Two signals g1(t) and g2(t) have average powers 4 and 5, respectively, and their zero-lag cross-correlation is 6. What is the average power of the sum signal g1(t) + g2(t)?
Electronics and Communication Engineering
Signals and Systems
Difficulty: Medium
Choose an option
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A9
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B21
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C3
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D15
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E11
Answer
Correct Answer: 21
Explanation
Introduction / Context:In signal analysis, the average power of a sum of signals depends on the individual powers and their correlation. This question checks understanding of how cross-correlation (at zero lag) contributes to the composite power of two simultaneous signals.
Given Data / Assumptions:
- P1 = average power of g1(t) = 4.
- P2 = average power of g2(t) = 5.
- R12(0) = zero-lag cross-correlation between g1 and g2 = 6.
- Signals are wide-sense stationary or time averages are meaningful for power calculation.
Concept / Approach:The time-average power of the sum is P{g1 + g2} = P1 + P2 + 2R12(0). The last term arises from the cross term 2g1(t)g2(t) when squaring and averaging. If R12(0) > 0, the sum’s power increases by twice the correlation value.
Step-by-Step Solution:
Start from (g1 + g2)^2 = g1^2 + g2^2 + 2g1 g2.Average both sides: P{g1 + g2} = P1 + P2 + 2 E[g1 g2].Given E[g1 g2] at zero lag = R12(0) = 6.So P{g1 + g2} = 4 + 5 + 2*6 = 21.Verification / Alternative check:
If the signals were uncorrelated (R12(0) = 0), power would be 9. Positive correlation makes it larger, consistent with 21.Why Other Options Are Wrong:
9: ignores correlation term.3 or 15: not consistent with formula P1 + P2 + 2R12(0).11: would correspond to R12(0) = 1, not 6.Common Pitfalls:
Confusing correlation (expected product) with covariance; omitting the factor 2.Final Answer:
21