Consider a continuous-time system with input $x\left(t\right)$ and output $y\left(t\right)$ given by
$y\left(t\right)=x\left(t\right)\mathrm{cos}\left(t\right)$
Consider the following state-space representation of a linear time-invariant system.
$\overset.x\left(t\right)=\begin{bmatrix}1&0\\0&2\end{bmatrix}x\left(t\right),y\left(t\right)=c^Tx\left(t\right),\;c=\begin{bmatrix}1\\1\end{bmatrix}\;\mathrm{and}\;x\left(0\right)=\begin{bmatrix}1\\1\end{bmatrix}$
The value of $y\left(t\right)$ for $t={\mathrm{log}}_{e}2$ is __________.
A moving average function is given by $y\left(t\right)=\;\frac1T\int\limits_{t-T}^tu\left(\zeta\right)\operatorname d\zeta$. If the input u is a sinusoidal signal of frequency $\frac{1}{2T}\mathrm{Hz},$ then in steady state, the output y will lag u (in degree) by ________.
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 of G as shown in Figure(b)?
For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?
The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integrationtime step is h. $\frac{{x}_{k+1}-{x}_{k}}{h}={y}_{k}$ $\frac{{y}_{k+1}-{y}_{k}}{h}=-{x}_{k}$ For this discrete-time system, which one of the following statements is TRUE?
$x\left(t\right)$ is nonzero only for ${T}_{x}<t<{{T}^{\text{'}}}_{x}$ , and similarly, $y\left(t\right)$ is nonzero only for ${T}_{y}<t<{{T}^{\text{'}}}_{y}$. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE?
Consider an LTI system with transfer function
$H\left(S\right)=\frac{1}{s\left(s+4\right)}$
If the input to the system is $\mathrm{cos}\left(3t\right)$ and the steady state output is $A\mathrm{sin}\left(3t+\alpha \right)$, then the value of $A$ is
Consider an LTI system with impulse response $h\left(t\right)={e}^{-5t}u\left(t\right)$. If the output of the system is $y\left(t\right)={e}^{-3t}u\left(t\right)-{e}^{-5t}u\left(t\right)$ then the input, $x\left(t\right)$, is given by
Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?
Let y[n] denote the convolution of h[n] and g[n], where h[n]=(1/2)^{n} u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1] equals
The input x(t) and output y(t) of a system are related as $y\left(t\right)={\int}_{-\infty}^{t}x\left(\tau \right)\mathrm{cos}\left(3\tau \right)d\tau .$The system is
Given two continuous time signals $x\left(i\right)={e}^{-t}$ and $y\left(i\right)={e}^{-2t}$ which exist for $t>0,$ the convolution $z\left(t\right)=x\left(t\right)*y\left(t\right)$ is
The system represented by the input-output relationship $\style{font-size:14px}{y\left(t\right)=\int\limits_{-\infty}^{5t}x\left(\tau\right)d\tau,t>0}$ is
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 system 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(t-$\tau $) is applied to a system with impulse response h(t-$\tau $), the output will be
A cascade of 3 Linear Time Invariant systems is causal and unstable. From this, we conclude that
A signal ${e}^{-\alpha t}\mathrm{sin}\left(\omega t\right)$ is the input to a real Linear Time Invariant system. Given K and $\phi$ are constants, the output of the system will be of the form $\style{font-size:18px}{Ke^{-\beta t}\sin\left(\nu t+\phi\right)}$ where
The impulse response of a causal linear time-invariant system is given as h(t) . Now consider the following two statements : Statement (I): Principle of superposition holds Statement (II):h(t)=0 for t<0 Which one of the following statements is correct ?
A system with x(t) and output y(t) is defined by the input-output relation :
$y\left(t\right)=\int\limits_{-\infty}^{-2t}x\left(t\right)d\tau$
The system will be
A signal $x\left(t\right)=\mathrm{sin}c\left(\alpha t\right)$ where α is a real constant $\left(\mathrm{sin}c\left(x\right)=\frac{\mathrm{sin}\left(\mathrm{\pi x}\right)}{\mathrm{\pi x}}\right)$ is the input to a Linear Time Invariant system whose impulse response h(t)=sin(βt),where β is a real constant. If min$\left(\alpha ,\beta \right)$denotes the minimum of α and β and similarly, $\mathrm{max}\left(\alpha ,\beta \right)$ denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the output of the system ?
Let a signal $a_1\sin\left(\omega_1t+\phi_1\right)$ be applied to a stable linear time invariant system. Let the corresponding steady state output be represented as $a_2F\left(\omega_2t+\phi_2\right)$. Then which of the following statement is true?
If u(t), r(t) denote the unit step and unit ramp functions respectively and u(t)*r(t) their convolution, then the function u(t+1)*r(t-2) is given by