal Multiple-Output Synthesis


Optimal multiple output synthesis1

Optimal Multiple-Output Synthesis

x

x

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(a) Optimal realization of

(b) Optimal realization of

f

f

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4

x

x

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(c) Optimal realization of

f

and

f

together

3

4


Optimal multiple output synthesis2

Optimal Multiple-Output Synthesis

x

1

x

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x

f

1

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x

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x

f

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f

f

(d) Combined circuit for

and

3

4


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