# Sampling

 Question 1
A band limited low-pass signal x(t) of bandwidth 5 kHz is sampled at a sampling rate $f_{s}$. The signal x(t) is reconstructed using the reconstruction filter H(f) whose magnitude response is shown below:

The minimum sampling rate $f_{s}$ (in kHz) for perfect reconstruction of x(t) is _______.
 A 10 B 11 C 12 D 13
GATE EC 2018   Signals and Systems
Question 1 Explanation:
Let us assume an arbitrary spectrum for x(t) as shown below:

The spectrum of the sampled signal can be given as,

For proper reconstruction of the signal,
\begin{aligned} f_{s}-5 & \geq 8 \\ f_{s} &\geq 8+5=13 \mathrm{kHz} \\ \text { So. } \quad f_{s(\mathrm{min})} &=13 \mathrm{kHz} \end{aligned}
 Question 2
The signal $x(t)=sin(14000\pi t)$, where t is in seconds is sampled at a rate of 9000 samples per second. The sampled signal is the input to an ideal low pass filter with frequency response H(f) as follows :
$H(f)=\left\{\begin{matrix} 1, & |f|\leq 12 kHz\\ 0, & |f|\geq 12 kHz \end{matrix}\right.$
What is the number of sinusoids in the output and their frequencies in kHz?
 A Number = 1, frequency = 7 B Number = 3, frequencies= 2,7,11 C Number = 2, frequencies = 2, 7 D Number = 2, frequencies = 2, 11
GATE EC 2017-SET-2   Signals and Systems
Question 2 Explanation:
\begin{aligned} x(t) &=\sin (14000 \pi t) ; f_{m}=7 \mathrm{kHz} \\ f_{s} &=9000 \text { samples per second } \\ &=9 \mathrm{kHz} \end{aligned}
The spectrum of the sampled signal can be given as shown below:

So, three sinusoids will be there at the output of the LPF and the frequencies of those sinusoids are 2 kHz, 7 kHz and 11 kHz.
 Question 3
Consider the signal $x(t)=cos(6\pi t)+sin(8\pi t)$, where t is in seconds. The Nyquist sampling rate (in samples/second) for the signal y(t)=x(2t+ 5) is
 A 8 B 12 C 16 D 32
GATE EC 2016-SET-3   Signals and Systems
Question 3 Explanation:
\begin{aligned} X(t) & =\cos (6 \pi t)+\sin (8 \pi t) \\ Y(t) & =x(2 t+5) \\ Y(t) & =\cos [6 \pi(2 t+5)+\sin (8 \pi(2 t+5)] \\ & =\cos (12 \pi t+30 \pi)+\sin (16 \pi t+40 \pi) \\ f_{m_{1}} & =6 \mathrm{Hz}_{i} \qquad f_{m 2}=8 \mathrm{Hz}\\ \text{Nyquist }&\text{sampling rate},\\ f_{s} &=2 f_{\max } \\ &=16 \text { samples/second } \end{aligned}
 Question 4
A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________
 A 5 B 12 C 18 D 24
GATE EC 2016-SET-1   Signals and Systems
Question 4 Explanation:
If x(t) is a message signal and y(t) is a sampled signal, then y(t) is related to x(t) as
$y(t)=x(t) \sum_{n=-\infty}^{\infty} \delta\left(t-n T_{s}\right)$
$r(f)=f_{s} \sum_{n=--\infty}^{\infty} X\left(f-n f_{s}\right)$
Spectrum of X(f) and Y(f) are as shown

Cut off frequency of LPF = 23 Hz
Hence, frequency at the output is 13 Hz
 Question 5
Consider a continuous-time signal defined as

$x(t)=(\frac{sin(\pi t/2)}{(\pi t/2)})*\sum_{n=-\infty }^{\infty }\delta (t-10n)$

where '*' denotes the convolution operation and t is in seconds. The Nyquist sampling rate (in samples/sec) for x(t) is _____.
 A 0.4 B 0.8 C 0.2 D 1
GATE EC 2015-SET-3   Signals and Systems
Question 5 Explanation:
$x(t)=\frac{\sin \left(\frac{\pi t}{2}\right)}{\left(\frac{\pi t}{2}\right)} * \sum_{n=-\infty}^{\infty} \delta[t-10 n]$

