# Basics of signals and Systems

 Question 1
Consider the signal $f(t)=1+2cos(\pi t)+3sin\left ( \frac{2 \pi}{3}t \right )+4cos\left (\frac{\pi}{2}t+\frac{\pi}{4} \right )$, where t is in seconds. Its fundamental time period, in seconds, is ____________
 A 8 B 12 C 16 D 20
GATE EC 2019   Signals and Systems
Question 1 Explanation:
\begin{aligned} f(t)=1+2 \cos (\pi t)&+3 \sin \left(\frac{2 \pi}{3} t\right)+4 \cos \left(\frac{\pi}{2} t+\frac{\pi}{4}\right) \\ \omega_{1}&=\pi \\ \omega_{2}&=\frac{2 \pi}{3} \\ \omega_{3}&=\frac{\pi}{2} \\ \omega_{0}&=G C D\left(\pi, \frac{2 \pi}{3}, \frac{\pi}{2}\right)=\frac{\pi}{6} \end{aligned}
Fundamental period,
$N=\frac{2 \pi}{\omega_{0}}=\frac{2 \pi}{(\pi / 6)}=12$
 Question 2
Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:
 A $\frac{d^{3}y}{dt^{3}} + a_{1} \frac{d^{2}y}{dt^{2}} + a_{2}\frac{dy}{dt} + a_{3}y =$ $b_{3}u+b_{2}\frac{du}{dt}+b_{1}\frac{d^{2}u}{dt^{2}}$ (with initial rest conditions) B $y(t)=\int_{0}^{t}e^{a(t-r)}\beta u(\tau)d \tau$ C $y= au +b$, $b \neq 0$ D $y=au$
GATE EC 2018   Signals and Systems
Question 2 Explanation:
$y=a u+b, b \neq 0$ is a non-linear system.
 Question 3
The input $x(t)$ and the output $y (t)$ of a continuous-time system are related as $y(t)=\int_{t-T}^{t}x(u)du$. 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 EC 2017-SET-2   Signals and Systems
Question 3 Explanation:
Given that, $y(t)=\int_{t-T}^{t} x(u) d u$
since the given system satisfies both homogeneity and additivity properties, the system is linear.
Check for time invariance:
$y\left(t-t_{0}\right)=\int_{t-t_{0}-T}^{t-t_{0}} x(u) d u$
When the applied input is $x\left(t-t_{0}\right)$
\begin{aligned} y_{1}(t) &=\\ \int_{t-T}^{t} x\left(u-t_{0}\right) d u &=\int_{t-t_{0}-T}^{t-t_{0}} x(\tau) d \tau \\ &=y\left(t-t_{0}\right) \end{aligned}
$\Rightarrow$ System is time invariant
 Question 4
Consider a single input single output discrete-time system with x[n] as input and y[n] as output, where the two are related as
$y[n]=\left\{\begin{matrix} n|x[n]| & for 0\leq n\leq 10\\ x[n]-x[n-1] & othrwise \end{matrix}\right.$
Which one of the following statements is true about the system?
 A It is causal and stable B It is causal but not stable C It is not causal but stable D It is neither causal nor stable
GATE EC 2017-SET-1   Signals and Systems
Question 4 Explanation:
Since present output does not depend upon future values of input, the system is causal and also every bounded input produces bounded output, so it is stable.
 Question 5
A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period $T_{s}$. In which one of the following cases is the sampled signal periodic?
 A $T=\sqrt{2}T_{s}$ B $T=1.2T_{s}$ C Always D Never
GATE EC 2016-SET-1   Signals and Systems
Question 5 Explanation:
A signal is said to be periodic if $\frac{T}{T_{s}}$ is a rational number.
Here, $T=1.2 T_{s}$
$\Rightarrow \frac{T}{T_{s}}=\frac{6}{5} \quad$ Which is a rational number
 Question 6
Two sequences $x_{1}[n] \; and \; x_{2}[n]$ have the same energy. Suppose $x_{1}[n] = \alpha 0.5^{n} u[n]$, where $\alpha$ is a positive real number and u[n] is the unit step sequence. Assume
$x_{2}[n]=\left\{\begin{matrix} \sqrt{1.5} &for n=0,1 \\ 0 & otherwise. \end{matrix}\right.$
Then the value of $\alpha$ is _________.
 A 1 B 1.5 C 2 D 2.5
GATE EC 2015-SET-3   Signals and Systems
Question 6 Explanation:
\begin{array}{l} \text { Energy of } x_{1}[n]=\sum_{n=-\infty}^{\infty}\left|x_{1}[n]\right|^{2} \\ \begin{aligned} =\sum_{n=0}^{\infty} \alpha^{2} &\left(\frac{1}{4}\right)^{n}=\alpha^{2} \cdot \frac{1}{1-\frac{1}{4}}=\alpha^{2} \cdot \frac{4}{3} \\ \text { Energy of } x_{2}[n] &=1.5+1.5=3 \\ \Rightarrow \quad \alpha^{2} \frac{4}{3} &=3 \\ \alpha^{2}=\frac{9}{4} \\ \quad \alpha=1.5 \end{aligned} \end{array}
 Question 7
The waveform of a periodic signal x(t) is shown in the figure.

