HW5_soln_2011

# HW5_soln_2011 - Homework-5 Solution 1(10p For this function...

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Homework-5 Solution 1. (10p) For this function, find a minimum sum-of-products solution, using the Quine-McCluskey method: F(a,b,c,d,e) = m (13,15,17,18,19,2 ̷ 0,21,23,25,27,29,31) + d(1,2,12,24) Group # : minterm Column 1 Column 2 Column 3 *(1,17) -0001 *(2,18) -0010 (17,19,21,23) 10—1 (17,19,25,27) 1-0-1 (17,21,25,29) 1—01 (17,19,21,23,25, 27,29,31) 1--- 1 1 00001 2 00010 17 10001 12 01100 18 10010 20 10100 24 11000 (17,19) 100-1 (17,21) 10-01 (17,25) 1-001 (18,19) 1001- *(12,13) 0110- (20,21) 1010- *(24,25) 1100- (13,15,29,31) -11-1 (19,23,27,31) 1—11 (21,23,29,31) 1-1-1 (25,27,29,31) 11—1 13 01101 19 10011 21 10101 25 11001 (13,15) 011-1 (13,29) -1101 (19,23) 10-11 (19,27) 1-011 (21,23) 101-1 (21,29) 1-101 (25,29) 11-01 (25,27) 110-1 15 01111 23 10111 27 11011 29 11101 (15,31) -1111 (23,31) 1-111 (27,31) 11-11 (29,31) 111-1 31 11111 ** The entries indicated by * can be dropped for next round since they just satisfy don’t care conditions. ** Check marks indicate that the entries are used for further reduction (hence, they are not prime implicants) The 5 variable set is ordered as {a,b,c,d,e}. Drawing the prime implicant chart Term/Variable 12 (d) 13 15 17 18 19 20 21 23 24 (d) 25 27 29 31 P1 : ae X X X X X X X X P2 : bce X X X X P3 : ab’c’d X X

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P4 : ab’cd’ X X P5 : abc’d’ X X P6 : a’bcd’ X X The boxes in green indicate essential prime implicates. The minimal SOP for this function is given by F = ae + bce + ab’c’d + ab’cd’ 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 (0,1,3,4,7,13,15,19,20,22,23,29,31) Minterms arranged by their group Column 1 Column 2 0 00000 (0,1) 0000- (0,4) 00-00 (3,7,19,23) -0-11 1 00001 4 00100 3 00011 20 10100 (1,3) 000-1 (4,20) -0100 (7,15,23,31) --111 (13,29,15,31) -11-1 7 00111 13 01101 19 10011 22 10110 (3,7) 00-11 (3,19) -0011 (20,22) 101-0 15 01111 23 10111 29 11101 (15,17) 0-111 (17,23) -0111 (13,29) -1101 (13,15) 011-1 (19,23) 10-11 (22,23) 1011- 31 1111 (15,31) -1111 (23,31) 1-111 (29,31) 111-1 ** Check marks indicate that the entries are used for further reduction (hence, they are not prime implicants) Drawing the prime implicant chart Term/Variable 0 1 3 4 7 13 15 19 20 22 23 29 31 P1 : b’de X X X X X P2 : cde X X X P3 : bce X
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