For example the function f a b c 14 56 7 nij

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Unformatted text preview: straightforward conversion to very nice gating networks which are more desirable from most implementation points of view. In their purest, nicest form they go into two-level networks, which are networks for which the longest path through which the signal must pass from input to output is two gates. Conversion Between Canonical Forms The complement of a function expressed as the sum-of-minterms equals the sum-ofminterms missing from the original function. This is because the original function is expressed by those minterms that make the function equal to 1, while its complement is a 1 for those minterms for which the function is a 0. For example, the function F (A, B, C) = (1,4, 5,6, 7) = nij +m4 +m5+m6 +m7 has a complement that can be expressed asf F (A, B, C) = (0, 2, 3) = m0 +m2 +m3 Now, if we take the complement of F, by De Morgan's theorem we obtain F back in a different form: F = m0 +m2 +m3 = m0 • m2 • m3 = M0-M2-M3 = II (0, 2, 3) The last conversion follows from the definition of minterms and maxterms as shown in Figure 6.16. From the figure, it is clear that the following relation holds true: That is, the maxterm with subscript/ is a complement of the minterm with the same subscript/, and vice-versa. The last example has demonstrated the conversion between a function expressed in sumof-minterms and its equivalent in product-of-maxterms. A similar argument will show that the conversion between the product-of-maxterms and the sum-of-minterms is similar. We now state a general conversion procedure: "To convert from one canonical form to another, interchange the symbol and list those numbers missing from the original form." For example, the function F (x, y, z) = TI (0, 2, 4, 5) is expressed in the product-of-maxterms form. Its conversion to sum-of-minterms is: F(x,y,z) = I(l,3,6,7) Note that in order to find the missing terms, one must realize that the total number of minterms or maxterms is always 2n, where n is the number of binary variables in the function. LOGIC GATES All operations within a co...
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