BEE2243 Digital systems

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BEE2243 Digital systems
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Lecture 2

• Variable Entered Karnaugh Map
• Hazard and Glitch
• Espresso Software

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VARIABLE-ENTERED KARNAUGH MAP
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Minimization of Boolean expressions

• In most cases a canonical Boolean function can be
  minimized
• reduction of the number of gates required to
  implement the corresponding circuit
• To minimize the Boolean function the two step are
• reduce the number of terms
• reduce the number of literal

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Example of Minimization
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Karnaugh Maps

• Boolean expressions can be graphically
  represented and simplified using Karnaugh map
• In a Karnaugh map the 2n minterms are represented
  on separate cells
• Boolean expressions may be represented on a
  Karnaugh map if they are expressed in canonical
  form
• The main feature of a Karnaugh map is that each
  square is logically adjacent to the square that
  is physically adjacent to it

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• Karnaugh map approach is not suitable for
  minimizing functions with more than six variables

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• 1 are entered in the cells which correspond to
  the function minterm. In the other cells a 0 (or
  nothing) is entered

example
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• Boolean function on the Karnaugh map can be
  simplified by using the property of adjacency
• In a four variable map, the top and the bottom
  rows and the left and right columns are logically
  adjacent

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Dont Care Conditions

• Sometime it is not possible to specify the output
  for some input combinations, called dont care
  conditions (they do not have relevant effect on
  the output)
• Functions that include dont care conditions are
  said
• incompletely specified
• In a Karnaugh map, dont care conditions are
• represented with -
• Dont care conditions can be useful to minimize
  the function. Indeed, we can assumed them 1 or 0
  depending on the convenience

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Example reduction with dont care conditions
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The Complementary Approach

• Sometime it is more convenient to group on a
  Karnaugh map the 0 rather than the 1.
• The resultant sum-of-products grouping 0 is the
  complement of the desired expression. Then if we
  complement by using DeMorgans theorem we found a
  product-of-sum expression

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Minimization of Multiple-Output Functions

• In minimizing multiple-output functions, the
  emphasis is on deriving product terms that can be
  shared among the function (instead of considering
  individual function)
• The resulting circuit has fewer gates which means
  lower area
• The determination of shared products term among
  many Boolean functions is an extremely
  complicated task (it can be efficiently performed
  by using a dedicated software)

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VEM (Variable-Entered Maps)

• The variable-Entered map, VEM, is useful to plot
  and n-variable function on and n-1 variable
  Karnaugh map
• One variable is used inside the plot, map-entered
  variable, and becomes one of the possible value
  of the cells
• The use of the map is similar to a simple
  Karnaugh map

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Map compressions of a three-variable function.

• A generic three-variable truth table.
• Conventional three-variable Karnaugh map.
• Compressed Karnaugh map of order 2 with x and y
  as the
• map variables and z as the map-entered variable.

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(d) Compressed Karnaugh map of order 2 with y and
x as the map-entered variable. (e)
Compressed Compressed Karnaugh map of order 2
with x and z as the map variables and y as
the map-entered variable. (f) Karnaugh map of
order 1 with x as the map variable and y and z
as the map-entered variables.
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Example of a three-variable function. (a) Truth
table. (b) Variable-entered map.
Literal Complement 0 1
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An example of a variable-entered map with
infrequently appearing variables.
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• Variable-entered maps grouping techniques.
• Grouping cells with the same literal.
• Grouping a 1-cell with both the z literal and
  the literal.
• Grouping a 1-cell with the z literal.

yz
xz
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Ex1. Obtaining a minimal sum from a map having
single-variable map entries. (a)Variable-entered
map. (b) Step 1. (c) Step 2.
wxz
yz
xy
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Ex2. Illustrating optimal groupings on a
variable-entered map.
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Ex3. Obtaining a minimal sum from a map having
single-variable map entries. (a)
Variable-entered map. (b) Step 1. (c) Step 2.
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Obtaining a minimal product from a map having
single-variable map entries.
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Question??

• Exercise1.