Mallard ECE 290_ Computer Engineering I - Spring 2009 - HWK #5 Solution

Mallard ECE 290_ Computer Engineering I - Spring 2009 - HWK #5 Solution

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3/29/09 8:31 PM Mallard ECE 290: Computer Engineering I - Spring 2009 - HWK #5 Solution Page 1 of 9 HWK #5 Solution Problem 5.1 For convenience, let's first write the functions in canonical SOP form and draw the K-maps: f(x,y,z) = AND(M1,M4,M6) = OR(m0,m2,m3,m5,m7) g(x,y,z) = OR(m1,m3,m5) h(x,y,z) = x XOR z = x'z + xz' = OR(m1,m3,m4,m6) yz 00 01 11 10 x 0 1 0 1 1 0 1 1 0 1 f yz 00 01 11 10 x 0 0 1 1 0 0 1 0 0 1 g yz 00 01 11 10 x 0 0 1 1 0 1 0 0 1 1 h a. With x, y, and z at the inputs of our 3:8 decoder, the decoder generates all 8 minterms. We can then simply OR the appropriate minterms together to realize each function. It is easy to implement h(x,y,z) = OR(m1,m3,m4,m6): use a 4-input OR gate. We only need a 3-input OR gate to implement g(x,y,z) = OR(m1,m3,m5). But we can easily use a 4- input OR gate by feeding "0" into the 4th OR input (or feed one of the 3 OR-gate inputs into the 4th input as well). How would we implement f? We can't use a 4-input OR gate, because f has 5 minterms. But we can use the NOR gate to implement f ' = OR(m1,m4,m6) and invert it. b. Any three-variable function can be implemented with a 4:1 MUX plus an inverter. We use y and z as the inputs to the select lines of the MUXes. So the only possible MUX data inputs are 0, 1, x, or x', and a single inverter is needed to compute x'. We can check the K-map to see the appropriate function to enter in each MUX input.
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3/29/09 8:31 PM Mallard ECE 290: Computer Engineering I - Spring 2009 - HWK #5 Solution Page 2 of 9 c. The ROM size is 8x3 (8 words with 3 bits per word). The ROM programming is analogous to the truth table. ROM Truth Table x y z f g h 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 Problem 5.2 We begin by drawing the K-maps for F, G, and H. Examining the K-maps for F, G, and H, we observe that it will take at least 4 product terms to implement F,
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Mallard ECE 290_ Computer Engineering I - Spring 2009 - HWK #5 Solution

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