Electrical and Computer Engineering

EE 202

Chapter 3 –

Gate-Level Minimization

Dr. Rhonda Kay Gaede

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ECE Department EE 202

3.2 The Map Method

_________ or __-maps: a diagram of squares containing _________

of a function to be minimized

Provides a __________ method for __________ Boolean expressions

Procedure:

1. _____ and _____ the 2, 3, or 4 variable K-map.

2. ______ values must be _________ such that only ___ ____

changes value between ________ blocks.

3. Group adjacent 1’s in ______ __ __ sets

4. Form the ___-__-______ using the variables of each grouped set

whose values __ ___ _____.

5. All squares containing 1’s must be used __ ____ once and can be

used ____ than once

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3.2 Two-Variable Map

x’y’

y

0

1

0 1

m00=m0 m01=m1

m10=m2 m11=m3

x’y

xy’ xy

x

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3.2 Two-Variable Function

( , ) (2,3)F x y

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3.2 Three-Variable Map

x yz

0

1

00 01 11 10

m0 m1 m3 m2

m4 m5 m7 m6

( , , ) (2,3,5)F x y z

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3.2 Three-Variable Functions

( , , ) (2,3,4,5)F x y z

' ' ' ' 'F xz yz x y z

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3.3 Four-Variable Map

AB

CD

m0

00 01 11 10

00

01

11

10

m1 m3 m2

m4 m5 m7 m6

m12

m8

m13 m15 m14

m9 m11 m10

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3.3 Four-Variable Function

( , , , ) (1,2,5,6,8,9,12,13,14)F A B C D

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3.3 Prime Implicants

' ' ' ' ' ' ' ' 'F A B C B CD A BCD AB C

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3.4 Product-Of-Sums (POS) Simplification

We know if we take the __________ of an expression F, all ___

change to ____ and ___ to ____.

If we group___ in a K-map and create a function (terms ORed

together) we will have a minimized ___

To find the minimized function F take the complement of F’, the

form will be in __________- ___ - ______ (_____)

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3.4 POS Simplification – Practice(1)

( , , ) (2,3,4)F x y z

sum-of-products (SOP) product-of-sums (POS)

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3.4 POS Simplification – Practice(2)

( , , ) ' 'F x y z x y y z

sum-of-products (SOP)

product-of-sums (POS)

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3.5 Don’t Care Conditions – Specification

____________ minterms whose output

values are unspecified are known as ____

_____. They are marked by d(me) or X

(book) in table, can be used ____, used __-

__ or ______ in circuit simplification

Example: Find a simplified Boolean expression

w x y z F

0 0 0 0 d

0 0 0 1 1

0 0 1 0 d

0 0 1 1 0

0 1 0 0 0

0 1 0 1 d

0 1 1 0 0

0 1 1 1 1

1 0 0 0 0

1 0 0 1 0

1 0 1 0 d

1 0 1 1 1

1 1 0 0 0

1 1 0 1 0

1 1 1 0 0

1 1 1 1 1

( , , , )F w x y z

( , , , )d w x y z

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3.5 Don’t Care Conditions – Example

( , , , )F w x y z

( , , , )d w x y z

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3.6 NAND and NOR Implementation

• In most cases digital circuits are constructed using

______ and ______ gates rather than ____ and ___

gates

• ______ and ____ gates are simpler to manufacture

• Any of the three primitive logic operations, ____, ___,

and _____ can be constructed using _____ or ____

gates

• We will develop a system to go from an ____, ___,

_____ circuit representation to the _____ and ____

representations

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3.6 NAND Representations

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3.6 NOR Representations

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3.6 NAND and NOR Equivalences

NOT gates at inputs can be drawn as circles or simply NOT gate:

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3.6 From AND-OR to NAND-NAND

1. Express function in _____

2. Replace ___ gates with ____________

3. Replace _____ gates with __________

4. _______________ literals connected directly to the _____________

5. Replace ___________ with ________

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3.6 From OR AND to NOR-NOR

1. Express function in _____

2. Replace ____ gates with ___________

3. Replace ___ gates with _______

4. ____________ literals connected directly to the ____________

5. Replace __________ with ______

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3.6 Don’t Care Conditions – Example (1)

Find simplified expressions for SOP, then implement as AND-OR

and NAND-NAND

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3.6 Don’t Care Conditions – Example(2)

Find simplified expressions for SOP, POS then implement as AND-OR,

OR-AND, NAND-NAND, and NOR-NOR

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3.8 Exclusive-OR Function

XOR is an odd function:

Even number of ones at inputs _____

Odd number of ones at inputs _____

Fx

y

x

y

zF

x y F

0 0

0 1

1 0

1 1

x y z F

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

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3.8 – Even 3-Bit Parity Generator

x

y

zP

x y z P

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

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3.8 Even 3-Bit Parity Checker

x

y

z

P

Err

x y z P Err

0 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0

1 1 0 1

1 1 1 0

1 1 1 1