# Linear Time Invariant Systems

 Question 1
Which of the following statement(s) is/are true?
 A If an LTI system is causal, it is stable B A discrete time LTI system is causal if and only if its response to a step input $u[n]$ is 0 for $\mathrm{n} \lt 0$ C If a discrete time LTI system has an impulse response $h[n]$ of finite duration the system is stable D If the impulse response $0 \lt |h[n]| \lt 1$ for all $n$, then the $L T I$ system is stable.
GATE EE 2023   Signals and Systems
Question 1 Explanation:
For causal system, impuse response
$h(n)=0 ; \quad n \lt 0$

Therefore, for step input also
$h(n)=0 ; n \lt 0$
 Question 2
Consider the system as shown below where $y(t)=x(e^t)$. The system is
 A linear and causal. B linear and non-causal. C non-linear and causal D non-linear and non-causal
GATE EE 2022   Signals and Systems
Question 2 Explanation:
We know, a linear system follows the law of superposition.
It is a combination of two laws: Both results are same, hence, it follows law of additivity.
(ii) Law of Homogeneity: Here also both results are same, hence it follows law of Homogeneity.
Therefore, System is linear.
We know, a causal system is independent of future values of input at each & every instant of time them system will be causal.
Given : $y(t)=x(e^t)$
Put $t=0$
$y(0)=x(e^0)=x(1)$
Because its depends on future value.
Therefore, system is non-causal.

 Question 3
Let a causal LTI system be governed by the following differential equation $y(t)+\frac{1}{4}\frac{dy}{dt}=2x(t)$, where $x(t)$ and $x(t)$ are the input and output respectively. Its impulse response is
 A $2e^{-\frac{1}{4}t}u(t)$ B $2e^{-{4}t}u(t)$ C $8e^{-\frac{1}{4}t}u(t)$ D $8e^{-{4}t}u(t)$
GATE EE 2022   Signals and Systems
Question 3 Explanation:
Given:
$y(t)+\frac{1}{4}\frac{dy}{dt}=2x(t)$
Taking Laplace transform,
$Y(s)+\frac{1}{4}(sY(s))=2X(s)$
Now, $H(s)=\frac{Y(s)}{X(s)}=\frac{2}{\frac{s}{4}+1}=\frac{8}{s+4}$
Taking inverse Laplace, transform,
$h(t)=8e^{-4t}u(t)$
 Question 4
If the input x(t) and output y(t) of a system are related as $y\left ( t \right )=\text{max}\left ( 0,x\left ( t \right ) \right )$, then the system is
 A linear and time-variant B linear and time-invariant C non-linear and time-variant D non-linear and time-invariant
GATE EE 2021   Signals and Systems
Question 4 Explanation:
\begin{aligned} y(t) &=\max (0, x(t)) \\ &=\left\{\begin{array}{cl} 0, & x(t) \lt 0 \\ x(t), & x(t)\gt 0 \end{array}\right. \end{aligned} Linearity check:
at input $x_{1}(t)=-2$, output $y_{1}(t)=0$
at input $x_{2}(t)=1$, output $y_{2}(t)=1$ $\therefore$ system is non-linear because it violates law of additivity.
Check for time-invariance :
Delayed $\mathrm{O} / \mathrm{P}:$
$y\left(t-t_{0}\right)=\left\{\begin{array}{cl} x\left(t-t_{0}\right), & x\left(t-t_{0}\right) \gt 0 \\ 0 & x\left(t-t_{0}\right)\lt 0 \end{array}\right.$
$\mathrm{O} / \mathrm{P}$ of system when input is $x\left(t-t_{0}\right)=f(t)$
$y_{1}(t)=\left\{\begin{array}{cl} f(t), & f(t) \gt 0 \\ 0, & f(t)\lt 0 \end{array}=\left\{\begin{array}{cl} x\left(t-t_{0}\right), & x\left(t-t_{0}\right)\gt 0 \\ 0, & x\left(t-t_{0}\right)\lt 0 \end{array}\right.\right.$
Therefore, system is time-invariant.
 Question 5
Which of the following options is true for a linear time-invariant discrete time system that obeys the difference equation:

$y[n]-ay[n-1]=b_0x[n]-b_1x[n-1]$
 A y[n] is unaffected by the values of $x[n - k]; k \gt 2$ B The system is necessarily causal. C The system impulse response is non-zero at infinitely many instants. D When $x[n] = 0, n \lt 0$, the function $y[n]; n \gt 0$ is solely determined by the function x[n].
GATE EE 2020   Signals and Systems
Question 5 Explanation:
\begin{aligned} y(n)-ay(n-1)&=b_{0}x(n)-b_{1}x(n-2) \\ &\text{By applying ZT,} \\ Y(z)-az^{-1}Y(z)&=b_{0}X(z)-b_{1}z^{-1}X(z)\\ \Rightarrow \, \, H(z)&=\frac{Y(z)}{X(z)}=\frac{b_{0}-b_{1}z^{-1}}{1-az^{-1}} \end{aligned}
By taking right-sided inverse ZT,
$h(n)=b_{0}a^{n}u(n)-b_{1}a^{n-1}u(n-1)$
By taking left-sided inverse ZT,
$h(n)=-b_{0}a^{n}u(-n-1)+b_{1}a^{n-1}u(-n)$
Thus system is not necessarily causal.
The impulse response is non-zero at infinitely many instants.

There are 5 questions to complete.