GATE EC LTI Systems questions

The response of a discrete time system $y[n]$ obeys the following relation: $$y[n]=\frac{5}{6} y[n-1]-\frac{1}{6} y[n-2]+x[n] .$$ The input to the system is $x[n]=\delta[n]-\frac{1}{3} \delta[n-1]$. Which of the following options is TRUE for $y[n]$?

Consider the discrete time system (S) with input $x[n]$ and output $y[n]$ as shown in the Figure. The two sub-systems represented by their impulse responses $h_{1}[n]$ and $h_{2}[n]$ are linear and time invariant. Which of the following statements is necessarily TRUE?

Consider the discrete-time system below with input $x[n]$ and output $y[n]$. In the figure, $h_{1}[n]$ and $h_{2}[n]$ denote the impulse responses of LTI Subsystems $1$ and $2$, respectively. Also, $\delta[n]$ is the unit impulse, and $b>0$. Assuming $h_{2}[n] \neq \delta[n]$, the overall system (denoted by the dashed box) is …

​​​​A causal and stable $\text{LTI}$ system with impulse response $h(t)$ produces an output $y(t)$ for an input signal $x(t)$. A signal $x(0.5 t)$ is applied to another causal and stable $\text{LTI}$ system with impulse response $h(0.5 t)$. The resulting output is $\_\_\_\_\_\_\_$.

​​​​​For a causal discrete-time LTI system with transfer function $$H(z)=\frac{2 z^{2}+3}{\left(z+\frac{1}{3}\right)\left(z-\frac{1}{3}\right)}$$ which of the following statements is/are true?

Consider a system with input $x(t)$ and output $y(t)=x\left(e^{t}\right)$. The system is

Let an input $x[n]$ having discrete-time Fourier transform $X\left(e^{j \Omega}\right)=1-e^{-j \Omega}+2 e^{-3 j \Omega}$ be passed through an LTI system. The frequency response of the LTI system is $H\left(e^{j \Omega}\right)=1-\frac{1}{2} e^{-j 2 \Omega}$. The output $y[n]$ of the system is

Let $x(t)=10 \cos (10.5 W t)$ be passed through an LTI system having impulse response $h(t)=\pi\left(\frac{\sin W t}{\pi t}\right)^{2} \cos 10 Wt$. The output of the system is

Let $\mathrm{x}_1(\mathrm{t})=\mathrm{u}(\mathrm{t}+1.5)-\mathrm{u}(\mathrm{t}-1.5)$ and $\mathrm{x}_2(\mathrm{t})$ is shown in the figure below. For $\mathrm{y}(\mathrm{t})=\mathrm{x}_1(\mathrm{t}) * \mathrm{x}_2(\mathrm{t})$, the $\int_{-\infty}^{\infty} \mathrm{y}(\mathrm{t}) \mathrm{dt}$ is ______________ (rounded off to the nearest integer).

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure. Statement $\text{I}:$ The system is necessarily a pure delay system for inputs which are bandlimited to $- \alpha \leq f \leq \alpha.$ Statement $\text{II}:$ For any wide-sense stationary input process with power spectral density $S_{x}(f),$ the …

Let $x_{1} (t) = e^{-t}u(t)$ and $x_{2}(t) = u(t) – u(t – 2),$ where $u(\cdot)$ denotes the unit step function. If $y(t)$ denotes the convolution of $x_{1}(t)$ and $x_{2}(t),$ then $\displaystyle \lim_{t \rightarrow \infty} y(t) =$ ______________ (rounded off to one decimal place).

The outputs of four systems $(S_{1}, S_{2}, S_{3},$ and $\text{S}_{4})$ corresponding to the input signal $\sin(t),$ for all time $t,$ are shown in the figure. Based on the given information, which of the four systems is/are definitely $\text{NOT LTI}$ (linear and time-invariant)?

For a unit step input $u[n]$, a discrete-time $\text{LTI}$ system produces an output signal $\left ( 2\delta \left [ n+1 \right ] +\delta \left [ n \right ]+\delta \left [ n-1 \right ]\right )$. Let $y[n]$ be the output of the system for an input $\left ( \left ( \dfrac{1}{2} \right )^n u\left [ n \right ]\right )$. The value of $y[0]$ is …

The output $y[n]$ of a discrete-time system for an input $x[n]$ is $$y\left [ n \right ]=\underset{-\infty \leq k\leq n}{\text{max}} \mid x\left [ k \right ] \mid$$ The unit impulse response of the system is

The pole-zero map of a rational function $G(s)$ is shown below. When the closed contour $\Gamma$ is mapped into the $G(s)$-plane, then the mapping encircles

Which one of the following pole-zero plots corresponds to the transfer function of an $\text{LTI}$ system characterized by the input-output difference equation given below? $$y\left [ n \right ]=\sum ^{3}_{k=0}\left ( -1 \right )^{k}x\left [ n-k \right ]$$

The loop transfer function of a negative feedback system is $$G\left ( s \right )H\left ( s \right )=\frac{K(s+11)}{s(s+2)(s+8)}.$$ The value of $K$, for which the system is marginally stable, is ___________.

A system with transfer function $G\left ( s \right )=\dfrac{1}{\left ( s+1 \right )\left ( s+a \right )},\:\:a> 0$ is subjected to an input $5 \cos3t$. The steady state output of the system is $\dfrac{1}{\sqrt{10}}\cos\left ( 3t-1.892 \right )$. The value of $a$ is _______.

The transfer function of a stable discrete-time $\text{LTI}$ system is $H\left ( z \right )=\dfrac{K\left ( z-\alpha \right )}{z+0.5}$, where $K$ and $\alpha$ are real numbers. The value of $\alpha$ (rounded off to one decimal place) with $\mid \alpha \mid > 1$, for which the magnitude response of the system is constant over all frequencies, …

Consider the following closed loop control system where $G\left ( s \right )=\dfrac{1}{s\left ( s+1 \right )}$ and $C\left ( s \right )=K\dfrac{s+1}{s+3}$. If the steady state error for a unit ramp input is $0.1$, then the value of $K$ is ______________.