# Linear Time Invariant Systems · Signals and Systems · GATE EE

Start Practice## Marks 1

GATE EE 2022

Let a causal LTI system be governed by the following differential equation $$y(t) + {1 \over 4}{{dy} \over {dt}} = 2x(t)$$, where x(t) and y(t) are th...

GATE EE 2022

Consider the system as shown below:
where y(t) = x(et). The system is...

GATE EE 2016 Set 1

Consider a continuous-time system with input x(t) and output y(t) given by
$$y\left(t\right)=x\left(t\right)\cos\left(t\right)$$. This system is

GATE EE 2015 Set 1

The impulse response g(t) of a system G, is as shown in Figure (a). What is the maximum
value attained by the impulse response of two cascaded blocks ...

GATE EE 2014 Set 2

Consider an LTI system with impulse response $$h\left(t\right)=e^{-5t}u\left(t\right)$$ . If the output of the system is
$$y\left(t\right)=e^{-3t}u\le...

GATE EE 2014 Set 2

Consider an LTI system with transfer function $$H\left(s\right)=\frac1{s\left(s+4\right)}$$.If the input to the system is cos(3t) and the steady state...

GATE EE 2014 Set 1

x(t) is nonzero only for $$T_x\;<\;t\;<\;T_x^1$$ , and similarly, y(t) is nonzero only for $$T_y\;<\;t\;<\;T_y^1$$ . Let
z(t) be convoluti...

GATE EE 2013

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

GATE EE 2013

Assuming zero initial condition, the response y(t) of the system given below to a
unit step input u(t) is
...

GATE EE 2013

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

GATE EE 2011

Given two continuous time signals $$x\left(t\right)=e^{-t}$$ and $$y\left(t\right)=e^{-2t}$$ which exist for t > 0,
the convolution z(t) = x(t)*y(t...

GATE EE 2011

A low–pass filter with a cut-off frequency of 30 Hz is cascaded with a high-pass
filter with a cut-off frequency of 20 Hz. The resultant system of fil...

GATE EE 2010

For the system $$\frac2{\left(s+1\right)}$$,
the approximate time taken for a step response to reach 98%
of its final value is

GATE EE 2010

The system represented by the input-output relationship $$y\left(t\right)=\int_{-\infty}^{5t}x\left(\tau\right)d\tau$$, t > 0 is

GATE EE 2009

A linear Time Invariant system with an impulse response $$h(t)$$ produces output $$y(t)$$ when input $$x(t)$$ is applied. When the input $$x\left( {t ...

GATE EE 2008

A signal $${e^{ - \alpha t}}\,\sin \left( {\omega t} \right)$$ is the input to a real Linear Time Invariant system. Given $$K$$ and $$\phi $$ are cons...

GATE EE 2008

A signal $${e^{ - \alpha t}}\,\sin \left( {\omega t} \right)$$ is the input to a real Linear Time Invariant system. Given $$K$$ and $$\phi $$ are cons...

GATE EE 2008

The impulse response of a causal linear time-invariant system is given as $$h(t)$$. Now consider the following two statements:
Statement-$$\left( {\r...

GATE EE 2007

Let a signal $${a_1}\,\sin \left( {{\omega _1}t + {\phi _1}} \right)$$ be applied to a stable linear time-invariant system. Let the corresponding ste...

GATE EE 2002

$$s(t)$$ is step response and $$h(t)$$ is impulse response of a system. Its response $$y(t)$$ for any input $$u(t)$$ is given by

GATE EE 1994

If $$f(t)$$ is the step-response of a linear time-invariant system, then its impulse response is given by ___________

GATE EE 1993

$$s(t)$$ is step response and $$h(t)$$ is impulse response of a system. Its response $$y(t)$$ for any input $$u(t)$$ is given by

## Marks 2

GATE EE 2015 Set 2

For linear time invariant systems, that are Bounded Input Bounded stable, which one of
the following statement is TRUE?

