Sampling · Signals and Systems · GATE ECE
Marks 1
1
A continuous -time function $$x\left( t \right)$$ 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?
GATE ECE 2016 Set 1
2
Consider the signal $$\,x\left( t \right)$$ $$\,\,\, = \,\,\,\cos \left( {6\pi t} \right)\,\, + \,\,\sin \left( {8\pi t} \right),$$ where $$\,t$$ is in seconds. The Nyquist sampling rate (in samples/second) for the signal $$y\left( t \right)\, = x\,\,\left( {2t + 5} \right)$$ is
GATE ECE 2016 Set 3
3
The signal $$\cos \left( {10\pi t + {\pi \over 4}} \right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\,\left( {{{\sin \left( {\pi t} \right)} \over {\pi t}}} \right)\,\cos \left( {40\pi t - {\pi \over 2}} \right).$$ The filter output is
GATE ECE 2015 Set 2
4
A modulated signal is $$y\left( t \right)\, = \,\,\,\,\,\,\,\,\,m\left( t \right)\,\cos \left( {40000\pi t} \right),$$ where the baseband signal $$m\left( t \right)\,$$ has frequency components less than 5 kHz only. The minimum required rate (in kHz) at which $$y\,\,\left( t \right)$$ should be sampled to recover $$m\,\,\left( t \right)$$ is ________.
GATE ECE 2014 Set 3
5
Let $$\,x\,\,\left( t \right)\,\,\, = \,\,\,\cos \,\,\,\left( {10\pi t} \right)\,\, + \,\,\cos \,\,\left( {30\pi t} \right)$$ 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.
GATE ECE 2014 Set 3
6
Consider two real valued signals, $$x\left( t \right)$$ band - limited to $$\,\left[ { - 500Hz,\,\,500Hz} \right]$$ and $$y\left( t \right)$$ band - limited to $$\,\left[ { - 1\,\,kHz,\,\,1kHz} \right].$$ For $$z\left( t \right)\,\, = \,x\left( t \right).y\left( t \right),$$ the Nyquist sampling frequency $$\left( {in\,\,kHz} \right)$$ is________.
GATE ECE 2014 Set 1
7
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
GATE ECE 2013
8
Consider a sampled signal $$y\left( t \right) = 5 \times {10^{ - 6}}\,x\left( t \right)\,\,\sum\limits_{n = - \infty }^{ + \infty } {\delta \left( {t - n{T_s}} \right)} $$
where $$x\left( t \right) = 10\,\,\cos \,\left( {8\pi \times {{10}^3}} \right)\,\,t$$ and
$${T_s} = 100\,\,\mu \sec .$$ When $$y\left( t \right)$$ is passed through an ideal low-pass filter with a cutoff frequency of 5 KHz, the output of the filter is
GATE ECE 2002
9
A band limited signal is sampled at the Nyquist rate. The signal can be recovered by passing the samples through
GATE ECE 2001
10
Flat top sampling of low pass signals
GATE ECE 1998
11
A 1.0 kHz signal is flat - top sampled at the rate of 1800 samples/sec and the samples are applied to an ideal rectangular LPF with cut - off frequency of 1100 Hz, then the output of the filter contains
GATE ECE 1995
12
Increased pulse-width in the flat-top sampling, leads to
GATE ECE 1994
Marks 2
1
Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $\left[-\omega_c, \omega_c\right]$, where $\omega_c$ is in rad/sec.
The signal $f(t)$ is uniformly sampled to obtain $y(t)=f(t) p(t)$. Here
$$ p(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-\tau-n T_s\right) $$
with $\delta(t)$ being the Dirac impulse, $T_s>0$, and $\tau>0$. The sampled signal $y(t)$ is passed through an ideal lowpass filter $h(t)=\omega_c T_s \frac{\sin \left(\omega_c t\right)}{\pi \omega_c t}$ with cutoff frequency $\omega_c$ and passband gain $T_s$.
The output of the filter is given by $\qquad$ .
GATE ECE 2025
2
A continuous time signal $x(t) = 2 \cos(8 \pi t + \frac{\pi}{3})$ is sampled at a rate of 15 Hz. The sampled signal $x_s(t)$ when passed through an LTI system with impulse response
$h(t) = \left( \frac{\sin 2 \pi t}{\pi t} \right) \cos(38 \pi t - \frac{\pi}{2})$
produces an output $x_o(t)$. The expression for $x_o(t)$ is ______.
GATE ECE 2024
3
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 lowpass filter with frequency response H(f) as following: $$H(f) = \left\{ {\matrix{
{1,} & {\left| f \right| \le \,12\,kHz} \cr
{0,} & {\left| f \right| > \,12\,kHz} \cr
} } \right.$$
What is the number of sinusoids in the output and their frequency inkHz?
