MIDF07_ANS

# MIDF07_ANS - ECSE-323 Department of Electrical and Computer...

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Question 1 :Boolean Logic Theory (10 points) (5 points) a) Poof that the Shannon Expansion Theorem F(A,B,C,…,Z) = f8e5 A F(0,B,C,…,Z) + A F(1,B,C,…,Z) can be generalized by replacing the + operator in the above expression by the operator, namely, that F(A,B,C,…,Z) = f8e5 A F(0,B,C,…,Z) A F(1,B,C,…,Z) is true. (5 points) b) By using conveniently ONLY the ExOR and AND operators, convert F(A,B,C) = AB + f8e5 C into an all positive function, i.e. a function that does not have any complemented variable. ___________________________________________________________________________ ANSWER a) For A = 0, F(A,B,C,…,Z) = f8e5 A F(0,B,C,…,Z) A F(1,B,C,…,Z) = 1 . F(0,B,C,…,Z) 0 = F(0,B,C,…,Z) 0 = F(0,B,C,…,Z). For A = 1, F(A,B,C,…,Z) = f8e5 A F(0,B,C,…,Z) A F(1,B,C,…,Z) = 0 1. F(1,B,C,…,Z) = 0 F(1,B,C,…,Z) = F(1,B,C,…,Z). The same result is obtained substituting A = 0 and A = 1 in the expression of the Shannon Expansion Theorem, F(A,B,C,…,Z) = f8e5 A F(0,B,C,…,Z) + A F(1,B,C,…,Z). Thus, since the Shannon Expansion Theorem is true, its generalization obtained by replacing the + operator, in the above expression, by the operator is also true. b) We expand F (A,B,C) = AB + f8e5 C with respect to C using the generalized Shannon Expansion Theorem. We obtain F(A,B,C) = f8e5 C [ 1 ] C [AB]. Replacing f8e5 C by 1 C, we obtain F(A,B,C) = 1 C CAB. ECSE-323 Digital Systems Design - Midterm Exam - Fall 2007 Your Name ______________________________________________________________ ______________________________________________________________________________
Question 2 : Application of Boolean Theory (10 points) (5 points) a) The function F(A,B,C,D) = m(0,2,6,16,20,22) has the following prime implicants: A B C D E - 0 1 1 0 - 0 0 0 0 1 0 - 0 0 1 0 1 - 0 0 0 - 1 0 0 0 0 - 0 Give all minimal sum-of-products expressions of F. (5 points) b) Using only the building block realizing the function f(a,b,c) = a b + f8e5 a c, draw a circuit that produces F(A,B,C) = A f8e5 C + f8e5 B C. (The optimal solution uses two such building blocks). ONLY TRUE INPUTS ARE AVIALABLE _________________________________________________________________________ ANSWER

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MIDF07_ANS - ECSE-323 Department of Electrical and Computer...

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