Because, most of the time we can save significant amount of logic.

⇐ ПредыдущаяСтр 3 из 3

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Method 1:

- use classic boolean algebra simplifications from Lecture 1.

- This method always works, but often takes lots of time.

Converting

Method 1:

Example

Method 1:

Method 2:

- use some advanced simplification rules.

- or apply duality which also works.

Method 2:

Continue from previous slide

Method 2:

Example

Back to DeMorgan’s Algebra

Example 1: Example 2:

Example 3: Example 4:

Example 5:

Method 3:

- use DeMorgan’s to get inverse F, then simplify using K-map.

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Method 1:

- use classic boolean algebra simplifications such as D2, from Lecture 1.

- This method always works, but often takes lots of time.

Method 1:

Method 2:

- use duality, simplify using familiar D1

Method 2:

Example

Method 2: Example

Method 2:

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Method 3:

- use K-maps

Karnaugh Maps (K-Maps) So what is Karnaugh Map?

A visual way to simplify logic expressions It gives the most simplified form of the expression

Goals of Circuit Minimization

(1) Minimize the number of primitive Boolean logic gates needed

to implement the circuit.

Ultimately, this also roughly minimizes the number of transistors, the chip

area, and the cost.

Also roughly minimizes the energy expenditure

among traditional irreversible circuits.

This will be our focus.

(2) It is also often useful to minimize the number of

combinational stages or logical depth of the circuit.

This roughly minimizes the delay or latency through the circuit, the time

between input and output.

Note: Goals (1) and (2) are often conflicting!

In the real world, a designer may have to analyze and optimize some

complex trade-off between logic complexity and latency.

Rules to obtain the most simplified expression

🞂 Simplification of logic expression using Boolean algebra is awkward because:

🞂 it lacks specific rules to predict the most suitable next step in the

simplification process

🞂 it is difficult to determine whether the simplest form has been achieved.

🞂 A Karnaugh map is a graphical method used to obtained the most simplified form of an expression in a standard form (Sum-of- Products or Product-of-Sums).

🞂 The simplest form of an expression is the one that has the minimum number of terms with the least number of literals (variables) in each term.

🞂 By simplifying an expression to the one that uses the minimum number of terms, we ensure that the function will be implemented with the minimum number of gates.

🞂 By simplifying an expression to the one that uses the least number of literals for each terms, we ensure that the function will be implemented with gates that have the minimum number of inputs.

Terminology/Definition

🞂 Implicant

🞂 Product term that could be used to cover minterms of a function

🞂 Prime Implicant

🞂 An implicant that is not part of another implicant

🞂 Essential Prime Implicant

🞂 A prime implicant that covers at least one minterm that is not contained in another prime implicant

🞂 Cover

🞂 A minterm that has been used in at least one group

Guidelines for Simplifying Functions

🞂 Each square on a K-map of n variables has n logically adjacent squares. (i.e. differing in exactly one variable)

🞂 When combing squares, always group in powers of 2m ,