Hw4_Solutions_2009_vivek

# Hw4_Solutions_2009_vivek - Homework#4 ECE 15a 1 p q = pq pq...

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Homework #4 ECE 15a Winter 2009 1. p q = pq’ + p’q 2. a. Expand the following diagram (only XOR and AND) to obtain the expression for p + q: p q p’q p q pq’ p q pq’ + p’q

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The above diagram can be expressed as, [p Θ (pq) ] Θ q ------------ 1 It can be shown that the expression 1, can be simplified to p+q: = [p’ (pq) + p (pq)’] Θ q = [0 + p(p’+q’)] Θ q = [pq’] Θ q = (pq’)’q + (pq’)q’ = (p’+q)q + pq’ = p’q + q + pq’ = q(p’+1)+pq’ = q + pq’ = q + p b. (p+q) (p’+q’) = pp’ + pq’ + p’q + qq’ = pq’ + p’q = p Θ q p q
3. a. f(A,B,C) = A’C’+B’+AC’ Show f, can be used as a NOT gate: f(A,B,C) = A’C’+B’+AC’ = C’(A’+A) + B’ = C’+B’ = (BC)’ ---------- 1 Note: Equation 1, represents NAND Gate, which is a functionally complete gate that can be used to construct AND, OR and NOT. In the following sections we still prove that the function ‘f’can be used to construct NOT, AND and OR gates. Consider B=C, Eqn. 1 becomes, C’ This shows that NOT gate can be implemented using the given function f. Not gate can be represented as follows: Given Function: NOT Gate f (a,b,c) a b c a’c’ + b’ + ac’ f (a,b,c) c c’ b

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Show f, can be used as a AND gate: It can be shown that inverting Eqn.1. using the not gate obtained above, AND gate can be formed using the function f. Show f, can be used as a OR gate: Since we can build OR, AND and NOT gates using f, the function is complete. b. Build a 2-input XOR gate from the f-gates defined above. The XOR gate can be constructed using AND, OR and NOT as follows: After simplification f (a,b,c) a’c’ + b’ + ac’ f (a,b,c) NAND Gate (bc)’ f (a,b,c) bc NOT Gate b’ b f (a,b,c) NOT Gate c f (a,b,c) NOT Gate c’ f (a,b,c) NAND Gate (b’c’)’ = b + c
Since all the gates in the above figure have been already realized in (a), the 2-input XOR function can also be realized. 4. y’z’+xy’+xz’ = xy’+x’y’z’+xyz’

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## This note was uploaded on 03/02/2009 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

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Hw4_Solutions_2009_vivek - Homework#4 ECE 15a 1 p q = pq pq...

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