GATE ECE

Signals and Systems

Continuous Time Linear Invariant System

Previous Years Questions

## Marks 1

Consider the following statements for continuous-time linear time invariant (LTI) system.
I. There is no bounded input bounded output (BIBO) stable s...

The input x(t) and the output y(t) of a continuous time system are related as
$$y\left( t \right) = \int\limits_{t - T}^t {x\left( u \right)du.} $$. ...

The impulse response of an LTI system can be obtained by

The result of the convolution $$x\left( { - t} \right) * \delta \left( { - t - {t_0}} \right)$$ is

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

Two system with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by...

The differential equation $$100{{{d^2}y} \over {dt}} - 20{{dy} \over {dt}} + y = x\left( t \right)$$ describes a system with an input x(t) and output ...

The impulse response h(t) of a linear time-invariant continuous time system is described by $$h\left( t \right) = \,\,\exp \left( {\alpha t} \right)u\...

The input and output of a continuous system are respectively denoted by x(t) and y(t). Which of the following descriptions corresponds to a causal sys...

Which of the following can be impulse response of a causal system?

Let x(t) be the input to a linear, time-invariant system. The required output is 4x(t - 2). The transfer function of the system should be

Convolution of x(t + 5) with impulse function $$\delta \left( {t\, - \,7} \right)$$ is equal to

The transfer function of a system is given by $$H\left( s \right) = {1 \over {{s^2}\left( {s - 2} \right)}}$$. The impulse response of the system is

A system with an input x(t) and an output y(t) is described by the relation: y(t) = t x(t). This system is

The transfer function of a zero - order - hold system is

The unit impulse response of a linear time invariant system is the unit step function u(t). For t>0, the response of the system to an excitation e-...

The transfer function of a linear system is the

Let h(t) be the impulse response of a linear time invariant system. Then the response of the system for any input u(t) is

Non - minimum phase transfer function is defined as the transfer function

Indicate whether the following statement is TRUE/FALSE: Give reason for your answer.
If G(s) is a stable transfer function, then $$F\left( s \right)...

## Marks 2

A continuous time signal x(t) = $$4\cos (200\pi t)$$ + $$8\cos(400\pi t)$$, where t is in seconds, is the input to a linear time invariant (LTI) filte...

The transfer function of a causal LTI system is H(s) = 1/s. If the input to the system is x(t) = $$\left[ {\sin (t)/\pi t} \right]u(t);$$ where u(t) i...

Consider the parallel combination of two LTI systems shown in the figure.
The impulse responses of the systems are
$${h_1}(t) = 2\delta (t + 2)\, ...

Consider an LTI system with magnitude response $$$\left| {H(f)} \right| = \left\{ {\matrix{
{1 - \,{{\left| f \right|} \over {20}},} & {\left| ...

The output of a standrad second-order system for a unit step input is given as $$y(t) = 1 - {2 \over {\sqrt 3 }}{e^{ - t}}\cos \left( {\sqrt 3 t - {\p...

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 no...

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

The input x(t) and output y(t) of a system are related as y(t) = $$\int\limits_{ - \infty }^t x (\tau )\cos (3\tau )d\tau $$.
The system is

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

A continuous time LTI system is described by $${{{d^2}y(t)} \over {d{t^2}}} + 4{{dy(t)} \over {dt}} + 3y(t)\, = 2{{dx(t)} \over {dt}} + 4x(t)$$.
Assu...

Consider a system whose input x and output y are related by the equation
$$$y(t) = \int\limits_{ - \infty }^\infty {x(t - \tau )\,h(2\tau )\,d\tau }...

A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s = - 2 and s = - 4, and one simple zero...

Let x(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, R3, ...

The frequency response of a linear, time-invariant system is given by H(f) = $${5 \over {1 + j10\pi f}}$$. The step response of the system is

Let g(t) = p(t) * p(t), where * denotes convolution and p(t) = u(t) - (t-1) with u(t) being the unit step function. The impulse response of filter mat...

The output y(t) of a linear time invariant system is related to its input x(t) by the following equation: y(t) = 0.5 x $$(t - {t_d} + T) + \,x\,(t - ...

A system described by the differential equation: $${{{d^2}y} \over {d{t^2}}} + 3{{dy} \over {dt}} + 2y = x(t)$$ is initially at rest. For input x(t) =...

A rectangular pulse train s(t) as shown in Fig.1 is convolved with the signal $${\cos ^2}$$ ($$4\pi \,{10^{3\,}}$$t). The convolved signal will be a
...

A causal system having the transfer function H(s) = $${1 \over {s + 2}}$$, is excited with 10 u(t). The time at which the output reaches 99% of its st...

The impulse response function of four linear system S1, S2, S3, S4 are given respectively by
$${h_1}$$(t), = 1;
$${h_2}$$(t), = U(t);
$${h_3}(t...

A linear time invariant system has an impulse response $${e^{2t}},\,\,t\, > \,0.$$ If the initial conditions are zero and the input is $${e^{3t}}$$...

Let u(t) be the unit step function. Which of the waveforms in Fig.(a) -(d) corresponds to the convolution of $$\left[ {u\left( t \right)\, - \,u\left(...

Match each of the items 1, 2 on the left with the most appropriate item A, B, C or D on the right.
In the case of a linear time invariant system
Li...

Match each of the items A, B and C with an appropriate item from 1, 2, 3, 4 and 5.
List - 1
(A) $${a_1}{{{d^{2y}}} \over {d{x^2}}} + {a_2}y{{dy} \ove...

The voltage across an impedance in a network is V(s) = Z(s) I(s), where V(s), Z(s) and $${\rm I}$$(s) are the Laplace Transforms of the corresponding ...

An excitation is applied to a system at $$t = T$$ and its response is zero for $$ - \infty < t < T$$.
Such a system is a

The response of an initially relaxed linear constant parameter network to a unit impulse applied at $$t = 0$$ is $$4{e^{ - 2t}}u\left( t \right).$$ Th...

The impulse response and the excitation function of a linear time invariant casual system are shown in Fig. a and b respectively. The output of the sy...

The transfer function of a zero-order hold is

## Marks 5

A deterministic signal x(t) = $$\cos (2\pi t)$$ is passed through a differentiator as shown in
Figure.
(a) Determine the autocorrelation Rxx ($$\tau ...

For the linear, time-invariant system whose block diagram is shown in Fig.(a),
with input x(t) and output y(t).
(a) Find the transfer function.
(b) Fo...

Fig.1, shows the block diagram representation of a control system. The system in block A has an impulse response $${h_A}(t) = {e^{ - t}}\,u(t)$$. The ...

Consider the following interconnection of the three LTI systems (Fig.1). $${h_1}(t)$$ , $${h_2}(t)$$ and $${h_3}(t)$$ are the impulse responses of the...