# hw3sol - ECEN 248 Introduction to Digital System Design...

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ECEN 248: Introduction to Digital System Design Department of Electrical Engineering Texas A&M University Assignment #3 Solutions 1. Suppose that a k -input gate has a delay of k seconds, and an area of k units. This data is valid for AND, OR, NAND, NOR gates of any number of inputs, for XOR gates of up to 2 inputs, and for INV gates. (a) What is the area and delay of a 12-input XOR function, implemented as a minimum SOP? (b) What is the area and delay of a 12-input XOR function, implemented using 2-input XOR gates? Solution. (a) As we saw in class, the output of an k input XOR gate is 1 whenever an odd number of its inputs are 1. Also, each of its product terms is such that it cannot be combined (using the combining theorem) with any other product term. Hence the size of the SOP of a XOR gate is the same as the size of the CSOP for such a gate. A k input XOR gate has 2 k - 1 product terms. Therefore a 12 input XOR has 2 11 product terms. This means we need 2 11 AND gates in the minimum SOP, each with 12 inputs. We also need each variable in its complemented form, which means we need 12 INV gates. Finally we need a single OR gate with 2 11 inputs.

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