Laplace Transform Practice Questions

Consider the following statements for continuous-time linear time invariant (LTI) systems. I. There is no bounded input bounded output (BIBO) stable system with a pole in the righthalf of the complex plane. II. There is non causal and BIBO stable system with a pole in the right half of the complex plane. Which one among the following is correct?

The transfer function of a causal LTI system is H(s) = 1/s. If the input to the system is $x(t) =[sin(t) / \pi t] u(t)$ , where u(t) is a unit step function, the system output y(t) as $t \rightarrow \infty$ is______.

Which one of the following is a property of the solutions to the Laplace equation: $\bigtriangledown ^{2} f= 0$ ?

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for $t\geq 0$ :

A signal $2cos(\frac{2\pi t}{3})-cos(\pi t)$ is the input to an LTI system with the transfer function $H(s)=e^{s}+e^{-s}$ . If $C_{k}$ denotes the $k^{th}$ coefficient in the exponential Fourier series of the output signal, then $C_{3}$ is equal to

The Laplace transform of the causal periodic square wave of period T shown in the figure below is

Consider the differential equation $\frac{dx}{dy}=10-0.2x$ with initial condition x(0) =1. The response x(t) for t > 0 is

Let $x(t)=\alpha s(t)+s(-t)$ with $s(t)=\beta e^{-4t}u(t)$ , where u(t) is unit step function. If the bilateral Laplace transform of x(t) is $X(s)=\frac{16}{s^{2}-16} \; \; -4 \lt Re\lbrace s\rbrace \lt 4$ ; then the value of $\beta$ is _______.

Input x(t) and output y(t) of an LTI system are related by the differential equation ${y}''(t)-{y}'(t)-6y(t)=x(t)$ . If the system is neither causal nor stable, the impulse response h(t) of the system is

The bilateral Laplace transform of a function

is

Let the signal f(t) = 0 outside the interval $[T_{1}, T_{2}],$ where $T_{1}$ and $T_{2}$ are finite. Furthermore, $|f(t)| \lt \infty$ . The region of convergence (RoC) of the signal's bilateral Laplace transform F(s) is

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. $\dot{y}(t)+5y(t)=u(t)$ when y(0) = 1 and u(t) is a unit step function, y(t) is

A causal LTI system has zero initial conditions and impulse response $h(t)$ . Its input $x(t)$ and output $y(t)$ are related through the linear constant-coefficient differential equation $\frac{d^{2}y(t)}{dt^{2}}+\alpha \frac{dy(t)}{dt}+\alpha ^{2}y(t)=x(t)$ Let another signal $g(t)$ be defined as $g(t)=\alpha ^{2}\int_{0}^{t}h(\tau )+\frac{dh(t)}{dt}+\alpha h(t)$ If $G(s)$ is the Laplace transform of $g(t)$ , then the number of poles of $G(s)$ is____.

The unilateral Laplace transform of $f (t)$ is $\frac{1}{s^{2}+s+1}$ . Which one of the following is the unilateral Laplace transform of $g(t) = t \cdot f(t)$ ?

Let h(t) denote the impulse response of a causal system with transfer function $\frac{1}{s+1}$ . Consider the following three statements. S1 : The system is stable S2: $\frac{h(t+1)}{h(t)}$ is independent of $t \gt 0$ S3 : A non-causal system with the same transfer function is stable For the above system

The input $-3e^{2t}u(t)$ , where u(t) is the unit step function, is applied to a system with transfer function $\frac{s-2}{s+3}$ . If the initial value of the output is -2, then the value of the output at steady state is _____.

A stable linear time invariant (LTI) system has a transfer function $H(S)=\frac{1}{s^{2}+s-6}$ . To make this system causal it needs to be cascaded with another LTI system having a transfer function $H_{1}(S)$ . A correct choice for $H_{1}(S)$ among the following options is

The impulse response of a system is $h\left ( t \right )= tu\left ( t \right )$ . For an input u(t-1), the output is

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?