Question 13.14: Comparison of Sum-of-Products and Product-of-Sums Designs Re...
Comparison of Sum-of-Products and Product-of-Sums Designs
Realize the function f described by the accompanying truth table using both 0 and 1 coverings in the Karnaugh map.
Known Quantities: Truth table for logic function.
\begin{array}{lll|l}x & y & z & f \\\hline 0 & 0 & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 1 & 0 & 1 \\0 & 1 & 1 & 1 \\1 & 0 & 0 & 1 \\1 & 0 & 1 & 1 \\1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 \\\hline\end{array}
Find: Realization in both sum-of-products and product-of-sums forms.
Analysis:
1. Product-of-sums expression. Product-of-sums expressions use zeros to determine the logical expression from a Karnaugh map. Figure 13.46 depicts the Karnaugh map covering with zeros, leading to the expression
f=(x+y+z) \cdot(\bar{x}+\bar{y})
2. Sum-of-products expression. Sum-of-products expressions use ones to determine the logical expression from a Karnaugh map. Figure 13.47 depicts the Karnaugh map covering with ones, leading to the expression
f=(\bar{x} \cdot y)+(\bar{x} \cdot \bar{y})+(\bar{y} \cdot z)
Comments: The product-of-sums solution requires the use of five gates (two OR, two NOT, and one AND), while the sum-of-products solution will use six gates (one OR, two NOT, and three AND). Thus, solution 1 leads to the simpler design.
