9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU1 Two Level Circuit Optimiztion An easier way to generate a minimal sum of products expression.

9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU2 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 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU3

Function representations  Truth Tables Any Boolean function can be represented by a Truth Table  Truth Tables have 1 line for each of the 2 n 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. 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU4

Illustration of this  Consider two function 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU5

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. 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU6

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 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU7

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 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU8

K-map for 2 variables  Karnaugh map for two variables Note adjacencies 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU9

For three variables  3 variables – 8 Truth Table entries – 8 variable K-map 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU10

4 variable K-map  The 4 variable Karnaugh map 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU11

9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU12

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 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU13

Minimizing  Step 2 find larges power of 2 groupings  Make sure all 1s are included (covered)  F(X,Y,Z)=X’ + Y’ 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU14

2 variable map examples  Easy to simplify if you can. 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU15

4 variable examples  Consider the map  What power  of 2 groups  exist? 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU16

First two  Is there a third?  Wraparound 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU17

Adjaceny  What cells are adjacent? 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU18

Another  Adjacency and wraparound  Top 00 cells are  adjacent to  bottom 10 cells minimal is F=A’C + B’C’ 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU19

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. 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU20

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 = m 7 + m 6 + m 3 + m 1 + repeat + repeat. 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU21

Class 7 summary assignment  Covered section 2-4  Problems for hand in  Problems for practice 2-15  Reading for next class: section 2-5 9/15/09 - L7 Two Level Circuit Optimization Copyright Joanne DeGroat, ECE, OSU22