# Laplace Transform

 Question 1
Which of the following statements is true about the two sided Laplace transform?
 A It exists for every signal that may or may not have a Fourier transform. B It has no poles for any bounded signal that is non-zero only inside a finite time interval. C The number of finite poles and finite zeroes must be equal. D If a signal can be expressed as a weighted sum of shifted one sided exponentials, then its Laplace Transform will have no poles.
GATE EE 2020   Signals and Systems
Question 1 Explanation:
It has no poles for any bounded signal that is nonzero in a finite time interval. This is true as we know for finite amplitude finite width signal ROC is entire s plane and ROC never includes any pole.
It implies for such signals there is no poles. Hence the correct answer is option (B).
 Question 2
The output response of a system is denoted as y(t), and its Laplace transform is given by
$Y(s)=\frac{10}{s(s^2+s+100\sqrt{2})}$
The steady state value of y(t) is
 A $\frac{1}{10\sqrt{2}}$ B $10\sqrt{2}$ C $\frac{1}{100\sqrt{2}}$ D $100\sqrt{2}$
GATE EE 2019   Signals and Systems
Question 2 Explanation:
\begin{aligned} &=\lim_{s \to 0}sY(s)\\ &=\lim_{s \to 0}\frac{10s}{s(s^2+s+100\sqrt{2})}\\ &=\frac{10}{100\sqrt{2}}=\frac{1}{10\sqrt{2}} \end{aligned}

 Question 3
A system transfer function is $H(s)=\frac{a_1s^2+b_1s+c_1}{a_2s^2+b_2s+c_2}$. If $a_1=b_1=0$, and all other coefficients are positive, the transfer function represents a
 A low pass filter B high pass filter C band pass filter D notch filter
GATE EE 2019   Signals and Systems
Question 3 Explanation:
\begin{aligned} H(s)&=\frac{c_1}{a_2s^2+b_2s+c_2}\\ & as \;\; a_1=b_1=0\\ &=\frac{c_1}{(1+s\tau _1)(1+s\tau _2)}\\ \text{Put }s&=0,\; H(0)=\frac{c_1}{c_2}\\ \text{Put }s&=\infty ,\; H(\infty )=0 \end{aligned}

which represents second order low pass filter.
 Question 4
The inverse Laplace transform of
$H(s)=\frac{s+3}{s^2+2s+1} \; for \; t\geq 0$ is
 A $3te^{-t}+e^{-t}$ B $3e^{-t}$ C $2te^{-t}+e^{-t}$ D $4te^{-t}+e^{-t}$
GATE EE 2019   Signals and Systems
Question 4 Explanation:
\begin{aligned} L^{-1}\left ( \frac{s+3}{s^2+2s+1} \right )&=L^{-1}\left ( \frac{s+1+2}{(s+1)^2} \right )\\ &=L^{-1}\left ( \frac{1}{s+1}+\frac{2}{(s+1)^2} \right )\\ &=e^{-t}+2te^{-t} \end{aligned}
 Question 5
Consider a linear time-invariant system with transfer function
$H(s)=\frac{1}{(s+1)}$
If the input is cos(t) and the steady state output is $A \cos (t+\alpha )$, then the value of A is _________.
 A 0.7 B 0.26 C 0.96 D 1.2
GATE EE 2016-SET-2   Signals and Systems
Question 5 Explanation:
\begin{aligned} H(s)&=\frac{1}{(s+1)}\\ \text{Put, }s&=j\omega \\ H(j\omega)&=\frac{1}{j\omega+1}\\ |H(j\omega)|&=\frac{1}{\sqrt{\omega ^2 +1}}\\ \because \; \text{ input }x(t)&=\cos (t)\\ \text{Here, } \omega &=1 \text{rad/sec}\\ \text{and }|x(t)|&=1\\ \text{hence, }&\text{steady state output}\\ y(t)&=|H(j\omega)|_{\omega=1} \cos (t+\angle H(j\omega)|_{\omega =1})\\ A&=|H(j\omega)|_{\omega =1}=\frac{1}{\sqrt{2}}=0.707 \end{aligned}

There are 5 questions to complete.