GATE EE 2013

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

GATE EE 2012

L et y[n] denote the convolution of h[n] and g[n], where $$h\left[n\right]=\left(1/2\right)^nu\left[n\right]$$ and g[n] is a causal
sequence. If y[0] ...

GATE EE 2012

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

GATE EE 2011

The response h(t) of a linear time invariant system to an impulse $$\delta\left(t\right)$$, under
initially relaxed condition is $$h\left(t\right)=e^{...

GATE EE 2010

Given the finite length input x[n] and the corresponding finite length output y[n]
of an LTI system as shown below, the impulse response h[n] of the s...

GATE EE 2009

A cascade of 3 Linear Time Invariant systems is casual and unstable. From this, we conclude that

GATE EE 2009

The $$z$$$$-$$ transform of a signal $$x\left[ n \right]$$ is given by $$4{z^{ - 3}} + 3{z^{ - 1}} + 2 - 6{z^2} + 2{z^3}.$$ It is applied to a system,...

GATE EE 2008

A system with input $$x(t)$$ and output $$y(t)$$ is defined by the input $$-$$ output relation:
$$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t...

GATE EE 2008

A signal $$x\left( t \right) = \sin c\left( {\alpha t} \right)$$ where $$\alpha $$ is a real constant $$\left( {\sin \,c\left( x \right) = {{\sin \lef...

GATE EE 2008

The transfer function of a linear time invariant system is given as $$G\left( s \right) = {1 \over {{s^2} + 3s + 2}}.$$ The steady state value of the ...

GATE EE 2007

$$X\left( z \right) = 1 - 3\,\,{z^{ - 1}},\,\,Y\left( z \right) = 1 + 2\,\,{z^{ - 2}}$$ are $$Z$$-transforms of two signals $$x\left[ n \right],\,\,y\...

GATE EE 2007

A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must

GATE EE 2007

Consider the discrete-time system shown in the figure where the impulse response of $$G\left( z \right)$$ is
$$g\left( 0 \right) = 0,\,\,g\left( 1 \r...

GATE EE 2007

A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must.
$$G\left( z \ri...

GATE EE 2006

A discrete real all pass system has a pole at $$z = 2\angle {30^ \circ };\,$$ it, therefore,

GATE EE 2006

A continuous-time system is described by $$y\left( t \right) = {e^{ - |x\left( t \right)|}},$$ where $$y(t)$$ is the output and $$x(t)$$ is the input....

GATE EE 2006

$$y\left[ n \right]$$ denotes the output and $$x\left[ n \right]$$ denotes the input of a discrete-time system given by the difference equation $$y\le...

GATE EE 2006

$$x\left[ n \right] = 0;\,n < - 1,\,n > 0,\,x\left[ { - 1} \right] = - 1,\,x\left[ 0 \right]$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\...

GATE EE 2004

In the system shown in Fig. the input $$x\left( t \right) = \sin t.$$ In the steady-state, the response $$y(t)$$ will be
...

GATE EE 2001

Given the relationship between the input $$u(t)$$ and the output $$y(t)$$ to be
$$y\left( t \right) = \int\limits_0^t {\left( {2 + t - \tau } \right){...

## Marks 4

GATE EE 1992

Match the following transfer functions and impulse responses
Transfer functions
$$\eqalign{
& \left( a \right)\,\,\,\,\,\,\,\,{1 \over {s\left( ...

## Marks 5

GATE EE 2002

A single input single output system with $$y$$ as output and $$u$$ as input, is described by
$$${{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + 10y ...

GATE EE 1997

A first order system is initially at rest and excited by a step input at time $$t=0.$$ Its output becomes $$1.1$$ $$V$$ is in $$4$$ seconds and eventu...

GATE EE 1992

The impulse response of a network is $$h\left( t \right) = 1$$ for $$0 \le t < 1$$ and zero otherwise. Sketch the impulse response of two such net...