Consider a causal LTI system defined by the following differential equation:\[\frac{d^2 y(t)}{dt^2} + 3 \frac{dy(t)}{dt} = \frac{dx(t)}{dt} + x(t).\](a) **Sketch the pole-zero plot for this system.**A pole-zero plot graphically represents the poles and zeros of a transfer function in the complex plane. In this system, the poles occur at the roots of the characteristic equation obtained by transforming the differential equation using the Laplace transform.(b) **Find and sketch the impulse response \( h(t) \) of this system.**The impulse response \( h(t) \) is the output of the system when the input is a Dirac delta function. This response can be found by calculating the inverse Laplace transform of the system's transfer function.(c) **Find the response of this system when the input is \( x(t) = 4e^{-3t}u(t) \).**The response of the system to this input can be solved using the convolution of \( x(t) \) with the impulse response \( h(t) \). This involves integrating the product of the input and the impulse response over time. The computation will require evaluating and simplifying the integral.

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Consider a causal LTI system defined by the following differential equation d'y(t) dy(t) dx(t) +3 dt² dt dt (a) Sketch the pole-zero plot for this system. (b) Find and sketch the impulse response h(t) of this system. (c) Find the response of this system when the input is r(t) = 4e-³tu(t). + x(t).

Consider a causal LTI system defined by the following differential equation d'y(t) dy(t) dx(t) +3 dt² dt dt (a) Sketch the pole-zero plot for this system. (b) Find and sketch the impulse response h(t) of this system. (c) Find the response of this system when the input is r(t) = 4e-³tu(t). + x(t).

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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Question

Consider a causal LTI system defined by the following differential equation:

\[
\frac{d^2 y(t)}{dt^2} + 3 \frac{dy(t)}{dt} = \frac{dx(t)}{dt} + x(t).
\]

(a) **Sketch the pole-zero plot for this system.**

A pole-zero plot graphically represents the poles and zeros of a transfer function in the complex plane. In this system, the poles occur at the roots of the characteristic equation obtained by transforming the differential equation using the Laplace transform.

(b) **Find and sketch the impulse response \( h(t) \) of this system.**

The impulse response \( h(t) \) is the output of the system when the input is a Dirac delta function. This response can be found by calculating the inverse Laplace transform of the system's transfer function.

(c) **Find the response of this system when the input is \( x(t) = 4e^{-3t}u(t) \).**

The response of the system to this input can be solved using the convolution of \( x(t) \) with the impulse response \( h(t) \). This involves integrating the product of the input and the impulse response over time. The computation will require evaluating and simplifying the integral.

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