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## Chapter 5

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**Chapter 5**Boolean Algebra and Reduction Techniques 1**5-9 Karnaugh Mapping**• Used to minimize the number of gates • Reduce circuit cost • Reduce physical size • Reduce gate failures • Requires SOP form • Karnaugh Mapping • Graphically shows output level for all possible input combinations • Moving from one cell to an adjacent cell, only one variable changes 31**Karnaugh Mapping**• Steps for K-map reduction: • Transform the Boolean equation into SOP form • Fill in the appropriate cells of the K-map • Encircle adjacent cells in groups of 2, 4 or 8 • Adjacent means a side is touching, NOT diagonal. • Watch for the wraparound • Find each term of the final SOP equation by determining which variables remain the same within circles 33**Figure 5.88 Encircling adjacent cells in a Karnaugh map.**These are the variables () that remain the same within each circle.**Discussion Point**• Use a K-map to simplify the circuit.**5-10 System Design Applications**• Use Karnaugh Mapping to reduce equations • Use AND-OR-INVERT gates to implement logic**Figure 5.96 (a) Simplified equation derived from a**Karnaugh map; (b) implementation of the odd-number decoder using an AOI.**Summary**• Several logic gates can be connected together to form combinational logic. • There are several Boolean laws and rules that provide the means to form equivalent circuits. • Boolean algebra is used to reduce logic circuits to simpler equivalent circuits that function identically to the original circuit.**Summary**• DeMorgan’s theorem is required in the reduction process whenever inversion bars cover more than one variable in the original Boolean equation. • NAND and NOR gates are sometimes referred to as universal gates, because they can be used to form any of the other gates.**Summary**• AND-OR-INVERT (AOI) gates are often used to implement sum-of-products (SOP) equations. • Karnaugh mapping provides a systematic method of reducing logic circuits. • Combinational logic designs can be entered into a computer using schematic block design software or VHDL.**Summary**• Using vectors in VHDL is a convenient way to group like signals together similar to an array. • Truth tables can be implemented in VHDL using vector signals with the selected signal assignment statement. • Quartus II can be used to determine the simplified equation of combinational circuits.