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 ^ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

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)}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Output power is

4 α^{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

If mean of

X (t) = α

then output mean

1 / α

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

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

Suggest Corrections

Similar questions

Q. If x(t) is signal applied as input to LTI system with impulse response h(t). The output of the system is y(t) then value of

y (t) |_{t = 5}

is ______.

Q. 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

Q. A stationary random process X(t) is applied to an LTI system whose impilse response is

h (t) = e^{- 2 t^{2}}

The output process of this LTI system is Y(t). If the mean value of the process X(t) is 2, then the mean value of the process Y(t) will be ______

Q. Consider the following table for an LTI system having an impulse response h(t).

Input	Output
x(t)	y(t)
$\frac{d}{d t} x (t)$	$- 3 y (t) + e^{- 2 t} u (t)$

where,

x (t) = 2 e^{- 3 t} u (t - 1)

Then the impulse response h(t) of the system is

Q. A singal x(t) =

\frac{5 s i n 5 t}{π t}

is applied as input to a system with h(t) =

10 \frac{s i n 25 t}{π t}

, if y(t) is output, then

Join BYJU'S Learning Program

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 $τ \leq 0$ ;