IKI10201 04aSimplification of Boolean Functions - PowerPoint PPT Presentation

Title: IKI10201 04aSimplification of Boolean Functions


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IKI10201 04a-Simplification of Boolean Functions

• Bobby Nazief
• Semester-I 2005 - 2006

The materials on these slides are adopted from
those in CS231s Lecture Notes at UIUC, which is
derived from Howard Huangs work and developed by
Jeff Carlyle.
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Excercise

• 3.10 Minimize the number of operators in the
  following Boolean expressions
• xy xy xy
• (x x)y xy y xy (y x)(y y)
  y x
• (x y)(x y)
• x yy x
• xy xy xz
• x(y y) xz x xz (x x)(x z)
  x z
• yz xy xz yz
• yz yz xy xz (y y)z x(y
  z) z xy xz

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Road Map
Logic Gates Flip-flops
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Boolean Algebra
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6
Finite-StateMachines
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Sequential DesignTechniques
Logic DesignTechniques
CombinatorialComponents
StorageComponents
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Binary Systems Data Represent.
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5
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Register-TransferDesign
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Generalized FSM
ProcessorComponents
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Expression simplification

• Before we look at designing circuits we should
  look at simplifying expressions
• Simplified expressions lead to simplified
  circuits
• Normal mathematical expressions can be simplified
  using the laws of boolean algebra
• We can also use additional tools to do
  expression simplification, such as K-map
  (Karnaugh Map)

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Map Representation

• A two-variable function has four possible
  minterms. We can re-arrange these minterms into a
  Karnaugh map (K-map).
• Now we can easily see which minterms contain
  common literals.
• Minterms on the left and right sides contain y
  and y respectively.
• Minterms in the top and bottom rows contain x
  and x respectively.

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A three-variable K-map

• For a three-variable expression with inputs x, y,
  z, the arrangement of minterms is more tricky
• Another way to label the K-map (use whichever you
  like)

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Why the funny ordering?

• With this ordering, any group of 2, 4 or 8
  adjacent squares on the map contains common
  literals that can be factored out.
• Adjacency includes wrapping around the left and
  right sides
• Well use this property of adjacent squares to do
  our simplifications.

xyz xyz xz(y y) xz ? 1 xz
xyz xyz xyz xyz z(xy xy
xy xy) z(y(x x) y(x
x)) z(yy) z
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On K-map

• K-maps define Boolean functions.
• K-maps are equivalent to Truth Tables or Boolean
  Expressions.
• K-maps aid visually identifying Prime Implicants
  (blocks of 2, 4, 8, ... adjacent squares whose
  values are 1) and Essential Prime Implicants
  (Prime Implicants that contain a 1-minterm that
  is not included in any other Prime Implicants) in
  each Boolean function.
• K-maps are good for manual simplification of
  Boolean functions.

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Map method of simplifications

• Imagine a two-variable sum of minterms
• xy xy
• Both of these minterms appear in the top row of a
  Karnaugh map, which means that they both contain
  the literal x.
• What happens if you simplify this expression
  using Boolean algebra?

xy xy x(y y) Distributive x ?
1 y y 1 x x ? 1 x
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More two-variable examples

• Another example expression is xy xy.
• Both minterms appear in the right side, where y
  is uncomplemented.
• Thus, we can reduce xy xy to just y.
• How about xy xy xy?
• We have xy xy in the top row, corresponding
  to x.
• Theres also xy xy in the right side,
  corresponding to y.
• This whole expression can be reduced to x y.

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Example K-map simplification

• Lets consider simplifying f(x,y,z) xy yz
  xz.
• First, you should convert the expression into a
  sum of minterms form, if its not already.
• The easiest way to do this is to make a truth
  table for the function, and then read off the
  minterms.
• You can either write out the literals or use the
  minterm shorthand.
• Here is the truth table and sum of minterms for
  our example

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Making the example K-map

• Next up is drawing and filling in the K-map.
• Put 1s in the map for each minterm, and 0s in the
  other squares.
• You can use either the minterm products or the
  shorthand to show you where the 1s and 0s belong.
• In our example, we can write f(x,y,z) in two
  equivalent ways.
• In either case, the resulting K-map is shown
  below.

