2 - QM Method, Decoder, MUX, Programmable Logic, State Machine, FF

2 - QM Method, Decoder, MUX, Programmable Logic, State Machine, FF

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23 Quine-McClusky Method (Q-M Method) Example 27: Use the Q-M method to find a minimal sum of products of the following function. ) 15 , 11 , 10 , 9 , 8 , 7 , 6 , 4 , 1 ( ) , , , ( m Z Y X W f = Index Minterm W X Y Z Minterm W X Y Z Minterm W X Y Z 1 1 0 0 0 1 (1, 9) - 0 0 1 PI 6 (8, 9, 10, 11) 1 0 - - PI 1 4 0 1 0 0 (4, 6) 0 1 - 0 PI 5 8 1 0 0 0 (8, 9) 1 0 0 - 2 6 0 1 1 0 (8, 10) 1 0 - 0 9 1 0 0 1 (6, 7) 0 1 1 - PI 4 10 1 0 1 0 (9, 11) 1 0 - 1 3 7 0 1 1 1 (10, 11) 1 0 1 - 11 1 0 1 1 (7, 15) - 1 1 1 PI 3 4 15 1 1 1 1 (11, 15) 1 - 1 1 PI 2 PI Chart 1 4 6 7 8 9 10 11 15 1 EPI × × 2 PI × × 3 PI × × 4 PI × × 5 EPI × 6 EPI × Reduced Chart 7 15 2 PI × 3 PI × × 4 PI × XYZ Z Y X Z X W X W PI EPI EPI EPI f + + + = + + + = 3 6 5 1
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24 Row and Column dominance Definition: A row i is said to dominate another row j if i covers every column covered by j and does not contain any more literals. ( Row Dominance ) A column u is said to dominate another column v if u has an X in every row in which v has an X. ( Column Dominance ) The resulting chart is further reduced using the following rules: Rule1: Remove the dominated rows. Rule2: Remove the dominating columns. 2 1 PI × 2 PI × D B A D C A PI PI D C B A f + = - + - = + = 0 00 000 0 ) , , , ( 1 * 2 or D C B D C A PI PI D C B A f + = - + - = + = 010 00 0 ) , , , ( 3 * 2
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25 Example 28: ) 6 , 1 ( ) 5 , 4 , 3 , 2 ( d m f + = Index Minterm A B C Minterm A B C 1 1 0 0 1 (1, 3) 0 - 1 1 PI 2 0 1 0 (1, 5) - 0 1 2 PI 4 1 0 0 (2, 3) 0 1 - 3 PI 2 3 0 1 1 (2, 6) - 1 0 4 PI 5 1 0 1 (4, 5) 1 0 - 5 PI 6 1 1 0 (4, 6) 1 - 0 6 PI PI Chart 2 3 4 5 1 PI × 2 PI × 3 PI × × 4 PI × 5 PI × × 6 PI × Reduced Table 2 3 4 5 * 3 PI * 5 PI B A B A PI PI f + = + = * 5 * 3 1 PI is dominated by 3 PI 2 PI is dominated by 5 PI 4 PI is dominated by 3 PI 6 PI is dominated by 5 PI
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26 Example 29: Simplify the following system using Q-M Method. ) 15 , 4 , 3 ( ) 14 , 13 , 11 , 9 , 8 , 7 , 6 , 2 ( ) , , , ( 4 3 2 1 d m X X X X f + = I n d e x M i n t e r m X 1 X 2 X 3 X 4 M i n t e r m X 1 X 2 X 3 X 4 M i n t e r m X 1 X 2 X 3 X 4 1 2 0 0 1 0 (2,3) 0 0 1 - (2,3,6,7) 0 - 1 - PI 3 4 0 1 0 0 (4,6) 0 1 - 0 PI 1 (3,7,11,15) - - 1 1 PI 4 8 1 0 0 0 (8,9) 1 0 0 - PI 2 (3,11,7,15) - - 1 1 2 3 0 0 1 1 (3,7) 0 - 1 1 (6,7,14,15) - 1 1 - PI 5 6 0 1 1 0 (3,11) - 0 1 1 (6,14,7,15) - 1 1 - 9 1 0 0 1 (6,7) 0 1 1 - (9,11,13,15) 1 - - 1 PI 6 3 7 0 1 1 1 (6,14) - 1 1 0 (9,13,11,15) 1 - - 1 11 1 0 1 1 (9,11) 1 0 - 1 13 1 1 0 1 (9,13) 1 - 0 1 14 1 1 1 0 (7,15) - 1 1 1 4 15 1 1 1 1 (11,15) 1 - 1 1 (13,15) 1 1 - 1 (14,15) 1 1 1 - PI Chart The EPI’s covers all the minterms, thus 6 5 3 2 4 3 2 1 ) , , , ( EPI EPI EPI EPI X X X X f + + + = 4 1 3 2 3 1 3 2 1 4 3 2 1 ) , , , ( X X X X X X X X X X X X X f + + + =
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27 Multiple-Output Circuits Computer #1 X 1 X 2 X 3 Y 1 Y
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2 - QM Method, Decoder, MUX, Programmable Logic, State Machine, FF

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