Introduction to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL

6.4 TWO-VARIABLE KARNAUGH MAP

An example of a Karnaugh map for a two-variable function is shown in Figure 6.3. Each value of the truth table is represented in the Karnaugh map. A 1 appears in both columns of the top row. Therefore, there exists a single product term that can cause f to be equal to 1 when the input variables have values that correspond to either of these cells. These values have been circled and are identified as x₁ = 0, but x₂ equals 0 for the left column and 1 for the right. This implies that if x₁ = 0, then f = 1, regardless of the value of x₂. The product term represented by this circle is simply . Similarly, if x₂ is 1, then regardless of x₁, the function f will also equal 1. Hence, the minimum realization for the logic function f can be expressed as

Figure 6.3 Karnaugh Map Simplification of a Two-Variable Logic Function

Algebraic simplification yields the same logic expression. Therefore, to find a minimum implementation for a given logic function, it is necessary to find the smallest number of product terms that produce a value of 1 for all cases. At the same time, the number of these product terms should be as low as possible. Notice how some of the 1's are ...