ECE380_Fall2013_Chapter4 - INTRODUCTION TO DIGITAL LOGIC ECE 380 Optimized Implementation of Logic Functions Jacob Chakareski Department of Electrical

ECE380_Fall2013_Chapter4 - INTRODUCTION TO DIGITAL LOGIC...

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INTRODUCTION TO DIGITAL LOGIC ECE 380 Optimized Implementation of Logic Functions Jacob Chakareski Department of Electrical and Computer Engineering
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 2 Synthesis of Logic Functions Implementation via Sum of products form Product of sums form Algebraic manipulation to minimize cost Automatic optimization via CAD tools
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 3 Function Synthesis Example f (x 1 , x 2 , x 3 ) = Σ m (0, 2, 4, 5, 6)
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 4 Algebraic Simplification Iteratively apply combining identity f = m 0 + m 2 + m 4 + m 5 + m 6 = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 = x 1 x 3 + x 1 x 2 x 3 + x 1 x 3 (m 4 =m 4 +m 4 : T3) = x 3 + x 1 x 2 Simplify by grouping minterms _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 5 Group Suitable Minterms x 1 x 2 x 3 m 0 0 0 0 m 2 0 1 0 m 4 1 0 0 m 6 1 1 0 x 1 x 2 x 3 m 4 1 0 0 m 5 1 0 1
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 6 Karnaugh Map: Table & Cells x 2 (a) Truth table (b) Karnaugh map 0 1 0 1 m 0 m 2 m 3 m 1 x 1 x 2 0 0 0 1 1 0 1 1 m 0 m 1 m 3 m 2 x 1 Minterms in adjacent cells can be combined
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 7 Two-Variable K-Map x x 2 1 0 1 1 f x 2 x 1 + = 0 1 0 1 1 Truth table
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 8 Two-Variable K-Map: Min Cost x x 2 1 0 1 1 f x 2 x 1 + = 0 1 0 1 1 Minimum-cost implementation of f ¾ Smallest number of product terms 1, every time f = 1 ¾ Each product term = min-cost
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 9 Three-Variable K-Map x 1 x 2 x 3 00 01 11 10 0 1 (b) Karnaugh map x 2 x 3 0 0 0 1 1 0 1 1 m 0 m 1 m 3 m 2 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 m 4 m 5 m 7 m 6 x 1 (a) Truth table m 0 m 1 m 3 m 2 m 6 m 7 m 4 m 5 Gray Code for valuations
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 10 3-Variable K-Map: Example 1 f x 1 x 3 x 2 x 3 + = x 1 x 2 x 3 0 0 1 0 1 1 0 1 00 01 11 10 0 1 Function f from Slide 37 in Chapter 2 lecture
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 11 3-Variable K-Map: Example 2 Function f from Slide 3 x 1 x 2 x 3 1 1 0 0 1 1 0 1 f x 3 x 1 x 2 + = 00 01 11 10 0 1
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 12 3-Variable K-Map: Example 3 x 1 x 2 x 3 1 1 1 1 x 1 0 0 1 0 00 01 11 10 0 1 x 2 x 3 Function f ( x 1 , x 2 , x 3 ) = Σ m (0, 1, 2, 3, 7)
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 13 Four-Variable K-Map x 1 x 2 x 3 x 4 00 01 11 10 00 01 11 10 x 2 x 4 x 1 x 3 m 0 m 1 m 5 m 4 m 12 m 13 m 8 m 9 m 3 m 2 m 6 m 7 m 15 m 14 m 11 m 10
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 14 4-Variable K-Map: Example 1 x 1 x 2 x 3 x 4 0 00 01 11 10 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 00 01 11 10 f 1 x 2 x 3 x 1 x 3 x 4 + = _ _
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Jacob Chakareski: ECE 380 Optimized Implementation of Logic Functions 15 4-Variable K-Map: Example 2 x 1 x 2 x 3 x 4 0 00 01 11 10 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 00 01 11 10 f 2 x 3 x 1 x 4 + = Higher cost Lower cost
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