GATE ECE
Signals and Systems
Continuous Time Signal Laplace Transform
Previous Years Questions

## Marks 1

The bilateral Laplace transform of a function $$f\left( t \right) = \left\{ {\matrix{ {1\,if\,a \le t \le b} \cr {0\,otherwise} \cr } } \r... The input$$ - 3{e^{2t}}\,\,u\left( t \right)$$, where u(t) is the unit step function$$\, {{s - 2} \over {s + 3}}$$. If the initial value of the outpu... A continuous, linear time - invariant fiilter has an impulse response h(t) described by$$h\left( t \right) = \left\{ {\matrix{ {3\,for\,0 \le t \l...
If the Laplace transform of a signal y(t) is $$Y\left( s \right) = {1 \over {s\left( {s - 1} \right)}},$$ then its final value is
The Laplace transform of i(t) tends to $$I\left( s \right)\,\, = \,{2 \over {s\left( {1 + s} \right)}}$$ As $$t \to \infty$$ , the value of i(t) te...
Given that $$L\left[ {f\left( t \right)} \right]\, = \,$$ $${{s + 2} \over {{s^2} + 1}},$$ $$L\left[ {g\left( t \right)} \right] = {{{s^2} + 1} \ove...$$If\,\,L\left[ {f\left( t \right)} \right]\, = \,F\left( s \right),$$then$$L\left[ {f\left( {t - T} \right)} \right]$$is equal to If L$$\left[ {f\left( t \right)} \right]$$=$$\omega /\left( {{s^2} + {\omega ^2}} \right),$$then the value of$$\matrix{ {Lim\,f\,\left( t \righ...
The Laplace Transform of eat .cos$$\left( {\alpha t} \right).u\left( t \right)$$ is equal to
The final value theorem is used to find the
If L$$\left[ {f\left( t \right)} \right]$$ = $${{2\left( {s + 1} \right)} \over {{s^2} + 2s + 5}}$$, then $$f\left( {0 + } \right)\,$$ and $$f\left( \... The laplace transform of a unit ramp function starting at t=a, is ## Marks 2 The Laplace transform of the casual periodic square wave of period T shown in the figure below is ... Let x(t) = a s(t) +s(-t) with s(t) =$$\beta {e^{ - 4t}}u\left( t \right)$$, where u(t) is unit step function. If the bilateral Laplace transform of x... The solution of the differential equation$${{h\left( {t + 1} \right)} \over {h\left( t \right)}}\,\,\,\,\,{{{d^2}y} \over {d{t^{ \to 2}}}} + {{2\,dy}...
The unilateral Laplace transform of F(t) is $${1 \over {{s^2} + s + 1}}$$. Which one of the following is the unilateral Laplace transform of g(t) = $... A stable linear time invariant (LTI) system has a transfer function H(s) = $${1 \over {{s^2} + s - 6}}$$. To make this system casual it needs to be ca... A casual 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 - co... Let h(t) denote the impulse response of a casual system with transfer function $${1 \over {s + 1}}$$. Consider the following three statements. S1: Th... A system is described by the following differential equation, where u(t) is the input to the system and y(t) is output of the system $$\mathop y\limit... A system is described by the differential equation$$${{{d^2}y} \over {d{t^2}}} + 5{{dy} \over {dt}} + 6y\left( t \right) = x\left( t \right)$$Let x... If$$F\left( s \right) = L\left[ {f\left( t \right)} \right] = {{2\left( {s + 1} \right)} \over {{s^2} + 4s + 7}}$$then the initial and final values ... Given f(t) =$${L^{ - 1}}\left[ {{{3s + 1} \over {{s^3} + 4{s^2} + \left( {K - 3} \right)s}}} \right].$$If$$\matrix{ {Lim\,f\,\left( t \right) =...
Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of $$\int\limits_0^t {f\left( \tau \right)\,d\tau }$$ is
Consider the function f(t) having Laplace transform $$F\left( s \right) = {{{\omega _0}} \over {{s^2} + {\omega _0}^2}}\,\,\,\,\,\,{\mathop{\rm Re}\n... In what range should Re(s) remain so that the Laplace transform of the function e(a+2)t+5 exists? The Laplace transform of a continuous - time signal x(t) is$$X\left( s \right) = {{5 - s} \over {{s^2} - s - 2}}$$. If the Fourier transform of tyhis... The inverse Laplace transform of the function$${{s + 5} \over {\left( {s + 1} \right)\left( {s + 3} \right)}}$$is If$$F\left( s \right) = L\left[ {f\left( t \right)} \right] = {K \over {\left( {s + 1} \right)\,\left( {{s^2} + 4} \right)}}$$then$$\mat...
The Laplace transform of the periodioc function f(t) describe4d by the curve below, i.e., f\left( t \right) = \left\{ {\matrix{ {\sin \,t\,\,\,if...
The Laplace transform of a function f(t)u(t), where f(t) is periodic with period T, is A(s) times the Laplace transform of its first period. Then
Laplace transform of the functions t u(t) and u(t) sin(t) are respectively:

## Marks 5

A sinsoidal signal, v(t) = A sin(t), is applied to an ideal full-wave rectifier. Show that the Laplace Transform of the output can be written in the ...
Find the Laplace transform of the waveform x(t) shown in Fig.1. ...
EXAM MAP
Joint Entrance Examination