Efficient Logic Circuit Design: An Introduction to Karnaugh Mapping Techniques
Exploring Karnaugh Mapping, a vital technique for minimizing logic circuits to reduce gate count, cost, and physical size. The method simplifies Boolean equations into Sum of Products (SOP) form by visually representing possible input combinations. Key steps include filling K-map cells and encircling groups of adjacent cells to derive minimized SOP equations. The process is essential in designing efficient combinational logic circuits using AND-OR-INVERT gates and applying Boolean laws. Learn how to leverage Karnaugh Mapping for effective circuit design.
Efficient Logic Circuit Design: An Introduction to Karnaugh Mapping Techniques
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Presentation Transcript
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.