A signal g(t) is defined by $g(t)=x(\frac{t-1}{2})$. The average power of g(t) is _______.
 A 1 B 2 C 3 D 4
GATE EC 2015-SET-1   Signals and Systems
Question 7 Explanation:
$\begin{array}{l} x(t)=-3 t, \quad-1\lt t \lt 1 \\ x\left(\frac{t-1}{2}\right)=-\frac{3}{2}(t-1) \qquad-1 \lt t \lt 3 \\ \text { and } \quad T=6 \\ \text { Average power }=\frac{1}{6} \int_{-1}^{3}\left(-\frac{3}{2}(t-1)\right)^{2} d t=2 \end{array}$
 Question 8
A discrete time signal $x[n]=sin(\pi ^{2 }n)$, $n$ being an integer, is
 A periodic with period $\pi$ B periodic with period $\pi ^{2}$ C periodic with period $\pi ^/2$ D not periodic
GATE EC 2014-SET-1   Signals and Systems
Question 8 Explanation:
\begin{aligned} x[n] &=\sin \left(\pi^{2} n\right) \\ \omega_{0} &=\pi^{2} \\ \therefore \quad N &=\frac{2 \pi}{\omega_{0}} \cdot m \end{aligned}
where m is the smallest integer that converts $\frac{2 \pi}{\omega_{0}}$ into a integer value.
$\therefore \quad N=\frac{2 \pi}{\pi^{2}} \cdot m=\frac{2}{\pi} \cdot m$
So, there exists no such integer value of m which could make the N integer, so the system is not periodic.
 Question 9
The impulse response of a continuous time system is given by $h(t)= \delta (t-1)+\delta (t-3)$. The value of the step response at t = 2 is
 A $0$ B $1$ C $2$ D $3$
GATE EC 2013   Signals and Systems
Question 9 Explanation:
Step response = Integration of impulse response
$\int_{-\infty}^{t} \delta(t-1) d t=u(t-1)$
$\int_{-\infty}^{t} \delta(t-3) d t=u(t-3)$

\begin{aligned} \text{at}\quad t&=2\\ y(t)&=1 \end{aligned}
 Question 10
For a periodic signal $v\left ( t \right )=30\sin 100t+10\cos 300t+6\sin \left ( 500t+\frac{\pi }{4} \right )$, the fundamental frequency in rad/s is
 A 100 B 300 C 500 D 1500
GATE EC 2013   Signals and Systems
Question 10 Explanation:
$\begin{array}{l} \omega_{1}=100 \\ \omega_{2}=300 \\ \omega_{3}=500 \end{array}$
H.C.F. of $\left(\omega_{1}, \omega_{2} \text { and } \omega_{3}\right)=$ H.C.F. (100,300,500)
$\omega=100 \mathrm{rad} / \mathrm{sec}$
There are 10 questions to complete.