Questions & Answers of Definitions and properties of Laplace transform

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


A signal $2\cos\left(\frac{2\mathrm\pi}3t\right)-\;\cos\;\left(\mathrm{πt}\right)$ is the input to an LTI system with the transfer function


If Ck denotes the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to

The bilateral Laplace transform of a function ft=1if atb0Other wise is

Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, ft< . The region of convergence (RoC) of the signal’s bilateral Laplace transform F(s) is

Consider the differential equation dxdt = 10-0.2x with initial condition x(0) = 1. The response x(t) for t>0 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

Let x(t) = α s(t) +s(–t) with s(t) = βe-4t u(t) , where u(t) is unit step function . If the bilateral Laplace transform of x(t) is

Xs = 16s2- 16-4 <Re{s}<4;

Then the value of β 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


When y(0) =1 and u(t) is a unit step function, y(t) is

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

A system is described by the differential equation d 2 y d t 2 + 5 d y d t + 6 y ( t ) = x ( t ) .
Let x(t) be a rectangular pulse given by
x ( t ) = { 1 0 < t < 2 0 o t h e r w i s e

Assuming that y(0) = 0 and d y d t = 0 at t = 0, the Laplace transform of y(t) is

The unilateral Laplace transform of f (t) is 1s2+s+1. The unilateral Laplace transform of t f (t) is

If the unit step response of a network is 1-e-αt then its unit impulse response is

An input x (t) = exp (−2t)u(t) + δ (t − 6) is applied to an LTI system with impulse response h(t) = u(t) . The output is is

If FS=L[ft]=2s+1s2+4s+7  then the initial and final values of f(t) are respectively

A continuous time LTI system is described by


Assuming zero initial conditions, the response y(t) of the above system for the input x(t)=e-2t u(t) is given by

Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of $\style{font-size:14px}{\int\limits_0^tf\left(\tau\right)\operatorname{d}\tau}$ is

If the Laplace transform of a signal y(t) is Ys=1ss-1, then its final value is