GATE ECE 2017 Set 2
4
Consider a continuous-time signal defined as $$x(t) = \left( {{{\sin \,(\pi t/2)} \over {(\pi t/2)}}} \right)*\sum\limits_{n = - \infty }^\infty {\delta (t - 10n)} $$ Where ' * ' denotes the convolution operation and t is in seconds. The Nyquist sampling rate (in samples/sec) for x(t) is __________________.
GATE ECE 2015 Set 3
5
The Nyquist sampling rate for the signal $$s(t) = {{\sin \,(500\pi t)} \over {\pi \,t}} \times {{\sin \,(700\pi t)} \over {\pi \,t}}$$ is given by
GATE ECE 2010
6
The minimum sampling frequency (in samples /sec) required to reconstruct the following signal from its samples without distortion $$x(t) = 5{\left( {{{\sin \,\,2\,\pi \,1000\,t)} \over {\pi \,t}}} \right)^3} + 7{\left( {{{\sin \,\,2\,\pi \,1000\,t} \over {\pi \,t}}} \right)^2}$$
would be
GATE ECE 2006
7
A signal m(t) with bandwidth 500 Hz is first multiplied by a signal g(t) where $$g(t)\, = \,\,\sum\limits_{k = - \infty }^\infty {{{( - 10)}^k}\,\delta (t - 0.5x{{10}^{ - 4}}k)} $$
The resulting signal is then passed through an ideal low pass filter with bandwidth 1 kHz. The output of the low pass filter would be
The resulting signal is then passed through an ideal low pass filter with bandwidth 1 kHz. The output of the low pass filter would be
GATE ECE 2006
8
A 1 kHz sinusoidal signal is ideally sampled at 1500 samples /sec and the sampled signal is passed through an ideal low-pass filter with cut-off frequency 800 Hz. The output signal has the frequency
GATE ECE 2004
9
Let x(t) = $$\,2\cos (800\pi t) + \cos (1400\pi t)$$. x(t) is sampled with the rectangular pulse train shown in figure. The only spectral components (in KHz) present in the sampled signal in the frequency range 2.5 kHz to 3.5 kHz are
GATE ECE 2003
10
A signal x(t) = 100 cos $$(24\pi \times {10^3})$$ t is ideally sampled with a sampling period of 50 $$\mu \sec $$ and then passed through an ideal low pass filter with cutoff frequency of 15 KHz. Which of the following frequencies is/ are present at the filter output?
GATE ECE 2002
11
The Nyquist sampling interval, for the signal Sinc(700t) + Sinc(500t) is
GATE ECE 2001
12
The Nyquist sampling frequency (in Hz) of a signal given by $$16 \times {10^{4\,}}\,\sin {c^2}(400t)*{10^6}\,\sin {c^3}(100t)$$ is
GATE ECE 1999
13
A signal has frequency components from 300 Hz to 1.8 KHz. The minimum possible rate at which the signal has to be sampled is ______ (fill in the blank).
GATE ECE 1991
14
A 4 GHz carrier is DSB-SC modulated by a low pass message signal with maximum frequency of 2 MHz. The resultant signal is to be ideally sampled. The minimum frequency of the sampling impulse train should be:
GATE ECE 1990
15
A signal containing only two frequency components (3 kHz and 6 kHz) is sampled at the rate of 8 kHz, and then passed through a low pass filter with a cutoff frequency of 8 kHz. The filter output
GATE ECE 1988
Marks 4
Marks 5
1
A band limited signal x(t) with a spectrum X(f) as shown in Fig. a is processed as shown in Fig.b. p(t) is a periodic train of impulses as in Fig. c. The ideal band pass filter has a pass band from 26 KHz to 34 KHz.
(a) Calculate the Fourier series coefficients $${c_n}$$ in the Fourier expansion of p(t) in form $$p(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}} \,\exp \,\,(j\,n\,2\pi \,t/T)$$.
(b) Find the Fourier Transform of p(t).
(c) Obtain and sketch the spectrum of $${x_s}(t)$$.
(d) Obtain and sketch the spectrum of y(t).
(a) Calculate the Fourier series coefficients $${c_n}$$ in the Fourier expansion of p(t) in form $$p(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}} \,\exp \,\,(j\,n\,2\pi \,t/T)$$.
(b) Find the Fourier Transform of p(t).
(c) Obtain and sketch the spectrum of $${x_s}(t)$$.
(d) Obtain and sketch the spectrum of y(t).
GATE ECE 2000
2
A low pass signal m(t) band-limited to B Hz is sampled by a periodic rectangular pulse train, $${p_\tau }(t)$$ of period $${T_s}$$ = 1/(3B) sec. Assuming natural sampling and that the pulse amplitude and pulse width are A volts and 1/(30B) sec, respectively, obtain an expression for the frequency spectrum of the sampled signal $${m_s}$$(t)
GATE ECE 1993