GATE EC LTI Systems questions

Let $h(t)$ denote the impulse response of a causal system with transfer function $\frac{1}{s+1}.$ Consider the following three statements. $S1$: The system is stable. $S2$: $\frac{h(t+1)}{h(t)}$ is independent of $t$ for $t > 0$. $S3$: A non-causal system with the same transfer function is stable. For the above system,

The steady state error of the system shown in the figure for a unit step input is _________.

In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following transfer functions has this root locus?

A real-valued signal $x(t)$ limited to the frequency band $\mid f \mid \leq \frac{W}{2}$ is passed through a linear time invariant system whose frequency response is $$H(f) = \begin{cases} e^{-j 4 \pi f}, & \mid f \mid \leq \frac{W}{2} \\ 0, & \mid f \mid > \frac{W}{2} \end{cases}.$$ The output of the system is

The sequence $x[n] = 0.5^n \: u[n]$, where $u[n]$ is the unit step sequence, is convolved with itself to obtain $y[n]$. Then $\Sigma_{n= -\infty}^{+ \infty} y[n]$ is _________.

A stable linear time variant (LTI) system has a transfer function $H(s) = \frac{1}{s^2+s-6}$. To make this system casual it needs to be cascaded with another LTI system having a transfer function $H_1(s)$. A correct choice for $H_1(s)$ among the following options is

A casual LTI system has zero initial conditions and impulse response $h(t)$. Its input $x(t)$ and output $y(t)$ are related through the linear constant-coefficient differential equation $$\frac{d^2y(t)}{dt^2} + a \frac{dy(t)}{dt}+a^2y(t)=x(t).$$ Let another signal $g(t)$ be defined as $$g(t)=a^2 \int_0^t h(\tau) d \tau +\frac{dh(t)}{dt}+ah(t).$$ If …

Consider a transfer function $G_p(s) = \frac{ps^2+3ps-2}{s^2+(3+p)s+(2-p)}$ with $p$ a positive real parameter. The maximum value of $p$ until which $G_p$ remains stable is ___________.

The characteristic equation of a unity negative feedback system is $1+KG(s)=0$. The open loop transfer function $G(s)$ has one pole at $0$ and two poles at $-1$. The root locus of the system for varying $K$ is shown in the figure. The constant damping ratio line, for $\xi = 0.5$, intersects the root locus at point A. The distance from the origin …

Two systems with impulse responses $h_{1}(t)$ and $h_{2}(t)$ 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

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

The open-loop transfer function of a dc motor is given as $\dfrac{\omega(s)}{V_{a}(s)} = \dfrac{10}{1+10s}.$ When connected in feedback as shown below, the approximate value of $K_{a}$ that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open-loop system 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 state diagram of a system is shown below. A system is described by the state-variable equations $$\dot{X}= AX+Bu;\:\: y = CX+Du$$ The state-variable equations of the system shown in the figure above are

A system with transfer function $$G(s)=\frac{(s^2+9)(s+2)}{(s+1)(s+3)(s+4)}$$ is excited by $\sin(\omega t)$. The steady-state output of the system is zero at

The state variable description of an LTI system is given by $$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix}=\begin{pmatrix} 0 & a_1 & 0\\ 0 & 0 & a_2\\a_3 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}+\begin{pmatrix} 0\\ 0\\1 \end{pmatrix}u$$ …

Let $y[n]$ denote the convolution of $h[n]$ and $g[n]$, where $h[n]=(\frac{1}{2})^nu[n]$ and $g[n]$ is a casual sequence. If $y[0]=1$ and $y[1]=\frac{1}{2}$, then $g[1]$ equals

The transfer function of a compensator is given as $$G_c(s)=\frac{s+a}{s+b}$$ $G_c(s)$ is a lead compensator if

The transfer function of a compensator is given as $$G_c(s)=\frac{s+a}{s+b}$$ The phase of the above lead compensator is maximum at