\begin{aligned} f_{\max } &=0.2 \mathrm{Hz} \\ \Rightarrow \quad f_{s} &=2 f_{\max }=0.4 \mathrm{Hz} \end{aligned}
 Question 6
The signal $cos(10\pi t+\frac{\pi }{4})$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $(\frac{sin(\pi t)}{\pi t})cos(40\pi t-\frac{\pi }{2})$. The filter output is
 A $\frac{15}{2}cos(40\pi t-\frac{\pi }{4})$ B $\frac{15}{2}(\frac{sin(\pi t)}{\pi t})cos(10\pi t+\frac{\pi }{4})$ C $\frac{15}{2}cos(10\pi t-\frac{\pi }{4})$ D $\frac{15}{2}(\frac{sin(\pi t)}{\pi t})cos(40\pi t-\frac{\pi }{4})$
GATE EC 2015-SET-2   Signals and Systems
Question 6 Explanation:
$\begin{array}{l} x(t)=\cos \left(10 \pi t+\frac{\pi}{4}\right) \\ X(f)=\frac{1}{2}[\delta(f-5)+\delta(f+5)] e^{j \pi / 4} \end{array}$
After sampling x(t) by 15 Hz, we get
$X_{s}(f)=15 \sum_{n=-\infty}^{\infty} X(f-15 n)$
Given, $\quad h(t)=\left[\frac{\sin (\pi t)}{\pi t}\right] \cos \left(40 \pi t-\frac{\pi}{2}\right)$
$H(f)=\frac{1}{2}\left[\text{rect} \frac{(f-20)}{2 \pi}+\text{rect} \frac{(f+20)}{2 \pi}\right] e^{-j \frac{\pi}{2}}$
\begin{aligned} \text{Output }\quad Y(t)&=X_{s}(f) \times H(f) \\ y(t)&=\frac{15}{2} \cos \left(40 \pi t-\frac{\pi}{4}\right) \end{aligned}
 Question 7
For a given sample-and-hold circuit, if the value of the hold capacitor is increased, then
 A droop rate decreases and acquisition time decreases B droop rate decreases and acquisition time increases C droop rate increases and acquisition time decreases D droop rate increases and acquisition time increases
GATE EC 2014-SET-4   Signals and Systems
Question 7 Explanation:
We know that
\begin{aligned} Q&=C V=i \cdot t &\ldots(i)\\ t&=\frac{C V}{i} &\ldots(ii) \end{aligned}
From equation (ii)
$t \propto C$ and it is clear that if value of capacitor increases then the acquisition time increases. For capacitor the drop rate is given as $\frac{d v}{d t}$
\begin{aligned} i &=C \frac{d v}{d t} \\ \frac{d v}{d t} &=\frac{i}{C} \\ \frac{d v}{d t} & \propto \frac{1}{C} \end{aligned}
From above relation it is clear that if capacitor value increases drop rate decreases.
 Question 8
Let $x(t)=cos(10\pi t)+cos(30\pi t)$ be sampled at 20 Hz and reconstructed using an ideal low-pass filter with cut-off frequency of 20 Hz. The frequency/ frequencies present in the reconstructed signal is/are
 A 5 Hz and 15 Hz only B 10 Hz and 15 Hz only C 5 Hz, 10 Hz and 15 Hz only D 5 Hz only
GATE EC 2014-SET-3   Signals and Systems
Question 8 Explanation:
$x(t)=\cos (10 \pi t)+\cos (30 \pi t)$
Given sampling frequency, $f_{s}=20 \mathrm{Hz}$
$\omega_{s}=40 \pi \mathrm{rad} / \mathrm{sec}$

$Y(\omega)=\sum_{K=-\infty}^{\infty} X\left(\omega-K \omega_{s}\right)$
After sampling waveform will be

Applying an ideal low-pass filter of cut-off frequency of $20 \mathrm{Hz}$ or $40 \pi \; \mathrm{ rad} / \mathrm{sec},$ we get the frequencies in reconstructed signal as $10 \pi \text{ and } 30 \pi \; \mathrm{rad} / \mathrm{sec}$.
or $5 \mathrm{Hz}$ and $15 \mathrm{Hz}$
 Question 9
Consider two real valued signals, x(t) band-limited to [-500 Hz, 500 Hz] and y(t) band-limited to [-1 kHz, 1 kHz]. For $z(t) = x(t)\cdot y(t)$, the Nyquist sampling frequency (in kHz) is ____.
 A 1 B 2 C 3 D 4
GATE EC 2014-SET-1   Signals and Systems
Question 9 Explanation:
Multiplication in time domain = convolution in frequency domain.
$x_{1}(t) \cdot x_{2}(t) \leftrightarrow X_{1}(\omega) * X_{2}(\omega)$
So, highest frequency component contained by the convolved signal $z(t)=1500 \mathrm{Hz}$
$\therefore$ Nyquist rate $=2 \times 1500$
$=3000 \mathrm{Hz}=3 \mathrm{kHz}$
 Question 10
A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is
 A 5 kHz B 12 kHz C 15 kHz D 20 kHz
GATE EC 2013   Signals and Systems
Question 10 Explanation:
\begin{aligned} \left(f_{s}\right)_{\min } &=2 f_{m} \\ \left(f_{s}\right)_{\min } &=2 \times 5=10 \mathrm{kHz} \\ \mathrm{So}, \quad f_{s} & \geq 10 \mathrm{kHz} \end{aligned}
There are 10 questions to complete.