# Questions & Answers of Linear Time-Invariant (LTI) Systems

## Weightage of Linear Time-Invariant (LTI) Systems

Total 18 Questions have been asked from Linear Time-Invariant (LTI) Systems topic of Signals and Systems subject in previous GATE papers. Average marks 1.50.

Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems ($u\left(t\right)$ denotes the unit-step function)?

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

$\sum\limits_{k=1}^3a_k\;\cos\left(k\omega_0t\right),\;\mathrm{where}\;a_k\neq0\;,\omega_0\neq0.$

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to $t=0$). Match the excitation signals X, Y, Z with the corresponding time responses for $t\geq0;$
 X: Impulse $P:1-{e}^{-t/T}$ Y: Unit step $Q:t-T\left(1-{e}^{-t/T}\right)$ Z: Ramp ${\mathrm{R:e}}^{-t/T}$

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

The result of the convolution $\mathrm x(-\mathrm t)\;\ast\;\mathrm\delta(-\mathrm t-{\mathrm t}_0)$ is

The impulse response of an LTI system can be obtained by

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input $x\left[n\right]$, the response $y\left[n\right]$ is

The complex envelope of the bandpass signal $x\left(t\right)=-\sqrt{2}\left(\frac{\mathrm{sin}\left(\pi t/5\right)}{\pi t/5}\right)\mathrm{sin}\left(\pi t-\frac{\pi }{4}\right)$,centered about $f=\frac{1}{2}\mathrm{Hz}$, is

A continuous, linear time-invariant filter has an impulse response h(t) described by

When a constant input of value 5 is applied to this filter, the steady state output is_____.

For a function g(t), it is given that ${\int }_{-\infty }^{+\infty }g\left(t\right){e}^{-jwt}dt=\omega {e}^{-2{\omega }^{2}}$ for any real value $\omega$.
If $y\left(t\right)={\int }_{-\infty }^{t}g\left(t\right)d\tau ,$, then is

Consider a discrete-time signal

If y[n] is the convolution of x[n] with itself, the value of y[4] is _________.

The input $-3{e}^{2t}u\left(t\right)$, where $u\left(t\right)$ is the unit step function, is applied to a system with transfer function $\frac{s-2}{s+3}$. If the initial value of the output is −2, then the value of the output at steady state is _______.

The sequence $x\left[n\right]=0.{5}^{n}u\left[n\right],$ where $u\left[n\right]$ is the unit step sequence, is convolved with itself to obtain $y\left[n\right]$. Then ${\sum }_{n=-\infty }^{+\infty }y\left[n\right]$ is _______.

The DFT of a vector $\left[\begin{array}{cccc}a& b& c& d\end{array}\right]$ is the vector $\left[\begin{array}{cccc}\alpha & \beta & \gamma & \delta \end{array}\right]$ . Consider the product

$\left[\begin{array}{cccc}p& q& r& s\end{array}\right]$=$\left[\begin{array}{cccc}a& b& c& d\end{array}\right]$$\begin{array}{}\\ \\ \\ \end{array}\left[\begin{array}{cccc}a& b& c& d\\ d& a& b& c\\ c& d& a& b\\ b& c& d& a\end{array}\right]$

The DFT of the vector $\left[\begin{array}{cccc}p& q& r& s\end{array}\right]$ is a scaled version of

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

A system is defined by its impulse response h(n) = 2n u(n − 2). The system is

The impulse response $h\left(t\right)$ of a linear time-invariant continuous time system is described by $h\left(t\right)=exp\left(\alpha t\right)u\left(t\right)+exp\left(\beta t\right)u\left(-t\right)$, where $u\left(t\right)$ denotes the unit step function, and $\alpha$ and $\beta$ are real constants. This system is stable if