طراحی مدارهای منطقی

93-92نیمسال دوم

دانشگاه آزاد اسالمی واحد پرند

دانشگاه آزاد اسالمی واحد پرندطراحی مدارهای منطقی

بهینه سازی مدارهای منطقی: جدول کارنو

Optimization

Algebraic procedures problems The procedures are difficult to apply in a

systematic way It is difficult to tell when you have arrived at a

minimum solution

Karnaugh map (For 3 and 4 variables) Quine-McCluskey

Optimization

Cost directly related to

The number of gates

&

Gate inputs used

Optimization

Karnaugh Minimum cost two-level circuits composed of AND and OR gates

SOP: a group of AND gates feeding a single OR gate

POS: a group of OR gates feeding a single AND gate

Minimum SOP/POS • has minimum number of terms (Minimum number of gates)• has minimum number of literals (minimum number of gate inputs)

Karnaugh Map

Karnaugh map of a function

Specifies the value of the function for every

combination of values of the independent variables

• Like Truth table

Terms near each other with one bit difference

• Maximum terms in one group (2,4,8,… terms)

• Minimum groups

Karnaugh Map (2-variable)

Karnaugh Map (3-variable)

Karnaugh Map (3-variable)

Karnaugh Map (3-variable)

Karnaugh Map (3-variable)

Proving boolean algebra equations Consensus theorem

Karnaugh Map (3-variable)

Minimized SOP Not unique

Karnaugh Map (4-variable)

Karnaugh Map (4-variable)

Karnaugh Map (4-variable)

With Don’t cares

Group X’s if they simplify the function

Karnaugh Map (4-variable)

POS extraction

Karnaugh Map (EPI)

Using PIs and EPIs

Implicant

Prime Implicant (PI)

Essential Prime Implicant (EPI)

Karnaugh Map (EPI)

In the process of finding PIs:

don’t-cares are treated just like 1’s

Note PI composed entirely of X’s can never be part of

the minimum solution

Karnaugh Map (EPI)

Karnaugh Map (EPI)

Karnaugh Map (5-variable)