HWK4 - HWK #4 Solution Problem 4.1 (a) h(A,B,C,D)= OR(m 1...

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Unformatted text preview: HWK #4 Solution Problem 4.1 (a) h(A,B,C,D)= OR(m 1 ,m 2 ,m 6 ,m 8 ,m 13 ,m 14 ,m 15 ,d ,d 4 ,d 7 ,d 10 ) h(A,B,C,D)= AND(M 3 ,M 5 ,M 9 ,M 11 ,M 12 ,d ,d 4 ,d 7 ,d 10 ) (b) There are 2 minimal SOP expressions. I shall explain first how I obtained these; their K-maps are provided below. Looking for a minimal SOP expression, I first observed that only the PI ACD contains m 13 ,only PI A'B'C' contains m 1 , and only PI B'D' contains m 8 . So any minimal SOP expression will contain: B'D' + ACD + A'B'C' Minterm m 6 and m 14 is covered by both BC' and CD'. Both use a single don't care term but both are loops of size 4 squares, so they will both have a product of two literals, and the minimal sop can be represented with either one of these loops along with the essential PI. To summarize: the min SOP expressions are as follows: B'D' + ABD + A'B'C' + { BC or CD' } h(A,B,C,D)= B'D' + ABD + A'B'C' + CD' h(A,B,C,D)= B'D' + ABD + A'B'C' + BC There are 4 minimal POS expressions: (B' + C + D) (A' + B + D') { (A + C' + D') or (B + C' + D') { (A + B' + C) or (A + B' + D') } h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + C) (A + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + C) (B + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + D') (A + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + D') (B + C' + D') (c) Your two expressions in part (b) are equal iff every square covered in your SOP K- map is uncovered in your POS K-map. Why? In the SOP K-map: for every minterm inside a loop, the function is defined to be 1, and for every minterm not inside any loop the function is defined to be 0. The opposite is true in the POS K-map: for every maxterm inside a loop the function is defined to be 0, and for every maxterm not inside a loop the function is defined to be 1. For this example: The first SOP expression is equal to the first, third, and fourth POS expressions. The second POS expression is equal to the second SOP expression. See parts (d) and (e) for additional explanation. (d) There are 2 minimal SOP expressions - see part (b). The function has 4 don't cares. Both of the minimal SOP K-maps have: d =1, d 10 =1, and d 4 =0. But the values of d 7 differ: For the first map: d 7 =0 For the second map: d 7 =1 Hence, the minimal SOP expressions are not equal - because the same squares are not circled in both cases....
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This note was uploaded on 02/26/2010 for the course ECE 290 taught by Professor Brown during the Fall '08 term at University of Illinois at Urbana–Champaign.

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HWK4 - HWK #4 Solution Problem 4.1 (a) h(A,B,C,D)= OR(m 1...

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