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

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McGILL UNIVERSITY Department of Electrical and Computer Engineering ECSE-323 Fall 2007 MIDTERM EXAM Question Maximum Points Points Attained 1 10 2 10 3 15 4 15 5 15 6 10 Total 75 points Please write down your name: ANSWER KEY Please write your student ID: ______________________________________ Instructions/Please read carefully! This is a close book exam. No books or notes are allowed. You may use a standard calculator. All work is to be done on the attached sheets and under no circumstance are booklets or loose sheets to be used. Write your name at the top of every sheet. Read the question carefully. If something appears ambiguous, write down your assumption. The points have been assigned according to the formula that 1 point = 1 exam minute, so please pace yourself accordingly.
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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 ______________________________________________________________ ______________________________________________________________________________
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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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