GATE EC LTI Systems questions

Let $Y(s)$ be the unit-step response of a causal system having a transfer function $$G(s)= \dfrac{3-s}{(s+1)(s+3)}$$ that is ,$Y(s)=\dfrac{G(s)}{s}.$ The forced response of the system is

For an LTI system, the Bode plot for its gain is as illustrated in the figure shown. The number of system poles $N_{p}$ and the number of system zeros $N_{z}$ in the frequency range $1\: Hz \leq f \leq \:10^{7} Hz$ is

Consider a causal second-order system with the transfer function $$G(s)=\dfrac{1}{1+2s+s^{2}}$$ with a unit-step $R(s)=\dfrac{1}{s}$ as an input. Let $C(s)$ be the corresponding output. The time taken by the system output $c(t)$ to reach $94\%$ of its steady-state value $\underset{t\rightarrow \infty}{\lim}\:c(t),$ rounded off to two decimal …

A discrete-time all-pass system has two of its poles at $0.25\angle 0^{\circ}$ and $2\angle 30^{\circ}$. Which one of the following statements about the system is TRUE?

For a unity feedback control system with the forward path transfer function $$G\left ( s \right )=\dfrac{K}{s\left ( s+2 \right )}$$The peak resonant magnitude $M_{r}$ of the closed-loop frequency response is $2$. The corresponding value of the gain $\text{K}$ (correct to two decimal places) is _________.

The figure below shows the Bode magnitude and phase plots of a stable transfer function $G\left ( s \right )=\dfrac{n_{0}}{s^{3}+d_{2}s^{2}+d_{1}s+d_{0}}.$ Consider the negative unity feedback configuration with gain $k$ in the feedforward path. The closed loop is stable for $k < k_{0}.$ The maximum value of $k_{0}$ is _________.

Consider the following statements for continuous-time linear time invariant (LTI) systems. There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane. There is no causal and BIBO stable with a pole in the right half of the complex plane. Which one among the following is correct?

Consider a single input single output discrete-time system with $x[ n ]$ as input and $y [ n ]$ as output, where the two are related as$$y [ n ]= \begin{cases} n \mid x [ n ] \mid, & \text{ for } \ 0\leq n\leq 10\\ x [ n ] – x [ n-1 ], & \text{otherwise.}\end{cases}$$ Which one of the following statements is true about the system?

Two discrete-time signals $x[n]$ and $h[n]$ are both non-zero only for $n=0,1,2$, and are zero otherwise .It is given that $$x[0]=1, \: x[1]=2, \: x[2]=1, \: h[0]=1$$ Let $y[n]$ be the linear convolution of $x[n]$ and $h[n]$. Given that $y[1]=3$ and $y[2]=4$, the value of the expression $(10y[3]+y[4])$ is__________.

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) filter with the impulse response $$h(t)=\begin{cases} \frac{2 \sin (300\pi t)}{\pi t},& t\neq 0 \\ 600, & t=0. \end{cases}$$ Let $y(t)$ be the output of this filter. The maximum value of $\mid y(t) \mid$ is …

An LTI system with unit sample response $h[n]=5\delta [n]-7\delta [n-1]+7\delta [n-3]-5\delta [n-4]$ is a

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

Consider the parallel combination of two LTI systems shown in the figure. The impulse responses of the systems are $$\begin{array} {} h_1(t)=2\delta (t+2)-3\delta (t+1) \\ h_2(t)=\delta (t-2). \end{array}$$ If the input $x(t)$ is a unit step signal, then the energy of $y(t)$ is ____________

The transfer function of a linear time invariant system is given by $H(s) = 2s^4 – 5s^3 + 5s -2$. The number of zeroes in the right half of the $s$-plane is _________

If the signal $x(t) = \large \frac{\sin(t)}{\pi t}$$*$$\large \frac{\sin(t)}{\pi t}$ with $*$ denoting the convolution operation, then $x(t)$ is equal to

The direct form structure of an FIR (finite impulse response) filter is shown in the figure. The filter can be used to approximate a

A second-order linear time-invariant system is described by the following state equations $$\frac{d}{dt}x_1(t)+2x_1(t)=3u(t)$$ $$\frac{d}{dt}x_2(t)+x_2(t)=u(t)$$ where $x_1(t)$ and $x_2(t)$ are the two state variables and $u(t)$ denotes the input. If the output $c(t)=x_1(t)$, then the system is

The forward-path transfer function and the feedback-path transfer function of a single loop negative feedback control system are given as $$G(s)=\frac{K(s+2)}{s^2+2s+2}\;\text{and}\hspace{0.3cm}H(s)=1,$$ respectively. If the variable parameter $K$ is real positive, then the location of the breakaway point on the root locus diagram of the system is …

A wide sense stationary random process $X(t)$ passes through the LTI system shown in the figure. If the autocorrelation function of $X(t)$ is $R_X(\tau)$, then the autocorrelation function $R_Y(\tau)$ of the output $Y(t)$ is equal to

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