5.
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5. N bit numbers with base K
You are asked to convert the following N bit number with base K to the corresponding number in base K^2.
Xn-1 Xn-2 Xn-3 ... X1 X0
Which of these choices will help you achieve this?
Show answer & explanation
Answer: C. C) Summation ( (K^2)^i * (N/2) x ( KXm-1 + Xm-2) ) where m belongs to N, 1 < m < N
To make a combinational circuit hazard-free, all adjacent prime implicants in the K-map must overlap. By analyzing the K-map for the given function, two additional redundant prime implicants are required to cover the transitions between the existing essential prime implicants.
Step-by-step Derivation:
Step 1: Map the function F(A, B, C, D) = Σ(0, 4, 11, 13, 15) + d(1, 2, 3, 10) on a 4-variable K-map.
Step 2: Identify the Prime Implicants (PIs).
- Group 1: cells (0, 1, 2, 3) -> Term: A'B'
- Group 2: cells (0, 4) -> Term: A'C'D'
- Group 3: cells (11, 15, 10, 14-X) -> Term: AD (Wait, 14 is not given, check 11, 15, 13, 15)
- Group 3: cells (13, 15) -> Term: ABD
- Group 4: cells (11, 15) -> Term: ACD
- Group 5: cells (10, 11) -> Term: AB'C
Step 3: Minimize the function. The essential prime implicants are A'B', ABD, and AB'C (or ACD).
Step 4: Identify hazards. A static-1 hazard occurs when two prime implicants covering adjacent 1s are not overlapped by a third prime implicant. - Transition between A'B' and A'C'D' (at cell 0): Covered.
- Transition between ABD and ACD (at cell 15): Covered.
- Transition between AB'C and ACD (at cell 11): Covered.
- Transition between A'B' and AB'C (at cell 2/3/10/11): There is a gap between the A'B' group and the AB'C group. A redundant term (B'C) is needed to bridge them.
- Transition between A'C'D' and ABD: A redundant term (A'D or similar) is needed to bridge the gap across the map.
Step 5: By calculating the minimal cover and adding the necessary consensus terms to eliminate all hazards, 2 redundant terms are required.