Consider a random process Yt defined as Yt-Xt-Xt- where Xt is stationary non-periodic process and - is constant- If RX- is the autocorrelation of random process Xt- then the variance of Yt will be

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Standard XII

Biology

Blackman's Law of Limiting Factor

Consider a ra...

Question

Consider a random process $Y (t)$ defined as $Y (t) = X (t) - X (t + τ)$ where $X (t)$ is stationary non-periodic process and $^{'} τ^{'}$ is constant. If $R_{X} (τ)$ is the autocorrelation of random process $X (t)$ , then the variance of $Y (t)$ will be

2 R_{X} (0) - R_{X} (τ)

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2 R_{X} (0)

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2 [R_{X} (0) - R_{X} (τ)]

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2 R_{X} (τ)

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Solution

The correct option is C $2 [R_{X} (0) - R_{X} (τ)]$
$Y (t) = X (t) - X (t + τ)$ and $¯ ¯¯¯¯¯¯¯¯¯¯¯ ¯ X^{2} (t) = R_{X} (0)$
Also, we have to obtain the mean value of the random process $Y (t)$ as
$E [Y (t)] = 0$
$σ_{x}^{2} = E [Y^{2} (t)] - E [Y (t)]^{2} = E [[X (t) - X (t + τ)]^{2}]$
$= E [X^{2} (t)] - 2 E [X (t) . X (t + τ)] + E [X^{2} (t + τ)]$
$= R_{X} (0) - 2 R_{X} (τ) + R_{X} (0) = 2 R_{X} (0) - 2 R_{X} (τ)$
$= 2 [R_{X} (0) - R_{X} (τ)]$

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