barcode reader code in asp.net c# Figure 1333 Four- and eight-cell subcubes for an arbitrary logic function in Software

Encoder QR in Software Figure 1333 Four- and eight-cell subcubes for an arbitrary logic function

Figure 1333 Four- and eight-cell subcubes for an arbitrary logic function
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WX YZ YZ YZ YZ 0 1 0 0 WX 0 1 0 0 WX 0 1 0 0 W X 0 1 0 0
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Quite a simpli cation! If we consider, now, a map in which we place a 1 in the cells corresponding to the minterms W X Y Z, W X Y Z, W X Y Z, and W X Y Z, forming the previous expression, we obtain the Karnaugh map of Figure 1334 It can easily be veri ed that the map of Figure 1334 shows a single four-cell subcube corresponding to the term Y Z We have not established formal rules yet, but it de nitely appears that the map method for simplifying Boolean expressions is a convenient tool In effect, the map has performed the algebraic simpli cation automatically! We can see that in any subcube, one or more of the variables present will appear in both complemented and uncomplemented form in all their combinations with the other variables These variables can be eliminated As an illustration, in the eight-cell subcube case of Figure 1335, the full-blown expression would be: W X Y Z+W X Y Z+W X Y Z+W X Y Z +W X Y Z + W X Y Z + W X Y Z + W X Y Z However, if we consider the eight-cell subcube, we note that the three variables X, W , and Z appear both in complemented and uncomplemented form in all their combinations with the other variables and thus can be removed from the expression This reduces the seemingly unwieldy expression simply to Y ! In logic design terms, a simple inverter is suf cient to implement the expression The example just shown is a particularly simple one, but it illustrates how simple it can be to determine the minimal expression for a logic function It should be apparent that the larger a subcube, the greater the simpli cation that will result For subcubes that do not intersect, as in the previous example, the solution can be found easily, and is unique Sum-of-Products Realizations Although not explicitly stated, the logic functions of the preceding section were all in sum-of-products form As you know, it is also possible to realize logic functions in product-of-sums form This section discusses the implementation of logic functions in sum-of-products form and gives a set of design rules The next section will do the same for product-of-sums form logical expressions The following rules are a useful aid in determining the minimal sum-of-products expression:
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Figure 1334 Karnaugh map for the function W X Y Z+W X Y Z+ W X Y Z+W X Y Z
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WX YZ YZ YZ YZ 1 1 0 0
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F O C U S O N M E T H O D O L O G Y
Sum-of-Products Realizations 1 Begin with isolated cells These must be used as they are, since no simpli cation is possible 2 Find all cells that are adjacent to only one other cell, forming two-cell subcubes 3 Find cells that form four-cell subcubes, eight-cell subcubes, and so forth 4 The minimal expression is formed by the collection of the smallest number of maximal subcubes
13
Digital Logic Circuits
The following examples illustrate the application of these principles to a variety of problems
EXAMPLE 139 Logic Circuit Design Using Karnaugh Maps
Problem
A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 y 1 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1
Design a logic circuit that implements the truth table of Figure 1336
Solution
Known Quantities: Truth table for y(A, B, C, D) Find: Realization of y Assumptions: Two-, three-, and four-input gates are available Analysis: We use the Karnaugh map of Figure 1337, which is shown with values of 1
and 0 already in place We recognize four subcubes in the map; three are four-cell subcubes, and one is a two-cell subcube The expressions for the subcubes are: A B D for the two-cell subcube; B C for the subcube that wraps around the map; C D for the four-by-one subcube; and A D for the square subcube at the bottom of the map Thus, the expression for y is: y = A B D + B C + CD + AD
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