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110年 - 110 國立臺灣大學_碩士班招生考試_電子工程研究所丁組:邏輯設計#101259
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Problem 1: (20 points)
(b) (4%) Is the set consisting of only majority gate MAJ functionally complete? Justify your answer. (Note that the majority gate MAJ(a,b,c) with three inputs a, b, c equals 1 if and only if at least two of the inputs are 1.)
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(a) (4%) Show that the set ( AND, OR, NOT) of logic operations is functionally complete, that is, any Boolean function can be represented with this set of logic operations.
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(c) (4%) Is the set consisting of only minority gate MIN functionally complete? Justify your answer. (Note that the minority gate MIN(a,b,c) with three inputs a, b, c equals 1 if and only if at most one of the inputs is 1.)
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(d) (4%) Can any Boolean function be represented in the exclusive-sun-of products (ESOP) expression, which is the AND-XOR two-level expression (with first level AND and second level XOR)? Justify your answer.
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(e)(4%) Can any Boolean function be represented in the exclusive-product-of-suns (EPOS) expression, which is the XOR-AND two-level expression (with first level XOR and second level AND)? Justify your answer.
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(a) (10%) Prove by induction that it represents an odd-parity function, that is, it evaluates to 1 if and only if the assignment to variables X1,...,Xn contains an odd number of 1's. (For example, the assignment (X1, X2, X3, X4)= (1,0,0,0) contains one 1.)
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(b) (5%) How many product terms are there for its minimum sum-of-products (SOP) expression?
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(c) (5%) How many sum terms are there for its minimum product-of-sums (POS) expression?
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(a) (5%) Determine the minimum clock cycle that the circuit can operate.
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(b) (5%) Determine the constraint on the flip-flop hold time for the circuit to operate correctly.
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(c) (5%) Assume the combinational part of the circuit is to be re-implemented by a single read-only memory (ROM). Determine the size of the ROM in terms of 1) the number of words and 2) the bit-width of each word.
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110年 - 110 國立臺灣大學_碩士班招生考試_電子工程研究所丁組:邏輯設計#101259
110年 · #101259
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