Fourier transform and Laplace transform are the significant mathematical tools used for signal and system analysis. The limitation of Fourier transform is that it is not defined for some of the signals. Therefore for such signals analysis Laplace transform is used to overcome the drawback of Fourier transform. Due to availability of convergence factor Laplace transform is completely integrable and hence defined for those signals for which Fourier transform is not absolutely integrable.

The LTI is represented by differential equation. The system response is obtained by solving the given differential equation. Any differential equation is converted into simpler algebraic equation by Laplace transform.

The Laplace transform converts integro-differential equation in _________ domain.

Laplace transform is equivalent to Fourier transform when, the real term \((\sigma)\) of complex frequency \(S=\sigma+j\omega\) ,should be

If \(x(t)=e^{-ct}-e^{-dt}\,u(-t)\), Laplace transform of x(t) is

For two signals \(x_1(t)\) and \(x_2(t)\). The correct Laplace transform expression for \(x_1(t)\cdot x_2(t)\) is :

A function f(t) is given by e^–αt cos ωt. Its pole location in s-plane will be given by

Following statements are given for Laplace transforms :

(a) \(L[x(t)]=\displaystyle\int_{-\infty}^{\infty}x(t)e^{st}\,dt\)

(b) \(L[x(t)]=\displaystyle\int_{-\infty}^{\infty}x(t)e^{-st}\,dt\)

(c) \(L\left[te^{-at}u(t)\right]=\dfrac{1}{(s+a)^{2}}\)

(d) \(L\left[te^{-at}u(t)\right]=\dfrac{1}{(s-a)^{2}}\)

Out of the above, the following is the correct answer :

Match the following :

List - I	List - II
a) $(\dfrac{1}{s-\alpha})$	i)
b) $(\dfrac{1}{s+\alpha})$	ii)
c) $(\dfrac{1}{(s-\alpha)^{2}+\omega_{o}^{2}})$	iii)
d) $(\dfrac{1}{(s+\alpha)^{2}+\omega_{o}^{2}})$	iv)

Codes :

Assertion (A) : Laplace and Z transforms can be applied to the analysis of many unstable systems and consequently play an important role in the investigation of the stability or instability of the systems.

Reason (R) : The region of convergence of the Laplace transform of x(t) is given by the expression : \(\displaystyle\int_{-\infty}^{\infty}x(t)e^{-\sigma t}\,dt\ \ge\ \infty\)

Select your answer using the codes given below :

Which of the following is the final value of the impulse response of the system whose transfer function is

(2s + 1)/(s ⁴ + 8s ³  + 16s ²  + s)

Find the Laplace transform for the following time domain.

y(t) = -2te ^-t + 4e ^-t - 4e ^-2t

Match List I with List II

Choose the correct answer from the options given below:

The Laplace transform of sin h (at) is