# Questions & Answers of Laplace Transform and z-Transform

## Weightage of Laplace Transform and z-Transform

Total 21 Questions have been asked from Laplace Transform and z-Transform topic of Signals and Systems subject in previous GATE papers. Average marks 1.62.

The pole-zero plots of three discrete-time system P, Q and R on the z-plane are shown below.

Which one of the following is TRUE about the frequency selectivity of these system?

The Laplace Transform of $f\left(t\right)={e}^{2t}\mathrm{sin}\left(5t\right)u\left(t\right)$ is

Let $S=\sum_{n=0}^\infty\limits n\alpha^n\;\mathrm{where}\;\left|\mathrm\alpha\right|<1$. The value of $\alpha$ in the range $0<\alpha <1$ such that $S=2\alpha$ is _______.

Consider a causal LTI system characterized by differential equation $\frac{dy\left(t\right)}{dt}+\frac{1}{6}y\left(t\right)=3x\left(t\right)$. The response of the system to the input $x\left(t\right)=3{e}^{-\frac{t}{3}}u\left(t\right)$, where u(t) denotes the unit step function, is

The solution of the differential equation, for $t>0,y\text{'}\text{'}\left(t\right)+2y\text{'}\left(t\right)+y\left(t\right)=0$ with initial conditions $y\left(0\right)=0$ and $y\text{'}\left(0\right)=1$ is($u\left(t\right)$ denotes the unit step function),

Consider a linear time-invariant system with transfer function

$H\left(s\right)=\frac{1}{\left(s+1\right)}$

If the input is $\mathrm{cos}\left(t\right)$ and the steady state output is $A\mathrm{cos}\left(t+\alpha \right)$, then the value of $A$ is _________.

Consider a discrete time signal given by

The region of convergence of its Z-transform would be

The Laplace transform of $f\left(t\right)=2\sqrt{t/\pi}$  is  $s^{-3/2}$.  The Laplace transform of $g\left(t\right)=\sqrt{1/\pi t}$  is

The z-Transform of a sequence $x\left[n\right]$ is given as $X\left(z\right)=2z+4-4/z+3/z^2$. If $y\left[n\right]$ is the first difference of $x\left[n\right]$ , then $Y\left(z\right)$ is given by

Let $X\left(Z\right)=\frac{1}{1-{z}^{-3}}$ be the Z-transform of a causal signal $x\left[n\right]$. Then, the values of $x\left[2\right]$ and $x\left[3\right]$ are

The transfer function $\frac{{V}_{2}\left(s\right)}{{V}_{1}\left(s\right)}$ of the circuit shown below is

The impulse response of a continuous time system is given by h(t) =$\delta$(t -1) +$\delta$(t - 3) . The value of the step response at t = 2 is

If $x\left[n\right]={\left(1/3\right)}^{\left|n\right|}-{\left(1/2\right)}^{n}u\left[n\right]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

Let the Laplace transform of a function $f\left(t\right)$ which exists for $t>0$ be ${F}_{1}\left(s\right)$ and the Laplace transform of its delayed version $f\left(t-\tau \right)$ be ${F}_{2}\left(s\right)$. Let ${F}_{1}^{*}\left(s\right)$ be the complex conjugate of ${F}_{1}\left(s\right)$ with the Laplace variable set as $s=\sigma +j\omega$. If $G\left(s\right)=\frac{{F}_{2}\left(s\right).{F}_{1}*\left(s\right)}{{\left|{F}_{1}\left(s\right)\right|}^{2}},$ then the inverse Laplace transform of $G\left(s\right)$ is

Given f(t) and g(t) as shown below:

g(t) can be expressed as

Given f(t) and g(t) as shown below:

The Laplace transform of g(t) is

The z-transform of a signal x[n] is given by 4z-3 + 3z-1 + 2 - 6z2 + 2z3. It is applied to a system, with a transfer function H(z) = 3z-1 - 2. 2. Let the output be y(n). Which of the following is true?

H(z) is a transfer function of a real system. When a signal x[n]=(1+j)n is the input to such a system, the output is zero. Further, the Region Of convergence (ROC) of

$\left(1-\frac{1}{2}{z}^{-1}\right)$H(z) is the entire Z-plane (except z=0). It can then be inferred that H(z) can have a minimum of

Given the residue of X(z)zn-1 at z=a for $n\ge 0$ will be

X(z) = 1 - 3z-1, Y(z) = 1 + 2z-2 are Z transforms of two signals x[n], y[n] respectively. A linear time invariant system has the impulse response h[n] defined by these two signals as h[n] = x[n-1]*y[n] where * denotes discrete time convolution. Then the output of the system for the input $\delta$[n-1]