1. EEL 3705 / 3705L Digital Logic Design
Fall 2006 Instructor: Dr. Michael Frank Module #5: Combinational Logic Optimization (Thanks to Dr. Perry for the slides)
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M. Frank, EEL3705 Digital Logic, Fall 2006

2. M. Frank, EEL3705 Digital Logic, Fall 2006
Wednesday, October 4, 2006
Administrivia:
Midterm #1 grades posted, handing papers back today
Averages: 96%, 89%, 96% (98.5 points overall)
This week’s lab:
“Top secret code display” (w. K-maps)
Design project #1: Due this Friday!!
Test your designs at end of lab period, or in TA office hours
Don’t delay writing the large required amount of documentation!
Homework assignment #3: To be posted very soon, watch for it.
Plan for today:
Go over midterm solutions briefly
Start in on next lecture topic, 3.5:
Techniques for Combinational Logic Optimization
Emphasis on use of Karnaugh maps
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M. Frank, EEL3705 Digital Logic, Fall 2006

3. Topic 3.5: Techniques for Combinational Logic Optimization
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4. Topic 3.5 – Minimizing Circuits
Subtopics:
Simplification using Boolean identities
Not covering in depth this semester in lecture
Karnaugh Maps (CIO #5)
Focus of this lecture
Don’t care conditions
The Quine-McCluskey Method
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M. Frank, EEL3705 Digital Logic, Fall 2006

5. Goals of Circuit Minimization
(1) Minimize the number of primitive Boolean logic gates needed to implement the circuit.
Ultimately, this also roughly minimizes the number of transistors, the chip area, and the cost.
Also roughly minimizes the energy expenditure
among traditional irreversible circuits.
This will be our focus.
(2) It is also often useful to minimize the number of combinational stages or logical depth of the circuit.
This roughly minimizes the delay or latency through the circuit, the time between input and output.
Note: Goals (1) and (2) are often conflicting!
In the real world, a designer may have to analyze and optimize some complex trade-off between logic complexity and latency.
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M. Frank, EEL3705 Digital Logic, Fall 2006

6. Minimizing DNF Expressions
Using DNF (or CNF) expressions guarantees that you can always find some circuit that implements any desired Boolean function.
However, the resulting circuit may be far larger than is really required!
We would like to find the smallest sum-of-products expression that is equivalent to a given function.
This will yield a fairly small circuit.
However, circuits of other forms (not either CNF or DNF) might be even smaller for complex functions.
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7. Simplification of Switching Functions
Chapter 3
Simplification of Switching Functions
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8. M. Frank, EEL3705 Digital Logic, Fall 2006
Karnaugh Maps (K-Map)
A K-Map is a graphical representation of a logic function’s truth table
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9. Relationship to Venn Diagrams
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10. Relationship to Venn Diagrams
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11. Relationship to Venn Diagramsb
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12. Relationship to Venn Diagramsb
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13. Relationship to Venn Diagrams
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14. M. Frank, EEL3705 Digital Logic, Fall 2006
Two-Variable K-Map
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15. Three-Variable K-Map
Note: The bit sequences must always be ordered using a Gray code!
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16. Three-Variable K-Map
Note: The bit sequences must always be ordered using a Gray code!
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17. Three-Variable K-Map
Note: The bit sequences must always be ordered using a Gray code!
Edges are adjacent
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18. Four-variable K-Map
Note: The bit sequences must be ordered using a Gray code!
Note: The bit sequences must be ordered using a Gray code!
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19. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map
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20. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map
Edges are adjacent
Edges are adjacent
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21. Plotting Functions on the K-map
SOP Form
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22. Canonical SOP Form using shorthand notation Three Variable Example
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23. Three-Variable K-Map Example
Plot 1’s (minterms) of switching function 1 1 1 1
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24. Three-Variable K-Map Example
Plot 1’s (minterms) of switching function 1 1 1 1
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25. Four-variable K-Map Example1 1 1 1 1
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26. Simplification of Switching Functions using K-MAPS
Karnaugh Maps (K-Map)
Simplification of Switching Functions
using K-MAPS
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27. Terminology/Definition
Literal
A variable or its complement
Logically adjacent terms
Two minterms are logically adjacent if they differ in only one variable position
Ex:
and
m6 and m2 are logically adjacent
Note:
Or, logically adjacent terms can be combined
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M. Frank, EEL3705 Digital Logic, Fall 2006

28. Terminology/Definition
Implicant
Product term that could be used to cover minterms of a function
Prime Implicant
An implicant that is not part of another implicant
Essential Prime Implicant
A prime implicant that covers at least one minterm that is not contained in another prime implicant
Cover
A minterm that has been used in at least one group
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29. Guidelines for Simplifying Functions
Each square on a K-map of n variables has n logically adjacent squares. (i.e. differing in exactly one variable)
When combing squares, always group in powers of 2m , where m=0,1,2,….
In general, grouping 2m variables eliminates m variables.
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M. Frank, EEL3705 Digital Logic, Fall 2006

30. Guidelines for Simplifying Functions
Group as many squares as possible. This eliminates the most variables.
Make as few groups as possible. Each group represents a separate product term.
You must cover each minterm at least once. However, it may be covered more than once.
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M. Frank, EEL3705 Digital Logic, Fall 2006

