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Byju's Answer

Standard XII

Physics

1st Law of Thermodynamics

A zero mean w...

Question

$A zero mean white Gaussian process X (t), with two-sided power spectral density of \frac{N_{0}}{2}, is applied to a real LTI system whose impulse response is h (t) . The auto-correlation function of the corresponding output process R_{Y} (τ) can be expressed as$

\frac{N_{0}}{2} h^{2} (τ)

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\frac{N_{0}}{2} h (τ)

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\frac{N_{0}}{2} h (τ) * h (τ)

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\frac{N_{0}}{2} h (τ) * h (- τ)

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Solution

The correct option is D $\frac{N_{0}}{2} h (τ) * h (- τ)$
$S_{X} (f) = \frac{N_{0}}{2}$

$R_{X} (τ) = \frac{N_{0}}{2} δ (τ)$

$R_{Y} (τ) = R_{X} (τ) * h (τ) * h (- τ)$

$= \frac{N_{0}}{2} δ (τ) * h (τ) * h (- τ)$

$= \frac{N_{0}}{2} h (τ) * h (- τ)$

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Using the given pattern, find the missing numbers.

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A zero mean white Gaussian process X(t), with two-sided power spectral density of N02,is applied to a real LTI system whose impulse response is h(t).The auto-correlation function of the corresponding output process RY(τ) can be expressed as

The correct option is D N02h(τ)∗h(−τ)SX(f)=N02 RX(τ)=N02δ(τ) RY(τ)=RX(τ)∗h(τ)∗h(−τ) =N02δ(τ)∗h(τ)∗h(−τ) =N02h(τ)∗h(−τ)

$A zero mean white Gaussian process X (t), with two-sided power spectral density of \frac{N_{0}}{2}, is applied to a real LTI system whose impulse response is h (t) . The auto-correlation function of the corresponding output process R_{Y} (τ) can be expressed as$

The correct option is D $\frac{N_{0}}{2} h (τ) * h (- τ)$
$S_{X} (f) = \frac{N_{0}}{2}$

$R_{X} (τ) = \frac{N_{0}}{2} δ (τ)$

$R_{Y} (τ) = R_{X} (τ) * h (τ) * h (- τ)$

$= \frac{N_{0}}{2} δ (τ) * h (τ) * h (- τ)$

$= \frac{N_{0}}{2} h (τ) * h (- τ)$