Chapter 5, Problem 5.6E

Interpretation Introduction

(a)

Interpretation:

The value of $y\left(t\to \infty \right)$ for a step input of height $M$ is to be determined.

Concept introduction:

For a large value of time, the asymptotic value of $y\left(t\right)$ can be calculated using Final Value theorem (FVM) as shown below:

$\underset{t\to \infty }{\lim }y\left(t\right)=\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$   ... (1)

This theorem is applicable only if $\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$ exists for all values of $\mathrm{Re}\left(s\right)\ge 0$.

Interpretation Introduction

(b)

Interpretation:

The value of $y\left(t\to \infty \right)$ for a unit impulse input $\delta \left(t\right)$ is to be determined.

Concept introduction:

For a large value of time, the asymptotic value of $y\left(t\right)$ can be calculated using Final Value theorem (FVM) as shown below:

$\underset{t\to \infty }{\lim }y\left(t\right)=\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$   ... (1)

This theorem is applicable only if $\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$ exists for all values of $\mathrm{Re}\left(s\right)\ge 0$.

Interpretation Introduction

(c)

Interpretation:

The value of $y\left(t\to \infty \right)$ for input $\sin t$ is to be determined.

Concept introduction:

For a large value of time, the asymptotic value of $y\left(t\right)$ can be calculated using Final Value theorem (FVM) as shown below:

$\underset{t\to \infty }{\lim }y\left(t\right)=\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$   ... (1)

This theorem is applicable only if $\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$ exists for all values of $\mathrm{Re}\left(s\right)\ge 0$.

Interpretation Introduction

(d)

Interpretation:

The value of $y\left(t\to \infty \right)$ for a unit rectangular pulse with $h=1$ is to be determined.

Concept introduction:

For a large value of time, the asymptotic value of $y\left(t\right)$ can be calculated using Final Value theorem (FVM) as shown below:

$\underset{t\to \infty }{\lim }y\left(t\right)=\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$   ... (1)

This theorem is applicable only if $\underset{s\to 0}{\lim }\left[sY\left(s\right)\right]$ exists for all values of $\mathrm{Re}\left(s\right)\ge 0$.