A voltage V(t) is a Gaussian, ergodic random process with mean 0 and variance 4 V^2. If measured with a direct-current (DC) meter (i.e., time average reading), what value will the meter indicate?
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A0
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B4
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C2
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D2√2
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E√2
Answer
Correct Answer: 0
Explanation
Introduction / Context:Understanding what various meters measure for random processes is fundamental in measurement and instrumentation. A DC (average-responding) meter indicates the time-average (mean) value of the input signal. For zero-mean random processes, this has a straightforward implication.
Given Data / Assumptions:
- V(t) is Gaussian and ergodic.
- Mean μ = 0 V.
- Variance σ^2 = 4 V^2 (so rms = √σ^2 = 2 V, but rms is not what a DC meter reads).
- DC meter displays time average (mean) for an ergodic process.
Concept / Approach:For an ergodic process, the time average equals the ensemble average. A DC meter, which essentially performs a long-time average, therefore indicates the process mean. Since μ = 0, the meter reading is 0 V. Note this differs from an rms-responding meter, which would show 2 V for the given variance.
Step-by-Step Solution:
DC meter → time average → equals ensemble mean for ergodic processes.Given mean μ = 0 → meter reading = 0 V.Verification / Alternative check:
If an rms meter were used, the indicated value would be √(E[V^2]) = √(σ^2) = 2 V, highlighting the difference between average and rms measurements.Why Other Options Are Wrong:
4: This is the variance, not a DC average.2 or 2√2: These relate to rms or peak equivalents, not DC average.√2: No measurement mode here yields √2 V directly.Common Pitfalls:
Confusing mean (average) with rms; instruments can be average-responding or true-rms—know which is used.Final Answer:
0