Graduate Institute of Electronics Engineering, NTU. CH5 Karnaugh Maps. Lecturer: 吳安宇教授 Date:2006/10/20 ACCESS IC LAB


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1 CH5 Karnaugh Maps Lecturer: 吳安宇教授 Date:2006/0/20 CCESS IC L
2 Problems in lgebraic Simplification The procedures are difficult to apply in a systematic way. It is difficult to tell when you have arrived at a minimum solution. (minimum SOP, POS) => Karnaugh map (Kmap) is the solution. pp. 2
3 Two and ThreeVariable KMapsK 0 Minterm: m 0 m 2 m m 3 m i F => => One circle => eliminates one variable + = ( +) = pp. 3
4 Two and ThreeVariable KMapsK 3variable C C m 0 m 4 m m 5 => C m 3 m 7 m 2 m 6 pp. 4
5 Two and ThreeVariable KMapsK Ex: Kmap for F(a,b,c) = m(,3,5) = M(0,2,4,6,7) C C F = C + C + C = C + C => Minimum SOP form pp. 5
6 Two and ThreeVariable KMapsK Other combinations C 0 C C C C F = (F = C+C+ C +C ) F = C +C = C F = C pp. 6
7 Two and ThreeVariable KMapsK Ex: f(a,b,c) = abc + b c + a a a b c b c b c b c b c => Kmap simplification c a b a abc => F = a + b c + bc pp. 7
8 Two and ThreeVariable KMapsK Consensus theorem: XY+X Z +YZ = XY+X Z Y Z X X Y Z X X Z Y => Z Y pp. 8
9 b c c Graduate Institute of Electronics Engineering, NTU Same Simplification in KMapsK F = m(0,,2,5,6,7) Check no. of terms and no. of literals a a a b c c a F(a,b,c) = a b +bc +ac b F(a,b,c) = a c +b c+ab b pp. 9
10 FourVariable KMapsK Ex: Kmap of F(a,b,c,d) = acd + a b + d C D b D C d c a F = d + a b + ac pp. 0
11 FourVariable KMapsK Ex : F = m(,3,4,5,0,2,3) C D F = C + D + CD D 0 0 C Circle of 2 k => Eliminate k variables Ex 2: F = m(0,2,3,5,6,7,8,0,,4,5) F = C + D + D F = SOP ( 選 0) => F = POS (De Morgan) pp.
12 KMap with Don t t cares Ex 3: F(,,C,D) = m(,3,5,7,9)+ d(6,2,3) D X 0 X 0 X C F = D+C D (SOP) Ex 4: Obtain POS form of minimum f(a,b,c,d) Find the minimum SOP form of f (a,b,c,d) by looping the 0 s on a map of f(a,b,c,d) pp. 2
13 FourVariable KMapsK Ex: f = X Z + WYZ + W Y Z + X Y W X Y Z W f = () + (2) + (3) Z (2) (3) () Y = Y Z + W YX + WXZ De Morgan s Law f = [Y Z+W XY+WXZ ] =(Y+Z )(W+X +Y )(W +X +Z) X pp. 3
14 Prime Implicants Implicant: ny single or any group of s in the Kmap of F function. Prime Implicant: If it cannot be combined with another term to eliminate a variable. single on the Kmap represents a PI if it is not adjacent to any other s. Two adjacent s on a map form a PI if they are not contained in a group of 4 s. pp. 4
15 Prime Implicants Implicant: product term ( terms) 5 minterms: { C, C, C, C, C} 5 group of 2literal terms: {,, C, C, C} group of literal term: {} C Prime Implicant (PI): n implicant that is not covered by another implicant: {, C} pp. 5
16 Essential Prime Implicants Essential Prime Implicant (EPI): PI that covers at least one minterm that is not covered by another PI. => {, C} are also Essential PI. => n essential PI in the Kmap by noting that it covers at least one minterm that is circled only once. C Cover: set of PI s which covers all s. (all minterms) pp. 6
17 Essential Prime Implicants Ex: a b c d c ll PI s: {a b d, bc, acd, a c d, ab, b cd} F = b c + ac + a b d Ex: C D D a C ll PI s: {CD, D, C, C} Essential PI s: {D, C,C} f = D+ C+C pp. 7
18 Essential Prime Implicants Ex: C D D EPI EPI EPI C Essential PI s: { C, D, CD} Two PI s can cover CD: { D, CD} D f = C + D +CD+ or CD pp. 8
19 Essential Prime Implicants Ex: C D X 0 0 X X PI (3) F = ()+(2)+(3) Rule: () EPI EPI (2). Find all Essential PI s. 2. Find a minimum set of PI s to cover the remaining s on the map. pp. 9
20 Flowchart for determining a minimum Sum of Products based on EPIs and PIs pp. 20
21 Kmap for 5 Variables pp. 2
22 Other forms of 5variable 5 KmapK pp. 22
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