cs151 introduction to digital design chapter 2: combinational logic circuits 2-9 exclusive-or...

CS151Introduction to Digital Design

Chapter 2: Combinational Logic Circuits

2-9 Exclusive-OR Operator and Gates

1Created by: Ms.Amany AlSaleh

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Exclusive-OR and Exclusive-NOR Circuits

The eXclusive OR (XOR) function is an important Boolean function used extensively in logic circuits.

The XOR function may be;• implemented directly as an electronic circuit (truly a gate)

or• implemented by interconnecting other gate types (used

as a convenient representation) The eXclusive NOR function is the complement of

the XOR function By our definition, XOR and XNOR gates are

complex gates.

Created by: Ms.Amany AlSaleh

3

XOR and XNOR (Cont.) Uses for the XOR and XNORs gate include:

• Adders/subtractors/multipliers• Counters/incrementers/decrementers• Parity generators/checkers

Definitions• The XOR function is: • The eXclusive NOR (XNOR) function, otherwise

known as equivalence is: Strictly speaking, XOR and XNOR gates do no exist

for more that two inputs. Instead, they are replaced by odd and even functions.

YXYXYX

YXYXYX


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Exclusive-OR Circuit Exclusive-OR (XOR) produces a HIGH output whenever the

two inputs are at opposite levels. The XOR function means:

X OR Y, but NOT BOTH


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Exclusive-OR Circuit XOR function can also be implemented with AND/OR gates

(also NANDs).


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Exclusive-NOR Circuits Exclusive-NOR (XNOR) produces a HIGH output whenever the

two inputs are at the same level. Why is the XNOR function also known as the equivalence

function?


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XOR/XNOR (Cont.) The XOR function can be extended to 3 or more

variables. For more than 2 variables, it is called an odd function or modulo 2 sum (Mod 2 sum), not an XOR:

The complement of the odd function is the even function.

The XOR identities: X1XX0X 1XX0XX

XYYX ZYX)ZY(XZ)YX(

ZYXZYXZYXZYXZYX


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XOR/XNOR (Cont.)• 3-input exclusive-OR (XOR) logic gate:

• F= X Y Z

Fxyz

xF

zy

X Y Z F

0 0 0 00 0 1 10 1 0 10 1 1 01 0 0 11 0 1 01 1 0 01 1 1 1


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Odd and Even Functions What about the case of more than 2 variables? A B C = (AB’ + A’B) C’ + (AB + A’B’) C

= AB’C’ + A’BC’ + ABC + A’B’C

This function is equal to 1 only if one variable is equal to 1 or if all three variables are equal to 1. This implies that an odd number of variables must be one. This is defined as an odd function.

• F is the logical sum of the four minterms having an index with an odd number of 1’s F is called an odd function

• The other four minterms not included are 000, 011, 101 and 110, they have an index with even number of 1’s F’ is called an even function.

The complement of an odd function is an even function. In general, an n-variable exclusive-OR function is an odd

function defined as the logical sum of the 2n/2 minterms whose binary index have an odd number of 1’s.

100 010 001111


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Odd and Even Functions (Cont.) Map for 3-variable XOR function.

Minterms are said to be distance 2 from each other.


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Odd and Even Functions (Cont.) Map for 3- and 4- variable XOR functions.


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Example: Odd Function Implementation

Design a 3-input odd function F = X Y Zwith 2-input XOR gates

Factoring, F = (X Y) Z The circuit:

XY

ZF


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Example: Even Function Implementation

Design a 4-input even function F = (W X Y Z)’with 2-input XOR and XNOR gates

Factoring, F = ( (W X) (Y Z) )’ The circuit:

WX

YF

Z


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Odd and Even Functions Why?

• Implementation feasibility and low cost• Power in implementing Boolean functions• Convenient conceptual representation

Gate classifications• Primitive gate - a gate that can be described using a single

primitive operation type (AND or OR) plus an optional inversion(s).

• Complex gate - a gate that requires more than one primitive operation type for its description

Primitive gates will be covered first


cs151 introduction to digital design chapter 2: combinational logic circuits 2-9 exclusive-or...

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