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Gate-level MinimizationPowerPoint Presentation

Gate-level Minimization

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- Although truth tables representation of a function is unique, it can be expressed algebraically in different forms
- The procedure of simplifying Boolean expressions (in 2-4) is
difficult since it lacks specific rules to predict the successive steps in the simplification process.

- Alternative: Karnaugh Map (K-map) Method.
- Straight forward procedure for minimizing Boolean Function
- Fact: Any function can be expressed as sum of minterms
- K-map method can be seen as a pictorial form of the truth table.

y

y

x

m0

m1

m2

m3

x

Two-variable map

y

y

y

y

x

x

x

x

The three squares can be determined from the intersection

of variable x in the second row and variable y in the second

column.

- Any two adjacent squares differ by only one variable.
- M5 is row 1 column 01. 101= xy’z=m5
- Since adjacent squares differ by one variable (1 primed, 1 unprimed)
- From the postulates of Boolean algebra, the sum of two minterms in adjacent squares can be simplified to a simple AND
- For example m5+m7=xy’z+xyz=xz(y’+y)=xz

Example 1

Example 4

Given:

(a) Express F in sum of minterms.

(b) Find the minimal sum of products using K-Map

(a)

Example 4 (continued)

Three-variable K-Map: Observations

- One square represents one minterm a term of 3 literals
- Two adjacent squares a term of 2 literals
- Four adjacent squares a term of 1 literal
- Eight adjacent squares the function equals to 1

- Need to ensure that all Minterms of function are covered
- But avoid any redundant terms whose minterms are already covered
- Prime Implicant is product Term obtained by combining maximum possible number of adjacent squares
- If a minterm in a square is covered by only prime implicant then ESSENTIAL PRIME IMPLICANT

Non Essential prime implicant CD, B’C, AD and AB’

Essential prime implicant BD and B’D’

Four-variable K-Map: Observations

- One square represents one minterm a term of 4 literals
- Two adjacent squares a term of 3 literals
- Four adjacent squares a term of 2 literal
- Eight adjacent squares a term of 1 literal
- sixteen adjacent squares the function equals to 1

SUM of PRODUCT and PRODUCT OF SUM

Simplify the following Boolean function in:

(a) sum of products (b) product of sums

Combining the one’s:

(a)

Combining the zero’s:

Taking the the complement:

(b)

Implementation of Boolean Functions

- Draw the logic diagram for the following function: F = (a.b)+(b.c)

a

b

F

c

- Implement a circuit
- 2 Level
- More than two level
- SOP
- POS

- Implement a circuit using OR and Inverter Gates only
- Implement a circuit using AND and Inverter Gates only
- Implement a circuit using NAND Gates only
- Implement a circuit using NOR Gates only

COVERT AND TO NAND WITH AND INVER.

CONVERT OR TO NAND WITH INVERT OR. SINGLE BUBBLE WITH INVERTER

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