hw3s - ENGRD2300 Introduction to Digital Logic Fall 2008...

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Unformatted text preview: ENGRD2300: Introduction to Digital Logic Fall 2008 Homework 3 Solutions Problem 1. a) Use a K-map to find a minimal sum of products expression for the “short_number_detector” problem from Lab 2. Did you do better with the K-map than you did in lab? If you had the same answer for both, which was easier, the method you used for Lab 2 or K-maps? A 00 00 0 01 1 01 2 11 6 10 4 0 3 1 7 1 5 0 0 C 1 0 B 0 B·C’ + A’·B’·C b) Use a K-map to find a minimal sum of products expression for the “even-or-odd-vowels” problem from Lab 2. Did you do better with the K-map than you did in lab? If you had the same answer for both, which was easier, the method you used for Lab 2 or K-maps? N3 00 00 0 01 1 11 3 N1 10 2 01 4 11 12 10 8 0 5 0 0 7 0 13 9 0 0 11 N0 0 0 6 1 15 0 1 1 0 N2 1 10 1 1 1 N3·N2·N0 + N3’·N1·N0’ + N3·N2’·N1 ENGRD2300: Introduction to Digital Logic Problem 2. a) Use a K-map to find a minimal sum of products expression for the canonical sum W 00 00 0 01 1 11 3 Y 10 2 01 4 11 12 10 8 Fall 2008 0 5 1 1 7 0 13 9 0 1 11 Z 0 1 6 1 15 1 0 X 1 14 0 10 0 1 0 ∑WXYZ(3,4,5,7,9,13,14,15) = W’XY’ + W’YZ + WXY + WY’Z b) Use a K-map to find minimal product of sums expression for the canonical product W 00 00 0 01 1 11 3 Y 10 2 01 4 11 12 10 8 0 5 1 1 7 0 13 9 0 1 11 Z 0 1 6 1 15 1 0 X 1 14 0 10 0 1 0 ∏WXYZ(0,1,2,6,8,10,11,12) = (W’+Y+Z)(W+X+Y)(W+Y’+Z)(W’+X+Y’) ENGRD2300: Introduction to Digital Logic Problem 3. a) Use a K-map to find a minimal sum of products expression for the canonical sum A 00 00 0 01 1 11 3 C 10 2 01 4 11 12 10 8 Fall 2008 0 5 0 1 7 1 13 9 0 0 11 D 1 0 6 1 15 1 0 B 1 14 0 10 0 0 0 ∑ABCD(1,5,7,12,13,15) = A·B·C’ + A’·C’·D + B·D b) Prove algebraically that your answer to (a) is the same as ∑ABCD(1,5,7,12,13,15) = A’·B’·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’·D’ + A·B·C’·D + A·B·C·D = A’·B’·C’·D + A’·B·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’·D’ + A·B·C’·D + A·B·C·D = A’·C’·D·(B+B’) + A’·B·C’·D + A’·B·C·D + A·B·C’·D’ + A·B·C’·D + A·B·C·D = A’·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’·D’ + A·B·C’·D + A·B·C’·D + A·B·C·D = A’·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’·(D+D’) + A·B·C’·D + A·B·C·D = A’·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’ + A·B·C’·D + A·B·C·D = A·B·C’ + A’·C’·D + A’·B·C’·D + A’·B·C·D + A·B·C’·D + A·B·C·D = A·B·C’ + A’·C’·D + A’·B ·D ·(C’+C) + A·B·D·(C’+C) = A·B·C’ + A’·C’·D + A’·B·D + A·B·D = A·B·C’ + A’·C’·D + (A’+A)·B·D = A·B·C’ + A’·C’·D + B·D Hint: use your K-map to guide your algebraic derivation. ENGRD2300: Introduction to Digital Logic Fall 2008 Problem 4. Wakerly Exercise 4.55 F = Q2 P2′ + Q2 Q1 P1′ + Q2 Q0 P1′ P0′ + Q0 P2′ P1′ P0′ + Q1 P2′ P1′ + Q1 Q0 P2′ P0′ + Q2 Q1 Q0 P0′ Problem 5. a) Wakerly Exercise 4.59 (c). W 00 01 4 11 12 10 8 00 01 20 11 28 W 10 24 00 0 01 1 11 3 Y 10 2 1 5 1 1 7 0 13 9 0 00 16 01 17 11 19 Y 10 18 0 0 0 0 1 21 1 29 1 25 1 1 6 0 15 0 11 Z 1 23 1 31 1 27 Z 0 0 X 0 14 1 10 0 22 0 30 1 26 1 1 1 0 X V 1 1 F = V′ W′ X′ + W X′ Y + W Y Z′ + W′ X Y′ + V W Y′ ENGRD2300: Introduction to Digital Logic b) Wakerly Exercise 4.59(f). W 00 01 4 11 12 10 8 00 01 20 11 28 W 10 24 Fall 2008 00 0 01 1 11 3 Y 10 2 0 5 1 d 7 1 13 9 0 00 16 01 17 11 19 10 18 0 0 0 0 1 21 1 29 0 25 d 0 6 1 15 1 11 Z Y 0 23 d 31 1 27 Z 1 1 X 1 14 1 10 0 22 1 30 1 26 0 1 0 1 X V 1 0 F = V′ X + W Z + X Z′ ENGRD2300: Introduction to Digital Logic Problem 6. Wakerly Exercise4.60(c). W 00 01 4 11 12 10 8 00 01 20 11 28 W 10 24 Fall 2008 00 0 01 1 11 3 Y 10 2 0 5 1 1 7 1 13 9 0 00 16 01 17 11 19 10 18 1 1 1 1 1 21 1 29 0 25 0 0 6 1 15 0 11 Z Y 1 23 1 31 0 27 Z 0 1 X 1 14 0 10 0 22 1 30 0 26 1 1 W 0 0 X 1 W 0 00 01 36 11 44 10 40 Z 00 01 52 11 60 10 56 00 32 01 33 11 35 Y 10 34 0 0 0 1 0 37 0 45 0 41 00 48 01 49 11 51 Y 10 50 0 0 1 1 0 53 1 61 0 57 0 39 0 47 0 43 0 55 1 63 0 59 Z 0 38 0 46 0 42 0 54 1 62 0 58 U 1 X 0 0 0 X V 1 0 F = U′ X Y′ + U’ W X + V W X + V’ W’ Y Z’ + V W’ X’ Y + U′ V W′ Y′ [ or U′ V W′ X′ ] ...
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