A random process Xt is applied to a network with impulse response ht-te- t- while - is a positive constant- The cross correlation of Xt with the output Yt is RXY-te-u- For - 0-

R_Y(τ)=R_XY(τ).h( τ) =∫_ ∞^∞u(τ+r)(τ +x)e^ α (+x).xe^ α x dx =e^ ατ∫_ ∞^∞x(τ +x)e^ 2xαdx=e^ ατ∫_ ∞^∞(τ x+x^2)e^ 2xαdx =e^α x(1 ατ)4α ^3 R_Y(τ)=e^ α |τ|(1+∞)4α ^ ...

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Byju's Answer

Standard XII

Chemistry

Order of Reaction

A random proc...

Question

A random process $X (t)$ is applied to a network with impulse response $h (t) = t e^{- α (t)}$ , while $α$ is a positive constant. The cross correlation of $X (t)$ with the output $Y (t)$ is
$R_{X Y} (τ) = t e^{- α τ} u (τ) .$ For $τ \leq 0$ ;

| R_{X Y} (τ) | \leq \sqrt{R_{X} (0) . R_{Y} (0)}

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R_{Y} (α) = \frac{e^{- α | τ |} (1 + α τ)}{4 α^{3}}

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Output power is

4 α^{2}

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If mean of

X (t) = α

then output mean

1 / α

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Solution

The correct option is B $R_{Y} (α) = \frac{e^{- α | τ |} (1 + α τ)}{4 α^{3}}$
$R_{Y} (τ) = R_{X Y} (τ) . h (- τ)$
$= \int_{- \infty}^{\infty} u (τ + r) (τ + x) e^{- α (+ x)} . x e^{- α x} d x$
$= e^{- α τ} \int_{- \infty}^{\infty} x (τ + x) e^{- 2 x α} d x = e^{- α τ} \int_{- \infty}^{\infty} (τ x + x^{2}) e^{- 2 x α} d x$
$= e^{α x} \frac{(1 - α τ)}{4 α^{3}}$
$R_{Y} (τ) = e^{- α | τ |} \frac{(1 + \infty)}{4 α^{3}}$
$h (t) \leftrightarrow H (s)$
$t e^{- α t} u (t) \leftrightarrow \frac{1}{(s + α)^{2}} \Rightarrow H (0) = \frac{1}{α^{2}}$
$M_{y} = H (0) . M_{x}$
$= \frac{1}{α^{2}} . α = \frac{1}{α}$
$\Rightarrow {E [X (t) . Y (t + τ)]}^{2} \leq E [X^{2} (t)] \times E [Y^{(} t + τ)]$ $\leftarrow Sctwartz unequality theorem$
$∴ | R_{X Y} (τ) | \leq \sqrt{R_{X} (0) . R_{Y} (0)}$

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A random process X(t) is applied to a network with impulse response h(t)=te−α(t), while α is a positive constant. The cross correlation of X(t) with the output Y(t) is RXY(τ)=te−ατu(τ). For τ≤0;

A random process $X (t)$ is applied to a network with impulse response $h (t) = t e^{- α (t)}$ , while $α$ is a positive constant. The cross correlation of $X (t)$ with the output $Y (t)$ is
$R_{X Y} (τ) = t e^{- α τ} u (τ) .$ For $τ \leq 0$ ;