### Problem 6Prove that if a continuous-time white noise random process with autocorrelation function (ACF) \( r_V(\tau) = (N_0/2)\delta(\tau) \) is input to a linear time-invariant (LTI) system with impulse response \( h(\tau) \), then the ACF of the output random process is given by:\[r_X(\tau) = \frac{N_0}{2} \int_{-\infty}^{\infty} h(t)h(t+\tau)dt.\]#### Explanation- **Continuous-time white noise random process**: A type of signal that has equal intensity at different frequencies, giving it a constant power spectral density. - **Autocorrelation function (ACF)**: A mathematical representation of the correlation between different instances of a random process as a function of time lag (\( \tau \)).- **Impulse response \( h(\tau) \)**: Describes the output of an LTI system in time when presented with a very short input signal at time zero (impulse).- **Linear Time-Invariant (LTI) system**: A system with properties of linearity and time invariance, meaning its output is directly proportional to its input and parameters do not change over time. This problem requires showing the relationship between the input white noise process, the LTI system's impulse response, and the output process's ACF.

Answered: Image /qna-images/answer/54265c03-e34f-4a8c-8d6e-35c29c748483.jpg

Prove that if a continuous-time white noise random process with ACF rū(t) = (№₁/2)8(t) is input to an LTI system with impulse response h(t), then the ACF of the output random process is No rx(t) = N * h(t)h(t + t)dt. 2 <-00

Prove that if a continuous-time white noise random process with ACF rū(t) = (№₁/2)8(t) is input to an LTI system with impulse response h(t), then the ACF of the output random process is No rx(t) = N * h(t)h(t + t)dt. 2 <-00

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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Question

### Problem 6

Prove that if a continuous-time white noise random process with autocorrelation function (ACF) \( r_V(\tau) = (N_0/2)\delta(\tau) \) is input to a linear time-invariant (LTI) system with impulse response \( h(\tau) \), then the ACF of the output random process is given by:

\[
r_X(\tau) = \frac{N_0}{2} \int_{-\infty}^{\infty} h(t)h(t+\tau)dt.
\]

#### Explanation

- **Continuous-time white noise random process**: A type of signal that has equal intensity at different frequencies, giving it a constant power spectral density.

- **Autocorrelation function (ACF)**: A mathematical representation of the correlation between different instances of a random process as a function of time lag (\( \tau \)).

- **Impulse response \( h(\tau) \)**: Describes the output of an LTI system in time when presented with a very short input signal at time zero (impulse).

- **Linear Time-Invariant (LTI) system**: A system with properties of linearity and time invariance, meaning its output is directly proportional to its input and parameters do not change over time.

This problem requires showing the relationship between the input white noise process, the LTI system's impulse response, and the output process's ACF.

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