3934401021 - Chapter 2 Karnaugh Maps Meilin Liu Department...

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Chapter 2 Karnaugh Maps Meilin Liu Department of Computer Science Wright State University Homepage: http://www.wright.edu Email: meilin.liu@wright.edu
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CEG 260, Meilin Liu 2 Two-Variable Karnaugh Maps 1 A B 0 0 1 m0 m1 m2 m3 Alternate way of representing Boolean function All rows of truth table represented with a Karnaugh Map Each square represents a minterm Easy to convert between truth table, K-map, and SOP Number of 1’s in K-map equals number of minterms included in the function A B A=0,B=1 A=1,B=1 A=0,B=0 A=1,B=0 MSB LSB 1 A B 0 0 1 A’B’ A’B AB’ AB
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CEG 260, Meilin Liu 3 Two-Variable Karnaugh Maps A B F 0 0 1 0 1 1 1 0 0 1 1 0 F = Σ(m 0 ,m 1 ) = A’B’ + A’B 1 A B 0 0 1 1 1 0 0 F = A’B’ + A’B=A’(B’+B)=A’ Every time, when you combine two squares to form one rectangle, one literal is reduced F = Σ(m 0 ,m2) = A’B’ + AB’ A B 0 0 1 1 0 1 0 A=0 F = A’ F = B’ B=0
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CEG 260, Meilin Liu 4 Two-Variable Karnaugh Maps F = Σ(m1,m 2 ) = A’B + AB’ F = Σ(m 0 ,m1,m2) 1 A B 0 0 1 1 1 1 0 F = A’+B’ 1 A B 0 0 1 0 1 1 0 F = A’B+AB’
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CEG 260, Meilin Liu 5 Rules for K-Maps We can simplify boolean expression by circling 1’s in the K- map Each circle represents simplified product term Following circling, we can deduce minimized Sum-of-Product. Rules to consider: Every cell containing a 1 must be included at least once. The largest possible “power of 2 rectangle” must be enclosed. The 1’s must be enclosed in the smallest possible number of rectangles.
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CEG 260, Meilin Liu 6 Two-Variable Karnaugh Maps F = Σ(m1,m 2 ,m 3 ) = A’B + AB’+AB F = Σ(m 0 ,m1,m2,m3) 1 A B 0 0 1 1 1 1 1 F = 1 1 A B 0 0 1 0 1 1 1 F = A+B
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CEG 260, Meilin Liu 7 Three-Variable Karnaugh Maps Numbering scheme based on Gray–code e.g., 00, 01, 11, 10 Only a single bit changes in code for adjacent map cells This is necessary to observe the variable transitions 00 01 BC A 0 1 11 10 A C B m0 m1 m3 m2 m5 m7 m6 m4
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CEG 260, Meilin Liu 8
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3934401021 - Chapter 2 Karnaugh Maps Meilin Liu Department...

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