# GATE Questions & Answers of Linear Time Invariant and Causal Systems

## What is the Weightage of Linear Time Invariant and Causal Systems in GATE Exam?

Total 23 Questions have been asked from Linear Time Invariant and Causal Systems topic of Signals and Systems subject in previous GATE papers. Average marks 1.43.

Consider a continuous-time system with input $x\left(t\right)$ and output $y\left(t\right)$ given by

$y\left(t\right)=x\left(t\right)\mathrm{cos}\left(t\right)$

This system is

Consider the following state-space representation of a linear time-invariant system.

$\overset.x\left(t\right)=\begin{bmatrix}1&0\\0&2\end{bmatrix}x\left(t\right),y\left(t\right)=c^Tx\left(t\right),\;c=\begin{bmatrix}1\\1\end{bmatrix}\;\mathrm{and}\;x\left(0\right)=\begin{bmatrix}1\\1\end{bmatrix}$

The value of $y\left(t\right)$ for $t={\mathrm{log}}_{e}2$ is __________.

The output of a continuous-time, linear time-invariant system is denoted by $\{x(t)\}$ where $x\left(t\right)$ is the input signal. A signal $z\left(t\right)$ is called eigen-signal of the system T , when $\{z(t)\}=y\;z(t)$, where $\gamma$ is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE

A moving average function is given by $\style{font-family:'Times New Roman'}{y\left(t\right)=\;\frac1t\int_{t-T}^tu\left(\tau\right)}$. If the input u is a sinusoidal signal of frequency $\frac{1}{2T}\mathrm{Hz},$ then in steady state, the output $y$ will lag $u$ (in degree) by ________.

The impulse response g(t) of a system, G, is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure(b)?

For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?

$x\left(t\right)$ is nonzero only for ${T}_{x} , and similarly, $y\left(t\right)$ is nonzero only for ${T}_{y}. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE?

Consider an LTI system with transfer function

$H\left(S\right)=\frac{1}{s\left(s+4\right)}$

If the input to the system is $\mathrm{cos}\left(3t\right)$ and the steady state output is $A\mathrm{sin}\left(3t+\alpha \right)$, then the value of $A$ is

Consider an LTI system with impulse response $h\left(t\right)={e}^{-5t}u\left(t\right)$. If the output of the system is $y\left(t\right)={e}^{-3t}u\left(t\right)-{e}^{-5t}u\left(t\right)$ then the input, $x\left(t\right)$, is given by

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

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

The input x(t) and output y(t) of a system are related as The system is

Given two continuous time signals $x\left(i\right)={e}^{-t}$ and $y\left(i\right)={e}^{-2t}$ which exist for $t>0,$ the convolution $z\left(t\right)=x\left(t\right)*y\left(t\right)$ is

The system represented by the input-output relationship $\style{font-size:14px}{y\left(t\right)=\int\limits_{-\infty}^{5t}x\left(\tau\right)d\tau,t>0}$ is

Given the finite length input x[n] and the corresponding finite length output y[n] of an LTI system as shown below, the impulse response h[n] of the system is

A Linear Time Invariant system with an impulse response h(t) produces output y(t) when input x(t) is applied. When the input x(t-$\tau$) is applied to a system with impulse response h(t-$\tau$), the output will be

A cascade of 3 Linear Time Invariant systems is causal and unstable. From this, we conclude that

A signal ${e}^{-\alpha t}\mathrm{sin}\left(\omega t\right)$ is the input to a real Linear Time Invariant system. Given K and $\phi$ are constants, the output of the system will be of the form $\style{font-size:18px}{Ke^{-\beta t}\sin\left(\nu t+\phi\right)}$ where

The impulse response of a causal linear time-invariant system is given as h(t) . Now consider the following two statements :
Statement (I): Principle of superposition holds
Statement (II):h(t)=0 for t<0
Which one of the following statements is correct ?

A system with x(t) and output y(t) is defined by the input-output relation :

$y\left(t\right)=\int\limits_{-\infty}^{-2t}x\left(t\right)d\tau$

The system will be

A signal $x\left(t\right)=\mathrm{sin}c\left(\alpha t\right)$ where α  is a real constant $\left(\mathrm{sin}c\left(x\right)=\frac{\mathrm{sin}\left(\mathrm{\pi x}\right)}{\mathrm{\pi x}}\right)$ is the input to a Linear Time Invariant system whose impulse response h(t)=sin(βt),where β is a real constant. If min$\left(\alpha ,\beta \right)$denotes the minimum of α and β and similarly,$\mathrm{max}\left(\alpha ,\beta \right)$ denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the output of the system ?

Let a signal $a_1\sin\left(\omega_1t+\phi_1\right)$ be applied to a stable linear time invariant system. Let the corresponding steady state output be represented as $a_2F\left(\omega_2t+\phi_2\right)$. Then which of the following statement is true?