pLease help me for the question in the image 4) Consider the system with the following input-output relation with x(t) and y(t) being input andoutput respectively.2ty(t) = x(t)drDetermine whether the system is memoryless, time invariant, linear, causal and stable.

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Description: pLease help me for the question in the image 4) Consider the system with the following input-output relation with x(t) and y(t) being input andoutput respectively.2ty(t) = x(t)drDetermine whether the system is memoryless, time invariant, linear, caus

Answered: 4) Consider the system with the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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pLease help me for the question in the image

4) Consider the system with the following input-output relation with x(t) and y(t) being input and
output respectively.
2t
y(t) = x(t)dr
Determine whether the system is memoryless, time invariant, linear, causal and stable.

Expert Solution

Step 1

This is not memory less because it depends on the past.

This is not time invariant because

$\begin{array}{rcl} \int_{- \infty}^{2 t} x (τ - t_{0}) d τ & = & \int_{- \infty}^{2 t - t_{0}} x (τ) d τ \\ \neq & y (t - t_{0}) \\ = & \int_{- \infty}^{2 (t - t_{0})} x (τ) d τ \end{array}$

The function given is linear :

$a x_{1} (t) + b x_{2} (t) \to a y_{1} (t) + b y_{2} (t)$

It is not casual as it depends on time which is a future input.

$y (1) = \int_{- \infty}^{2} x (τ) d τ$

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