LTI system

Consider a discrete-time linear time-invariant (LTI) system S, where y[n] = S{x[n]} Let S{δ[n]} = {1, n ∈ {0, 1, 2} {0, otherwise where δ[n] is the discrete-time unit impulse funct...

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system S. The output is observed to be the continuous-time unit step signa...

Consider a discrete-time linear time-invariant (LTI) system, $\boldsymbol{S}$, where $$ y[n]=S\{x(\mathrm{n})\} $$ $$Let\,\,\,\, S\{\delta[n]\}=\left\{\begin{array}{lc} 1, & n \in\...

Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time-constant, in sec, of the system...

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system $S$. The output is observed to be the continuous-time unit step sig...

Which of the following statement(s) is/are true?

The magnitude and phase plots of an LTI system are shown in the figure. The transfer function of the system is

Consider the state-space description of an LTI system with matrices A = $\begin{bmatrix} 0 & 1 \ -1 & -2 \end{bmatrix}$, B = $\begin{bmatrix} 0 \ 1 \end{bmatrix}$, C = $\begin{bmat...

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 the input and output respectivel...

Let an input x(t) = 2 sin(10$$\pi$$t) + 5 cos(15$$\pi$$t) + 7 sin(42$$\pi$$t) + 4 cos(45$$\pi$$t) is passed through an LTI system having an impulse response, $$h(t) = 2\left( {{{\s...

Two discrete-time linear time-invariant systems with impulse responses $h_1\lfloor n\rfloor=\delta\lfloor n-1\rfloor+\delta\lfloor n+1\rfloor$ and $h_2[n]=\delta[n]+\delta[n-1]$ ar...

The symbols, a and T, represent positive quantities, and u(t) is the unit step function. Which one of the following impulse responses is NOT the output of a causal linear time-inva...

The characteristic equation of a linear time-invariant (LTI) system is given by $\Delta(s) = s^4 + 3s^3 + 3s^2 + s + k = 0$. The system is BIBO stable if

A continuous-time input signal $x(t)$ is an eigenfunction of an LTI system, if the output is

Let a causal LTI system be characterized by the following differential equation, with initial rest condition $\frac{d^2y}{dt^2} + 7\frac{dy}{dt} + 10y(t) = 4x(t) + 5\frac{dx(t)}{dt...

Let the signal $x(t) = \sum_{k=-\infty}^{+\infty} (-1)^k \delta (t - \frac{k}{2000})$ be passed through an LTI system with frequency response H($\omega$), as given in the figure be...

For a system having transfer function G(s) = $\frac{-s+1}{s+1}$, a unit step input is applied at time t = 0. The value of the response of the system at t = 1.5 sec (rounded off to...

Consider a causal and stable LTI system with rational transfer function H(z), whose corresponding impulse response begins at n = 0. Furthermore, H(1) = $\frac{5}{4}$. The poles of...

Let a causal $$LTI$$ system be characterized by the following differential equation, with initial rest condition $${{{d^2}y} \over {d{t^2}}} + 7{{dy} \over {dt}} + 10y\left( t \rig...

The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where x(t) is the input signal. A signal z(t) is called eigen-signal of the system T, when T{z(t...

A moving average function is given by $$y\left(t\right)=\frac1T\int_{t-T}^tu\left(\tau\right)d\tau$$. If the input u is a sinusoidal signal of frequency $$\frac1{2T}Hz$$, then in s...

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

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 output is $$A\sin\left(3t+\al...

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\left(t\right)-e^{-5t}u\left(t\ri...

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

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 response h(t) of a linear time invariant system to an impulse $$\delta\left(t\right)$$, under initially relaxed condition is $$h\left(t\right)=e^{-t}\;+\;e^{-2t}$$. The respons...

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^{ - t}} + {e^{ -...

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

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

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 \left( {\pi x} \right)} \over {\pi...

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 constants, the output of the syste...

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 output of this system for a un...

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 constants, the output of the syste...

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 output of the system for a unit...

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

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 steady state output be represented...

$$x\left[ n \right] = 0;\,n < - 1,\,n > 0,\,x\left[ { - 1} \right] = - 1,\,x\left[ 0 \right]$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 2$$ is the input...

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

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){e^{ - 3\left( {t - \tau } \rig...

A linear time-invariant system initially at rest, when subjected to a unit-step input, gives a response $$y\left( t \right) = t{e^{ - t}},\,\,t > 0.$$ The transfer function of the...

The output of a linear time invariant control system is $$c(t)$$ for a certain input $$r(t).$$ If $$r(t)$$ is modified by passing it through a block whose transfer function is $${e...

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