31. K-map Simplification Procedure
Plot the K-map
Circle all prime implicants on the K-map
Identify and select all essential prime implicants for the cover.
Select a minimum subset of the remaining prime implicants to complete the cover.
Read the K-map
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32. M. Frank, EEL3705 Digital Logic, Fall 2006
Example
Use a K-Map to simplify the following Boolean expression
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33. Three-Variable K-Map Example
Step 1: Plot the K-map 1 1 1 1 1
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34. Three-Variable K-Map Example
Step 2: Circle ALL Prime Implicants 1 1 1 1 1
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35. Three-Variable K-Map Example
Step 3: Identify Essential Prime Implicants PI
EPI PI 1 1
EPI 1 1 1
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36. Three-Variable K-Map Example
Step 4: Select minimum subset of remaining
Prime Implicants to complete the cover. PI
EPI 1 1
EPI 1 1 1
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37. Three-Variable K-Map Example
Step 5: Read the map. 1 1 1 1 1
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38. M. Frank, EEL3705 Digital Logic, Fall 2006
Solution
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39. M. Frank, EEL3705 Digital Logic, Fall 2006
Example
Use a K-Map to simplify the following Boolean expression
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40. Three-Variable K-Map Example
Step 1: Plot the K-map 1 1 1 1
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41. Three-Variable K-Map Example
Step 2: Circle Prime Implicants
Wrong!!
We really
should draw
A circle around
all four 1’s 1 1 1 1
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M. Frank, EEL3705 Digital Logic, Fall 2006

42. Three-Variable K-Map Example
Step 3: Identify Essential Prime Implicants
EPI
EPI
Wrong!!
We really
should draw
A circle around
all four 1’s 1 1 1 1
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M. Frank, EEL3705 Digital Logic, Fall 2006

43. Three-Variable K-Map Example
Step 4: Select Remaining Prime Implicants to
complete the cover.
EPI
EPI 1 1 1 1
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44. Three-Variable K-Map Example
Step 5: Read the map. 1 1 1 1
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45. M. Frank, EEL3705 Digital Logic, Fall 2006
Solution
Since we can still simplify the function
this means we did not use the largest
possible groupings.
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M. Frank, EEL3705 Digital Logic, Fall 2006

46. Three-Variable K-Map Example
Step 2: Circle Prime Implicants
Right! 1 1 1 1
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47. Three-Variable K-Map Example
Step 3: Identify Essential Prime Implicants
EPI 1 1 1 1
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48. Three-Variable K-Map Example
Step 5: Read the map. 1 1 1 1
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49. M. Frank, EEL3705 Digital Logic, Fall 2006
Solution
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50. M. Frank, EEL3705 Digital Logic, Fall 2006
Special Cases
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51. Three-Variable K-Map Example1 1 1 1 1 1 1 1
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52. Three-Variable K-Map Example
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53. Three-Variable K-Map Example1 1 1 1
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54. Four Variable Examples
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55. M. Frank, EEL3705 Digital Logic, Fall 2006
Example
Use a K-Map to simplify the following Boolean expression
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56. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map 1 1 1 1 1 1 1 1
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57. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map 1 1 1 1 1 1 1 1
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58. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map 1 1 1 1 1 1 1 1
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59. M. Frank, EEL3705 Digital Logic, Fall 2006
Example
Use a K-Map to simplify the following Boolean expression
D=Don’t care (i.e. either 1 or 0)
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60. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map 1 d 1 1 1 d 1 d 1 1
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61. M. Frank, EEL3705 Digital Logic, Fall 2006
Four-variable K-Map 1 d 1 1 1 d 1 d 1 1
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62. M. Frank, EEL3705 Digital Logic, Fall 2006
Five Variable K-Maps
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63. M. Frank, EEL3705 Digital Logic, Fall 2006
Five variable K-map
Use two four variable K-maps
A=1
A=0
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64. Use Two Four-variable K-Maps
A=0 map
A=1 map
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65. M. Frank, EEL3705 Digital Logic, Fall 2006
Five variable example
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66. Use Two Four-variable K-Maps
A=0 map
A=1 map 1 1 1 1 1 1 1 1
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67. Use Two Four-variable K-Maps
A=0 map
A=1 map 1 1 1 1 1 1 1 1
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68. M. Frank, EEL3705 Digital Logic, Fall 2006
Five variable example
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69. Plotting POS Functions
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70. K-map Simplification Procedure
Plot the K-map for the function F
Circle all prime implicants on the K-map
Identify and select all essential prime implicants for the cover.
Select a minimum subset of the remaining prime implicants to complete the cover.
Read the K-map
Use DeMorgan’s theorem to convert F to F in POS form
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71. M. Frank, EEL3705 Digital Logic, Fall 2006
Example
Use a K-Map to simplify the following Boolean expression
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72. Three-Variable K-Map Example
Step 1: Plot the K-map of F 1 1 1 1 1
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73. Three-Variable K-Map Example
Step 2: Circle ALL Prime Implicants 1 1 1 1 1
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74. Three-Variable K-Map Example
Step 3: Identify Essential Prime Implicants PI
EPI PI 1 1
EPI 1 1 1
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75. Three-Variable K-Map Example
Step 4: Select minimum subset of remaining
Prime Implicants to complete the cover. PI
EPI 1 1
EPI 1 1 1
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76. Three-Variable K-Map Example
Step 5: Read the map. 1 1 1 1 1
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77. M. Frank, EEL3705 Digital Logic, Fall 2006
Solution
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78. M. Frank, EEL3705 Digital Logic, Fall 2006
TPS Quiz
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