Prof. Sin-Min Lee

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CS147 Lecture 6
POS, K-map and Multiplexer

• Prof. Sin-Min Lee
• Department of Computer Science

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Karnaugh maps

• Last time we saw applications of Boolean logic to
  circuit design.
• The basic Boolean operations are AND, OR and NOT.
• These operations can be combined to form complex
  expressions, which can also be directly
  translated into a hardware circuit.
• Boolean algebra helps us simplify expressions and
  circuits.
• Today well look at a graphical technique for
  simplifying an expression into a minimal sum of
  products (MSP) form
• There are a minimal number of product terms in
  the expression.
• Each term has a minimal number of literals.
• Circuit-wise, this leads to a minimal two-level
  implementation.

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Review Standard forms of expressions

• We can write expressions in many ways, but some
  ways are more useful than others
• A sum of products (SOP) expression contains
• Only OR (sum) operations at the outermost level
• Each term that is summed must be a product of
  literals
• The advantage is that any sum of products
  expression can be implemented using a two-level
  circuit
• literals and their complements at the 0th level
• AND gates at the first level
• a single OR gate at the second level
• This diagram uses some shorthands
• NOT gates are implicit
• literals are reused
• this is not okay in LogicWorks!

f(x,y,z) y xyz xz
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Unsimplifying expressions
xy yz xz (xy ? 1) (yz ? 1) (xz ?
1) (xy ? (z z)) (yz ? (x x)) (xz ?
(y y)) (xyz xyz) (xyz xyz)
(xyz xyz) xyz xyz xyz xyz

• You can also convert the expression to a sum of
  minterms with Boolean algebra.
• Apply the distributive law in reverse to add in
  missing variables.
• Very few people actually do this, but its
  occasionally useful.
• In both cases, were actually unsimplifying our
  example expression.
• The resulting expression is larger than the
  original one!
• But having all the individual minterms makes it
  easy to combine them together with the K-map.

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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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The Sum-of-Products (SOP) Form
When two or more product terms are summed by
Boolean addition
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Conversion of a General Expression to SOP Form
Any logic expression can be change into SOP form
by applying Boolean Algebra techniques
Example
Try This
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The Standard SOP Form
Multiply
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The Products-of-Sum (POS) Form
When two or more sum terms are multiplied.
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The Standard POS Form
Rule 12!
Add
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Boolean Expression and Truth Table
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Converting SOP to Truth Table

• Examine each of the products to determine where
  
• the product is equal to a 1.
• Set the remaining row outputs to 0.

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Converting POS to Truth Table

• Opposite process from the SOP expressions.
• Each sum term results in a 0.
• Set the remaining row outputs to 1.

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Converting from Truth Table to SOP and POS
SOP
Inputs Inputs Inputs Output
A B C X
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
POS
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The Karnaugh Map
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The Karnaugh Map

• Provides a systematic method for simplifying
  Boolean expressions
• Produces the simplest SOP or POS expression
• Similar to a truth table because it presents all
  of the possible values of input variables

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The 3-Variable K-Map
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The 4-Variable K-Map
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K-Map SOP Minimization

• A 1 is placed on the K-Map for each product term
  in the expression.
• Each 1 is placed in a cell corresponding to the
  value of a product term

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Example
Map the following standard SOP expression on a
K-Map
Solution
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Example
Map the following standard SOP expression on a
K-Map
Solution
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Exercise
Map the following standard SOP expression on a
K-Map
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Answer
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K-Map Simplification of SOP Expressions

• A group must contain either 1, 2, 4, 8 or 16
  cells.
• Each cell in group must be adjacent to one or
  more cells in that same group but all cells in
  the group do not have to be adjacent to each
  other
• Always include the largest possible number 1s in
  a group in accordance with rule 1
• Each 1 on the map must be included in at least
  one group. The 1s already in a group can be
  included in another group as long as the
  overlapping groups include noncommon 1s

Goal
To maximize the size of the groups and to
minimize the number of groups
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Example Group the 1s in each K-Maps
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Determining the minimum SOP Expression from the
Map

• Groups the cells that have 1s. Each group of
  cells containing 1s create one product term
  composed of all variables that occur in only one
  form (either uncomplemented or complemented)
  within the group. Variable that occurs both
  uncomplemented and complemented within the group
  are eliminated. These are called contradictory
  variables.

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Example Determine the product term for the K-Map
below and write the resulting minimum SOP
expression
1
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Example Use a K-Map to minimize the following
standard SOP expression
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Example Use a K-Map to minimize the following
standard SOP expression
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Mapping Directly from a Truth Table
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Dont Care (X) Conditions

• A situation arises in which input variable
  combinations are not allowed
• Dont care terms either a 1 or a 0 may be
  assigned to the output

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Dont Care (X) Conditions
Example of the use of dont care conditions to
simplify an expression
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Exercise Use K-Map to find the minimum SOP from
1
2
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SOP
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POS
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Multiplexers

• A combinational circuit that selects info from
  one of many input lines and directs it to the
  output line.
• The selection of the input line is controlled by
  input variables called selection inputs.
• They are commonly abbreviated as MUX.

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Combinational circuit implementation using MUX

• We can use Multiplexers to express Boolean
  functions also.
• Expressing Boolean functions as MUXs is more
  efficient than as decoders.
• First n-1 variables of the function used as
  selection inputs last variable used as data
  inputs.
• If last variable is called Z, then each data
  input has to be Z, Z, 0, or 1.

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