Sampling


Question 1
Let x_{1}(t) and x_{2}(t) be two band-limited signals having bandwidth B=4 \pi \times 10^{3} \mathrm{rad} / \mathrm{s} \mathrm{each}. In the figure below, the Nyquist sampling frequency, in rad/s, required to sample y(t), is

A
20 \pi \times 10^{3}
B
40 \pi \times 10^{3}
C
8 \pi \times 10^{3}
D
32 \pi \times 10^{3}
GATE EC 2023   Signals and Systems
Question 1 Explanation: 
Given that, x_{1}(t) and x_{2}(t) are two bandlimited signals having bandwidth B=4 \pi \times 10^{3} \mathrm{rad} / \mathrm{sec}.

and y(t)=x_{2}(t) \cos \left(12 \pi \times 10^{3} t\right)+x_{1}(t) \cos \left(4 \pi \times 10^{3} t\right)

So, \text{Nyquist Rate }=2\omega _{max}=2[16 \pi \times 10^3]=32 \pi \times 10^3\; rad/sec
Question 2
Consider a real-valued base-band signal x(t), band limited to \text{10 kHz}. The Nyquist rate for the signal y\left ( t \right )=x\left ( t \right )x\left ( 1+\dfrac{t}{2} \right ) is
A
\text{15 kHz}
B
\text{30 kHz}
C
\text{60 kHz}
D
\text{20 kHz}
GATE EC 2021   Signals and Systems
Question 2 Explanation: 






\mathrm{NR}=2 \times f_{\mathrm{max}}=2 \times 15=30 \mathrm{kHz}


Question 3
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 3 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 4
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 4 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 5
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 5 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}


There are 5 questions to complete.