CSC270 Homework 3 2011

--D. Thiebaut 15:18, 9 February 2011 (EST)

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This assignment is due a week from Friday 2/11, i.e on Friday 2/18, at midnight. You can work on this assignment in groups of two if you desire.

Problem #1

How can we recognize the presence of an XOR gate in a Karnaugh map? In other word, what pattern or patterns of covers will be indicative of the fact that the resulting function can be simplified using an XOR gate?

Problem #2

Use Karnaugh maps to help you find the simplest expression of the following functions. Make sure that if you can simplify (use fewer gates) the result given to you by the Karnaugh map, then you should do so!

• f(a, b, c, d) = Σ( 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15 )

• g(a, b, c, d) = Σ( 0, 1, 2, 3, 12, 13, 14, 15 )

• h(a, b, c, d) = Π( 2, 3, 6, 8, 10, 11, 12, 14, 15 )

Problem #3

• We saw in class that a decoder is a circuit that generates all the minterms of the inputs (except for the 74LS42 decoder which generates only 10 of the 16 minterms).
• We also saw that a decoder can have active-high outputs, or active-low outputs.

Question 1

Assume that you have available to you an active-high output, 4-to-16 decoders. Use it along with as few other gates as possible (and, or, not, nand, nor, or xor) to implement the function g below:

g(a, b, c, d) = Σ( 0, 1, 2, 3, 12, 13, 14, 15 )

Question 2

Same as Question 1, but assume this time that the decoder that you have has active-low outputs. Of course you could use an inverter to invert all the minterms of interest and use your answer to Question 1, and it would work. But it could be that you could get a simpler answer with fewer gates using the fact that the outputs are active-low... Figure out what is the most economical diagram in this case. The most economical is the one with the fewest Integrated Circuits (ICs). We'll assume that the 4-to-16 decoder fits in 1 IC.

Submission

• Same procedure as for Homework #2, but this time label you file hw3.pdf, and submit as follows

 rsubmit hw3 hw3.pdf

from your 270b-xx account.