GATE EC 2023 Set 1 Q16 — LTI Systems

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?

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