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Complex logic operations
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The term complex, when used to describe a logic operation, does not necessarily mean complicated. A more apt adjective might be composite. But there are cases when logic operations are indeed quite complicated. No matter how messy a particular logic operation might appear, it can always be broken down into the elementary operations defined above.
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562 Basic digital principles
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30-4 Four-input AND gate (A) and four-input NOR gate (B).
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Suppose you want to find an arrangement of logic gates for some complex logic operation. This can be done in two ways. You can use either truth tables or Boolean algebra. For any complex logic operation, there might be several solutions, some requiring more gates than others. A digital design engineer has the job not only of finding an arrangement of gates for a given complex operation, but of finding the scheme that will yield the desired result with the least number of gates.
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Working with truth tables
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Truth tables are, in theory, infinitely versatile. It is possible to construct or break down logic operations of any complexity by using these tables, provided you have lots of paper and a fondness for column-and-row matrix drawing. Computers can also be programmed to work with truth tables, although the displays and printouts get horrible to deal with if the logic functions are messy.
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Building up
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You can build complex logic operations up easily by means of a truth table. An example of such a building process is shown in Table 30-5.
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Table 30-5.
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X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 X 0 0 1 1 1 1 1 1 Y
Truth table for -(X + Y) + XZ.
(X 1 1 0 0 0 0 0 0 Y) XZ 0 0 0 0 0 1 0 1 (X Y) 1 1 0 0 0 1 0 1 XZ
Working with truth tables 563 There are three variables X, Y, and Z. Each can be either 0 or 1 (low or high). All the possible combinations are listed by writing binary numbers XYZ upwards from 000 through 111. This forms the first column of the truth table. If there are n total variables, there will be 2n possible combinations. The second column lists the values of X Y, the OR function. Some rows are duplicates of each other; you write all the resultants down anyway. The third column lists the negations of the values in the second column; this is the NOR operation (X Y). The fourth column shows the values of XZ. The fifth column is the disjunction (OR operation) of the values in the third and fourth column. It renders values for the complex logic operation (X Y) XZ.
Breaking down
Suppose you were called upon to break down a logical operation, rather than to build it up. This is the kind of problem encountered by engineers. In the process of designing a certain digital circuit, the engineer is faced with figuring out what combination of logic gates will yield the complex operation, say, XY (XZ) YZ. Proceed by listing all the possible logic states for the three variables X, Y, and Z, exactly as in Table 30-5. Then find the values of XY (X AND Y), listing them in the fourth column. Next, find values XZ and list them in a fifth column. Negate these values to form a sixth column, depicting (XZ). (You might be able to perform the AND and NOT operations together in your head, skipping over the XZ column. But be careful! It s easy to make errors, and in a digital circuit, one error can be catastrophic.) Next, find values YZ and list them in a seventh column. Finally, perform the OR operation on the values in the columns for XY, (XZ), and YZ. A multiple-valued OR is 0 only if all the individual variables are 0; if any or all of the inputs are 1, then the output is 1. This process yields Table 30-6.
Table 30-6.
X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 XY 0 0 0 0 0 0 1 1
Truth table for XY
XZ 0 0 0 0 0 1 0 1 (XZ) 1 1 1 1 1 0 1 0
(XZ)
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