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Consider a random process X(t) = 3V(t) − 8, where (t) is a zero mean stationary random process with autocorrelation RV(τ) = 4e−5|τ|. The power in X(t) is ________
Correct answer is '100'. Can you explain this answer?
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Consider a random processX(t) = 3V(t) − 8, where (t) is a zero m...
Understanding the Random Process X(t)
The random process is defined as:
X(t) = 3V(t) - 8
where V(t) is a zero mean stationary random process.
Power Calculation of X(t)
To find the power of the process X(t), we need to determine its mean square value.
1. **Mean of X(t)**:
Since V(t) has a zero mean, the mean of X(t) is:
- E[X(t)] = E[3V(t) - 8] = 3E[V(t)] - 8 = 0 - 8 = -8
2. **Mean Square Value of X(t)**:
The mean square value is calculated as:
- E[X(t)^2] = E[(3V(t) - 8)^2]
- Expanding this:
- E[X(t)^2] = E[9V(t)^2 - 48V(t) + 64]
- Since E[V(t)] = 0, the middle term vanishes:
- E[X(t)^2] = 9E[V(t)^2] + 64
3. **Finding E[V(t)^2]**:
The power of the random process V(t) is equal to its autocorrelation at zero lag:
- R_V(0) = E[V(t)^2] = 4e^0 = 4
4. **Substituting E[V(t)^2] Back**:
- E[X(t)^2] = 9(4) + 64 = 36 + 64 = 100
Conclusion
Thus, the power in the process X(t) is:
- **Power of X(t)** = 100
The random process is defined as:
X(t) = 3V(t) - 8
where V(t) is a zero mean stationary random process.
Power Calculation of X(t)
To find the power of the process X(t), we need to determine its mean square value.
1. **Mean of X(t)**:
Since V(t) has a zero mean, the mean of X(t) is:
- E[X(t)] = E[3V(t) - 8] = 3E[V(t)] - 8 = 0 - 8 = -8
2. **Mean Square Value of X(t)**:
The mean square value is calculated as:
- E[X(t)^2] = E[(3V(t) - 8)^2]
- Expanding this:
- E[X(t)^2] = E[9V(t)^2 - 48V(t) + 64]
- Since E[V(t)] = 0, the middle term vanishes:
- E[X(t)^2] = 9E[V(t)^2] + 64
3. **Finding E[V(t)^2]**:
The power of the random process V(t) is equal to its autocorrelation at zero lag:
- R_V(0) = E[V(t)^2] = 4e^0 = 4
4. **Substituting E[V(t)^2] Back**:
- E[X(t)^2] = 9(4) + 64 = 36 + 64 = 100
Conclusion
Thus, the power in the process X(t) is:
- **Power of X(t)** = 100
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Community Answer
Consider a random processX(t) = 3V(t) − 8, where (t) is a zero m...
Concept:
ACF is defined as:
Rx(τ) = E[x(t) x(t + τ)] = E[x(t) x(t - τ)]
Properties of ACF:
1. Rx(-τ) = Rx (τ)
2. Rx (0) = E [x2(t)] = Power of x(t)
Calculation:
Given:
x(t) = 3 V(t) – 8, Rv(τ) = 4e-5|τ|
E [V(t)] = 0
We know that,
Power of x(t) = E[x2(t)]
= E[9V2(t) + 64 – 48 E[V(t)]]
= 9E [V2(t)] + 64 – 48 E[V(t)]
Now,
E[V2(t)] = Rv(0) = 4
So,
Power of x(t) = (9 × 4) + 64 – 0
Power of x(t) = 100
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Consider a random processX(t) = 3V(t) − 8, where (t) is a zero mean stationary random process with autocorrelationRV(τ) = 4e−5|τ|. The power inX(t)is ________Correct answer is '100'. Can you explain this answer?
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