GATE EE
Signals and Systems
Linear Time Invariant Systems
Previous Years Questions

## Marks 1

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...
Consider the system as shown below: where y(t) = x(et). The system is...
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
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 ...
Consider an LTI system with impulse response $$h\left(t\right)=e^{-5t}u\left(t\right)$$ . If the output of the system is $$y\left(t\right)=e^{-3t}u\le... Consider an LTI system with transfer function$$H\left(s\right)=\frac1{s\left(s+4\right)}$$.If the input to the system is cos(3t) and the steady state... x(t) is nonzero only for$$T_x\;<\;t\;<\;T_x^1$$, and similarly, y(t) is nonzero only for$$T_y\;<\;t\;<\;T_y^1$$. Let z(t) be convoluti... Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is ... Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by... The impulse response of a system is h(t) = tu(t). For an input u(t − 1), the output is Given two continuous time signals$$x\left(t\right)=e^{-t}$$and$$y\left(t\right)=e^{-2t}$$which exist for t > 0, the convolution z(t) = x(t)*y(t... A low–pass filter with a cut-off frequency of 30 Hz is cascaded with a high-pass filter with a cut-off frequency of 20 Hz. The resultant system of fil... For the system$$\frac2{\left(s+1\right)}$$, the approximate time taken for a step response to reach 98% of its final value is The system represented by the input-output relationship$$y\left(t\right)=\int_{-\infty}^{5t}x\left(\tau\right)d\tau$$, t > 0 is A linear Time Invariant system with an impulse response$$h(t)$$produces output$$y(t)$$when input$$x(t)$$is applied. When the input$$x\left( {t ...
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...
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...
The impulse response of a causal linear time-invariant system is given as $$h(t)$$. Now consider the following two statements: Statement-$$\left( {\r... Let a signal$${a_1}\,\sin \left( {{\omega _1}t + {\phi _1}} \right)$$be applied to a stable linear time-invariant system. Let the corresponding ste...$$s(t)$$is step response and$$h(t)$$is impulse response of a system. Its response$$y(t)$$for any input$$u(t)$$is given by If$$f(t)$$is the step-response of a linear time-invariant system, then its impulse response is given by ___________$$s(t)$$is step response and$$h(t)$$is impulse response of a system. Its response$$y(t)$$for any input$$u(t)$$is given by ## Marks 2 For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE? The impulse response of a continuous time system is given by h(t) =$$\delta$$(t − 1) +$$\delta$$(t − 3). The value of the step response at t = 2 is The input x(t) and output y(t) of a system are related as$$\int_{-\infty}^tx\left(\tau\right)\cos\left(3\tau\right)d\tau$$.The system is L et y[n] denote the convolution of h[n] and g[n], where$$h\left[n\right]=\left(1/2\right)^nu\left[n\right]$$and g[n] is a causal sequence. If y ... The response h(t) of a linear time invariant system to an impulse$$\delta\left(t\right)$$, under initially relaxed condition is$$h\left(t\right)=e^{...
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...
A cascade of 3 Linear Time Invariant systems is casual and unstable. From this, we conclude that
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,...
A signal $$x\left( t \right) = \sin c\left( {\alpha t} \right)$$ where $$\alpha$$ is a real constant $$\left( {\sin \,c\left( x \right) = {{\sin \lef... The transfer function of a linear time invariant system is given as$$G\left( s \right) = {1 \over {{s^2} + 3s + 2}}.$$The steady state value of the ... A system with input$$x(t)$$and output$$y(t)$$is defined by the input$$-$$output relation:$$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t...
$$X\left( z \right) = 1 - 3\,\,{z^{ - 1}},\,\,Y\left( z \right) = 1 + 2\,\,{z^{ - 2}}$$ are $$Z$$-transforms of two signals $$x\left[ n \right],\,\,y\... A signal is processed by a causal filter with transfer function$$G(s).$$For a distortion free output signal waveform,$$G(s)$$must Consider the discrete-time system shown in the figure where the impulse response of$$G\left( z \right)$$is$$g\left( 0 \right) = 0,\,\,g\left( 1 \r...
A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must. $$G\left( z \ri... A discrete real all pass system has a pole at$$z = 2\angle {30^ \circ };\,$$it, therefore,$$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...
$$x\left[ n \right] = 0;\,n < - 1,\,n > 0,\,x\left[ { - 1} \right] = - 1,\,x\left[ 0 \right]$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\... A continuous-time system is described by$$y\left( t \right) = {e^{ - |x\left( t \right)|}},$$where$$y(t)$$is the output and$$x(t)$$is the input.... In the system shown in Fig. the input$$x\left( t \right) = \sin t.$$In the steady-state, the response$$y(t)$$will be ... Given the relationship between the input$$u(t)$$and the output$$y(t)$$to be$$y\left( t \right) = \int\limits_0^t {\left( {2 + t - \tau } \right){...

Match the following transfer functions and impulse responses Transfer functions \eqalign{ & \left( a \right)\,\,\,\,\,\,\,\,{1 \over {s\left( ... ## Marks 5 A single input single output system withy$$as output and$$u$$as input, is described by$$\${{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + 10y ...
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...
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...
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