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K-maps from truth tables

• You can also fill in the K-map directly from a
  truth table.
• The output in row i of the table goes into square
  mi of the K-map.
• Remember that the rightmost columns of the K-map
  are switched.

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Grouping the minterms together

• The most difficult step is grouping together all
  the 1s in the K-map.
• Make rectangles around groups of one, two, four
  or eight 1s.
• All of the 1s in the map should be included in at
  least one rectangle.
• Do not include any of the 0s.
• Each group corresponds to one product term. For
  the simplest result
• Make as few rectangles as possible, to minimize
  the number of products in the final expression.
• Make each rectangle as large as possible, to
  minimize the number of literals in each term.
• Its all right for rectangles to overlap, if that
  makes them larger.

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Reading the MSP from the K-map

• Finally, you can find the Minimal Sum of Product
  (MSP).
• Each rectangle corresponds to one product term.
• The product is determined by finding the common
  literals in that rectangle.
• For our example, we find that xy yz xz yz
  xy.

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Practice K-map 1

• Simplify the sum of minterms m1 m3 m5 m6.
• The final MSP here is xz yz xyz.

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Four-variable K-maps

• We can do four-variable expressions too!
• The minterms in the third and fourth columns, and
  in the third and fourth rows, are switched
  around.
• Again, this ensures that adjacent squares have
  common literals.
• Grouping minterms is similar to the
  three-variable case, but
• You can have rectangular groups of 1, 2, 4, 8 or
  16 minterms.
• You can wrap around all four sides.

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Example Simplify wyz wz xyz wy

• First, the K-map
• The resulting MSP is wz wz yz or wz wz
  wy.

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Example Simplify wxyzwxywxzwxywxyz

• First, the K-map

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Dont-Care Conditions

• You dont always need all 2n input combinations
  in an n-variable function.
• If you can guarantee that certain input
  combinations never occur.
• If some outputs arent used in the rest of the
  circuit.
• We mark dont-care outputs in truth tables and
  K-maps with Xs.
• Within a K-map, each X can be considered as
  either 0 or 1. You should pick the interpretation
  that allows for the most simplification.

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Dont CarePractice

• Find a MSP for
• f(w,x,y,z) ?(0,2,4,5,8,14,15), d(w,x,y,z)
  ?(7,10,13)
• This notation means that input combinations wxyz
  0111, 1010 and 1101 (corresponding to minterms
  m7, m10 and m13) are unused.

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Dont CareSolution to Practice

• Find a MSP for
• f(w,x,y,z) ?(0,2,4,5,8,14,15), d(w,x,y,z)
  ?(7,10,13)

All prime implicants are circled. We can treat
Xs as 1s if we want, so the red group includes
two Xs, and the light blue group includes one
X. The only essential prime implicant is xz.
(An essential prime implicant is one that covers
a minterm that is covered by no other prime
implicants.) The MSP is xz wxy wxy. It
turns out some of the groups are redundant we
can cover all of the minterms in the map without
them.
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Example Seven Segment Display
Input digit encoded as 4 bits ABCD
Table for e
b
Assumption Input represents a legal digit (0-9)
e
d
CD BD
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Example Seven Segment Display
a
Table for a
f
b
Assumption Input represents a legal digit (0-9)
g
e
c
d
A C BD BD
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K-map Summary

• K-maps are an alternative to algebra for
  simplifying expressions.
• The result is a minimal sum of products (MSP),
  which leads to a minimal two-level circuit.
• Its easy to handle dont-care conditions.
• K-maps are really only good for manual
  simplification of small expressions... but thats
  good enough for IKI10201!
• Things to keep in mind
• Remember the correct order of minterms on the
  K-map.
• When grouping, you can wrap around all sides of
  the K-map, and your groups can overlap.
• Make as few rectangles as possible, but make each
  of them as large as possible. This leads to
  fewer, but simpler, product terms.
• There may be more than one valid solution.