An Introduction to Digital Logic by A. Potton (auth.)

By A. Potton (auth.)

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De Morgan's theorem provides the required link between the AND operations in the original expression and the required NOR form. From the above analysis, the required operation is given by F=P· C· {K+G) where =P · C · X =P+C+X =P+C+ (K+G) X = (K +G) Since the final expression involves no AND type operations, it may be implemented using NOR gates only. 4. 3 A second design example A second simple design example will now be considered. 3. Examination of the truth table indicates that the system operation may be described in boolean terms by F=A·B·C·D+A·B·C·D+ A·B·C·D In designing this system it will be assumed that a range of gates is to be used in which only positive logic NAND operations are available.

Show a suitably modified system. 6 A logic system has three inputs A, 8, C. The output F is required to be 1 if the total number of l's at the inputs is an even number (zero is regarded as an even number). Design the system using not more than the following complement of gates: 1 1 2 3 two-input AND gate three-input AND gate two-input OR gates NOT gates. 4 Karnaugh Maps Karnaugh maps together with boolean algebra provide the logic system designer with the two major 'tools of the trade'. Map techniques have certain advantages when looking for relationships which enable the total number of gates required to implement a given logical operation to be minimised.

20 shows the various cell loopings. 9 'Don't care' situations We now suppose that in the previous problem, the specification for the system is again changed this time such that the value ofF for the condition N 1 = N 2 is irrelevant, that is it may be 0 or 1. 4. 21, the 'don't care' cells containing a dagger may be assumed to contain 0 or 1 to suit ones own convenience. This leads to some simplification of the logical expression describing the circuit operation. 10 Minimisation with functions of more than four variables The Karnaugh map technique can be extended to handle functions of five, six or more variables by using maps consisting of arrays of cells in three or more dimensions.

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