9/15/09 - L7 Two Level Circuit Optimization Copyright 2009 - Joanne DeGroat, ECE, OSU1 Two Level...

9/15/09 - L7 Two Level Circuit Optimization

Copyright 2009 - Joanne DeGroat, ECE, OSU 1

Two Level Circuit OptimiztionAn easier way to generate a minimal sum of products expression.



Class 7 outline Cost Functions Map Structures Maps

Two Variable Three Variable Four Variable

Material from section 2-4 of text

Intro and Cost Function The complexity of the gates to physically

implement a Boolean function is typically a 1-to-1 relationship with the Boolean expression of the function. If the expression has 3 terms ANDed together, the

implementation has a 3-input AND gate. If the expression has 4 AND terms Ored together

for the final output the implementation has a 4-input OR gate



Function representations Truth Tables

Any Boolean function can be represented by a Truth Table Truth Tables have 1 line for each of the 2n possible input

combinations Are a full specification of the function

Sum-of-Products There is just one maximal sum-of-products representation This is a representation where each term contain each

literal of the function, i.e., specifies each line of the Truth Table where the value of the function is 1.



Illustration of this Consider two function



Cost Functions A cost function is a metric to place a value

on the ‘cost’ to implement a logic function. The lowest cost will be the

minimal sum-of-products or minimal product-of-sums representation.

These representation of the function with have the fewest number of terms with the fewest number of literals in each term.



Different cost factors Literal cost – the sum of the number of literals

required for expression of the function. Gate inputs – the number of inputs to gates in

the implementation This is a cost function that is sometimes used

today. CAD tools that automatically generate the

implementation have proprietary cost functions



Manual minimization Usually done with Karnaugh maps K-maps are an extension of Venn Diagrams Consider a function of 2 inputs F(A, B) Have 4 regions A B AB and A’B’ Note adjacencies

A adjacent to AB and A’B’ AB adjacent to A and B



K-map for 2 variables Karnaugh map for two variables

Note adjacencies



For three variables 3 variables – 8 Truth Table entries –

8 variable K-map



4 variable K-map The 4 variable Karnaugh map



Minimizing functions using maps Form the largest power of 2 group of adjacent

1 that you can. 1s on K-maps can be used multiple times.

Example: 3 variable map Simplify F(A,B,C)=∑m(0,1,2,3,4,5) Note minterm positions Step 1 – enter 1s onto map



Minimizing Step 2 find larges power of 2 groupings

Make sure all 1s are included (covered) F(X,Y,Z)=X’ + Y’



2 variable map examples Easy to simplify if you can.



4 variable examples Consider the map What power of 2 groups exist?



First two Is there a third?

Wraparound



Adjaceny What cells are adjacent?



Another Adjacency and wraparound

Top 00 cells are adjacent to bottom 10 cells

minimal is F=A’C + B’C’



In general Map all 1s of the function to the K-map Choose the largest power of 2 groups until all the 1 of the

function are covered For a 4 variable function Group 8 1s – will have 1 literal Group 4 1s – will have 2 literals Group 2 1s – will have 3 literals No group – a lone 1 – term will have all 4 literals

Sometimes will have a choice on how to cover that last 1. Make the choice that results in a term with the fewest literals. Sometimes either of 2 answers are equal.



Simplifying functions Simplify XY+X’Z+YZ Expand to generate minterms

= XY(Z+Z’) + X’Z(Y+Y’) + (X+X’)YZ = XYZ+XYZ’ + X’YZ + X’Y’Z + XYZ+X’YZ = m7 + m6 + m3 + m1 + repeat + repeat

.



Class 7 summary assignment Covered section 2-4 Problems for hand in

2-16 2-17 Problems for practice

2-15

Reading for next class: section 2-5



9/15/09 - L7 Two Level Circuit Optimization Copyright 2009 - Joanne DeGroat, ECE, OSU1 Two Level...

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