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HW5_solutions_last

# HW5_solutions_last - Homework 5 Solutions 2012 ECE 15A...

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Homework 5 Solutions ECE 15A 2012 1. (10p) For this function, find a minimum sum-of-products solution, using the Quine-McCluskey method: F ( a, b, c, d, e ) = m (0 , 2 , 3 , 7 , 10 , 11 , 16 , 17 , 18 , 20 , 23 , 31) + d (4 , 15 , 21) Column I Column II Column III group 0 0 00000 (0,2) 000-0 (0,2,16,18) -00-0 (0,16) -0000 (0,16,4,20) -0-00 (0,4) 00-00 group 1 2 00001 (2,3,10,11) 0-01- 4 00100 (2,3) 0001- (16,17,20,21) 10-0- 16 01000 (2,10) 0-010 (2,18) -0010 (16,17) 1000- (16,18) 10-00 (16,20) 01-00 (4,20) -0100 group 2 3 00011 (3,7) 00-11 (3,11,7,15) 0- -11 10 01010 (3,11) 0-011 17 10001 (10,11) 0101- 18 10010 (17,21) 10-01 20 10100 (20,21) 1010- group 3 7 00111 (7,23) -0111 (7,23,15,31) - -111 11 01011 (21,23) 101-1 21 10101 (7,15) 0-111 (11,15) 01-11 group 4 15 01111 (15,31) -1111 23 10111 (23,31) 1-111 group 5 31 11111 1

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0 2 3 7 10 11 16 17 18 20 23 31 (0,2,16,18)* b c e X X X X (0,16,4,20) b d e X X X (2,3,10,11)* a c d X X X X (16,17,20,21)* ab d X X X (3,11,7,15) a de X X X (7,23,15,31)* cde X X X (21,23) ab ce X * indicates essential prime implicant f ( a, b, c, d, e ) = a c d + ab d + b c e + cde 2
2. (10p) Find all prime implicants of the following function using the Quine-McCluskey method and then find all minimum sum-of-products solutions using Petrick’s method. F ( a, b, c, d, e ) = m (0 , 2 , 4 , 5 , 10 , 11 , 13 , 15 , 16 , 18 , 20 , 21 , 26 , 27 , 29 , 31) Column I Column II Column III group 0 0 00000 (0,2) 000-0 (0,2,16,18) -00-0 (0,4) 00-00 (0,4,16,20) -0-00 (0,16) 000-0 group 1 2 00010 (2,10) 0-010 (2,10,18,26) - -010 4 00100 (2,18) -0010 (4,5,20,21) -010- 16 10000 (4,5) 0010- (4,20) -0100 (16,18) 100-0 (16,20) 10-00 group 2 5 00101 (5,13) 0-101 (5,13,21,29) - -101 10 01010 (5,21) -0101 (10,11,26,27) -101- 18 10010 (10,11) 0101- 20 10100 (10,26) -1010 (18,26) 1-010 (20,21) 1010- group 3 11 01011 (11,15) 01-11 (11,15,27,31) -1-11 13 01101 (11,27) -1011 (13,15,29,31) -11-1 21 10101 (13,15) 011-1 26 11010 (13,29) -1101 (21,29) 1-101 (26,27) 1101- group 4 15 01111 (15,31) -1111 27 11011 (27,31) 11-11 29 11101 (29,31) 111-1 group 5 31 11111 0 2 4 5 10 11 13 15 16 18 20 21 26 27 29 31 P0 (0,2,16,18) b c e X X X X P1 (0,4,16,20) b d e X X X X P2 (2,10,18,26) c de X X X X P3 (4,5,20,21) b cd X X X X P4 (5,13,21,29) cd e X X X X P5 (10,11,26,27) bc d X X X X P6 (11,15,27,31) bde X X X X P7 (13,15,29,31) bce X X X X This table can be reduced noting that the table repeats itself starting at column 16: 3

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0 2 4 5 10 11 13 15 P0 (0,2,16,18) b c e X X P1 (0,4,16,20) b d e X X P2 (2,10,18,26) c de X X P3 (4,5,20,21) b cd X X P4 (5,13,21,29) cd e X X P5 (10,11,26,27) bc d X X P6 (11,15,27,31) bde X X P7 (13,15,29,31) bce X X
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