# Sampling

 Question 1
Consider the two continuous-time signals defined below:

$x_{1}(t)=\left\{\begin{matrix} |t|, & -1\leq t \leq 1 \\ 0 & otherwise \end{matrix}\right.$,

$x_{2}(t)=\left\{\begin{matrix} 1-|t|, & -1\leq t \leq 1\\ 0, & otherwise \end{matrix}\right.$

These signals are sampled with a sampling period of T=0.25 seconds to obtain discretetime signals $x_{1}[n] \; and \; x_{2}[n]$, respectively. Which one of the following statements is true?
 A The energy of $x_{1}[n]$ is greater than the energy of $x_{2}[n]$ B The energy of $x_{1}[n]$ is greater than the energy of $x_{2}[n]$ C $x_{1}[n] nor x_{2}[n]$ have equal energies. D Neither $x_{1}[n] nor x_{2}[n]$ is a finite-energy signal
GATE EE 2018   Signals and Systems
Question 1 Explanation:
$x_1(t)=\left\{\begin{matrix} |t|, & -1 \leq t \leq 1\\ 0, & \text{ otherwise} \end{matrix}\right.$

$T_s=$ sampling time-period = 0.25 sec
$x_1(n)=\{ 1,0.75,0.5,0.25,0,0.25,0.5,0.75,1\}$

$x_2(t)=\left\{\begin{matrix} 1-|t|, & -1 \leq t \leq 1\\ 0, & \text{ otherwise} \end{matrix}\right.$

$x_2(n)=\{ 0,0.25,0.5,0.75,1,0.75,0.5,0.25,0\}$

Since, $x_1(n)$ is having one more non-zero sample of amplitude '1' as compared to $x_2(n)$. Therefore, energy of $x_1(n)$ greater than energy of $x_2(n)$.
 Question 2
The output y(t) of the following system is to be sampled, so as to reconstruct it from its samples uniquely. The required minimum sampling rate is
 A 1000 samples/s B 1500 samples/s C 2000 samples/s D 3000 samples/s
GATE EE 2017-SET-2   Signals and Systems
Question 2 Explanation:

From the above block diagram,
$Z(t)=x(t) \cos 1000 \pi t$
By using modulation property of fourier transform,
$Z(\omega )=\frac{1}{2}[X(\omega + 1000 \pi t)+X(\omega - 1000 \pi t)]$

Now, $h(t)=\frac{\sin 1500 \pi t}{\pi t}=1500 Sa(1500 \pi t)$

Thus, $H(\omega )$ is a low pass filter and it will pass frequency, component of $Z(\omega )$ upto $1500 \pi$ rad/sec.

Therefore, maximum frequency component of $y(t)$ is
$\omega _m =1500 \pi$ rad/sec or $f_m =750 Hz$
So, the minimum sampling rate for $y(t)$ is
$f_{s\; min}=2f_m=1500Hz=1500$ samples/sec
 Question 3
Let $x_{1}(t)\leftrightarrow X_{1}(\omega ) \; and \; x_{2}(t) \leftrightarrow X_{2}(\omega \omega )$ be two signals whose Fourier Transforms are as shown in the figure below. In the figure, $h(t)=e^{-2|t|}$ denotes the impulse response.

For the system shown above, the minimum sampling rate required to sample y(t), so that y(t) can be uniquely reconstructed from its samples, is
 A $2B_{1}$ B $2(B_{1}+B_{2})$ C $4(B_{1}+B_{2})$ D $\infty$
GATE EE 2016-SET-2   Signals and Systems
Question 3 Explanation:
Given that,
Bandwidth of $X_1(\omega )=B_1$
Bandwidth of $X_2(\omega )=B_2$
system has $h(t)=e^{-2|t|}$ and input to the system is $x_1(t) \cdot x_2(t)$
The bandwidth of $x_1(t) \cdot x_2(t)$ is $B_1+B_2$.
The bandwidth of output will be $B_1+B_2$.
So, sampling rate will be $2(B_1+B_2)$.
 Question 4
A sinusoid x(t) of unknown frequency is sampled by an impulse train of period 20 ms. The resulting sample train is next applied to an ideal lowpass filter with cutoff at 25 Hz. The filter output is seen to be a sinusoid of frequency 20 Hz. This means that x(t)
 A 10Hz B 60 Hz C 30 Hz D 90 Hz
GATE EE 2014-SET-3   Signals and Systems
Question 4 Explanation:
Given, impulse train of period 20 ms.
Then, sampling frequency $=\frac{1}{20 \times 10^{-3}}=50Hz$
If the input signal $x(t)= \cos \omega _m (t)$ having spectrum

The filtered out sinusoidal signal has 20 Hz frequency, the sampling must be under sampling. The output signal which is an under sampled signal with sampling frequency 50 Hz is

and $50-f_m=20Hz\;\Rightarrow \;f_m=30Hz$
 Question 5
For the signal
$f(t) = 3 \sin 8 \pi t + 6 \sin 12 \pi t + \sin 14 \pi t$,
the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is _____.
 A 7 B 14 C 18 D 9
GATE EE 2014-SET-3   Signals and Systems
Question 5 Explanation:
\begin{aligned} f_{m_1} &=4Hz \\ f_{m_2} &=6Hz \\ f_{m_3} &=7Hz \end{aligned}
Then minimum sampling frequency satisfying the nyquist criterion is 7*2=14Hz.
 Question 6
A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency in kHz which is not valid is
 A 5 kHz B 12 kHz C 15 kHz D 20 kHz
GATE EE 2013   Signals and Systems
Question 6 Explanation:
\begin{aligned} (f_s)_{min}&=2f_m \\ (f_s)_{min}&=2 \times 5=10kHz \\ f_s&\geq 10kHz \end{aligned}
 Question 7
The frequency spectrum of a signal is shown in the figure. If this is ideally sampled at intervals of 1 ms, then the frequency spectrum of the sampled signal will be

 A A B B C C D D
GATE EE 2007   Signals and Systems
Question 7 Explanation:

Given that, sampling interval =1 msec
i.e. $T_s=1 \; msec=10^{-3}\; sec$
Therefore sampling frequency
$f_s=\frac{1}{T_s}=\frac{1}{10^{-3}}=1 kHz$
after sampling new signal in frequency domain
$U_T(f)=\frac{1}{T_s}\sum_{n=-\infty }^{\infty }U(f-nf_s)$
Therefore, spectrum of sampled signal will be

There are 7 questions to complete.