ECE 616 Advanced FPGA Designs - Electrical and Computer Engineering University of Western Ontario

ECE 616 Advanced FPGA DesignsElectrical and Computer Engineering University of Western Ontario... 01/31/24 15Boolean Algebra • Basic mathematics used for logic design • Laws and theorems

ECE 616 Advanced FPGA Designs

Electrical and Computer Engineering

University of Western Ontario

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General

1 Welcome remark

2 Digital and analog

3 VLSI: ASIC and FPGA

4 Overview

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Information

1 Text book in library:

 M J S Smith, Application-Specific Integrated

Circuits, Addison-Wesley, 1997 ISBN: 0201500221.

 Digital Systems Design Using VHDL, Charles H Roth, Jr.,

PWS Publishing, 1998 (ISBN: 0-534-95099-X).

2 Class notes and lab manual:

www.engga.uwo.ca/people/wwang

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Digital and Analog

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t (

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Basic Logic Gates

Cin '

XY '

YCin '

X Cin

' Y ' X

XYCin '

XYCin Cin

' XY YCin

' X

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Full Adder (cont’d)

Module Truth table

Y ' X )(

' Cin Y

' X )(

' Cin '

Y X

)(

Cin Y

X (

) Cin Y

' X )(

Cin '

Y X

)(

' Cin Y

X )(

Cin Y

X (

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Boolean Algebra

• Basic mathematics used for logic design

• Laws and theorems can be used to

simplify logic functions

– Why do we want to simplify logic functions?

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Laws and Theorems of Boolean Algebra

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Laws and Theorems of Boolean Algebra

) XYCin '

XYCin (

) XYCin Cin

' XY (

) XYCin YCin

' X (

XYCin '

XYCin Cin

' XY YCin

' X Cout

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Theorems to Apply to Exclusive-OR

X 0

' X 1

0 X

1 '

X

X Y

Y

) Z Y

( X

Z )

Y X

XZ XY

) Z Y

(

' Y ' X XY

Y '

X '

Y X

)' Y X

– each square corresponds to one

of the 16 possible minterms

– 1 - minterm is present;

0 (or blank) – minterm is absent;

– X – don’t care

• the input can never occur, or

• the input occurs but the output is not specified

– adjacent cells differ in only one value =>

can be combined

Location

of minterms

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Karnaugh Maps (cont’d)

• Example

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Sum-of-products Representation

• Function consists of a sum of prime implicants

• Prime implicant

– a group of one, two, four, eight 1s on a map

represents a prime implicant if it cannot be combined

with another group of 1s to eliminate a variable

• Prime implicant is essential if it contains a 1

that is not contained in any other prime implicant

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Selection of Prime Implicants

Two minimum

forms

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Procedure for min Sum of products

• 1 Choose a minterm (a 1) that has not been

covered yet

• 2 Find all 1s and Xs adjacent to that minterm

• 3 If a single term covers the minterm and all

adjacent 1s and Xs, then that term is an essential prime implicant, so select that term

• 4 Repeat steps 1, 2, 3 until all essential prime

implicants have been chosen

• 5 Find a minimum set of prime implicants that

cover the remaining 1s on the map If there is more than one such set, choose a set with a minimum

number of literals

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Products of Sums

• F(1) = {0, 2, 3, 5, 6, 7, 8, 10, 11}

F(X) = {14, 15}

– Altera’s MAX+plus II and the UP1 Educational board:

A User’s Guide, B E Wells, S M Loo

– Altera University Program Design Laboratory Package