Transmission of Signal Through Continuous Time LTI Systems · Signals and Systems · GATE ECE
Marks 1
1
Let x1(t) = e$$-$$t u(t) and x2(t) = u(t) $$-$$ u(t $$-$$ 2), where u( . ) denotes the unit step function. If y(t) denotes the convolution of x1(t) and x2(t), then $$\mathop {\lim }\limits_{t \to \infty } y(t)$$ = __________ (rounded off to one decimal place).
GATE ECE 2022
2
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 23Hz. The fundamental frequency (in Hz) of the output is _____________________.
GATE ECE 2016 Set 1
3
A real - values signal x(t) limited to the frequency band $$\left| f \right| \le {W \over 2}$$ is passed through a linear time invariant system whose frequency response is
$$H(f) = \left\{ {\matrix{ {{e^{ - j4\pi f}},} & {\left| f \right| \le \,{W \over 2}} \cr {0,} & {\left| f \right| > \,{W \over 2}} \cr } } \right.$$
$$H(f) = \left\{ {\matrix{ {{e^{ - j4\pi f}},} & {\left| f \right| \le \,{W \over 2}} \cr {0,} & {\left| f \right| > \,{W \over 2}} \cr } } \right.$$
The output of the system is
GATE ECE 2014 Set 4
4
Let g(t) = $${e^{ - \pi {t^2}}}$$, and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is
GATE ECE 2013
5
Assuming zero initial condition, the response y (t) of the system given below to a unit step input u(t) is
GATE ECE 2013
6
A system with the transfer function $${{Y(s)} \over {X(s)}} = {s \over {s + p}}\,\,$$ has an output
$$y(t) = \cos \left( {2t - {\pi \over 3}} \right)\,$$ for the input signal
$$x(t) = p\cos \left( {2t - {\pi \over 2}} \right)$$. Then, the system parameter 'p' is
$$y(t) = \cos \left( {2t - {\pi \over 3}} \right)\,$$ for the input signal
$$x(t) = p\cos \left( {2t - {\pi \over 2}} \right)$$. Then, the system parameter 'p' is
GATE ECE 2010
7
Consider the pulse shape s(t) as shown. The impulse response h(t) of the filter matched to this pulse is
GATE ECE 2010
8
A low-pass filter having a frequency response $$H(j\omega )$$ = $$A(\omega ){e^{j\Phi (\omega )}}$$, does not product any phase distortion if
GATE ECE 2006
9
In the system shown below,
x(t) = (sint)u(t). In steady-state, the response y(t) will be
x(t) = (sint)u(t). In steady-state, the response y(t) will be
GATE ECE 2006
10
A linear phase channel with phase delay $${\tau _p}$$ and group delay $${\tau _g}$$ must have
GATE ECE 2002
11
The input to a channel is a band pass signal. It is obtained by linearly modulating a sinusoidal carrier with a signal- tone signal. The output of the channel due to this input is given by y(t) = (1/100) cos$$(100t - {10^{ - 6}})\,$$ cos$$({10^6}t - 1.56)$$. The group delay $$({t_g})$$ and the phase delay $$({t_p})$$, in seconds, of the channel are
GATE ECE 1999
12
A rectangular pulse of duration T is applied to a filter matched to this input. The output of the filter is a
GATE ECE 1996
Marks 2
1
Let $$x(t) = 100\cos (10.5Wt)$$ be passed through an LTI system having impulse response $$h(t) = \pi {\left( {{{\sin Wt} \over {\pi t}}} \right)^2}\cos 10Wt$$. The output of the system is
GATE ECE 2023
2
The outputs of four systems (S1, S2, S3 and S4) corresponding to the input signal sin(t), for all time t, are shown in the figure.
Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?
GATE ECE 2022
3
A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for $$t\, \ge 0$$:
X: Impulse P: 1 $$ - {e^{ - t/T}}$$
Y: Unit step Q: t $$ - T(1 - {e^{ - t/T}})$$
Z: Ramp R: $${e^{ - t/T}}$$
X: Impulse P: 1 $$ - {e^{ - t/T}}$$
Y: Unit step Q: t $$ - T(1 - {e^{ - t/T}})$$
Z: Ramp R: $${e^{ - t/T}}$$
GATE ECE 2016 Set 1
4
A continuous-time filter with transfer function $$\,H(S) = {{2s + 6} \over {{s^2} + 6s + 8}}$$ is converted to a discrete time filter with transfer function $$G(Z) = {{2{z^2} - 0.5032\,z} \over {{z^2} - 0.5032z + k}}$$ so that the impulse response of the continuous-time filter, sampled at 2 Hz, is identical at the sampling instants to the impulse response of the discrete time filter, The value of k is _________________________.
GATE ECE 2016 Set 2
5
In the system shown in Figure (a), m(t) is a low-pass signal with bandwidth W Hz. The frequency response of the band-pass filter H(f) is shown in Figure (b). If it is desired that the output signal z(t) = 10x(t), the maximum value of W (in Hz) should be strictly less than ___________________________
GATE ECE 2015 Set 1
6
The 3 - dB bandwidth of the low - pass signal $${e^{ - 1}}$$ u(t), where u(t) is the unit step function, is given by
GATE ECE 2007
7
Consider the signal x(t) shown in Fig. Let h(t) denote the impulse response of the filter matched to x(t), with h(t) being non-zero only in the interval 0 to 4 sec. The slope of h(t) in the interval 3 < t < 4 sec is
GATE ECE 2004
8
A system has poles at 0.01 Hz, 1 Hz and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz. The approximate phase of the system response at 20 Hz is
GATE ECE 2004
9
The system under consideration is an RC low -pass filter (RC-LPF) with R = 1.0 $$k\Omega $$ and C = 1.0 $$\mu F$$.
Let $${t_g}$$ (f) be the group delay function of the given RC-LPF and $${f_2}$$ = 100 Hz. Then $${t_g}$$$${(f_2)}$$ in ms, is
GATE ECE 2003
10
The system under consideration is an RC low -pass filter (RC-LPF) with R = 1.0 $$k\Omega $$ and C = 1.0 $$\mu F$$.
Let H(t) denote the frequency response of the RC-LPF. Let $${f_1}$$ be the highest frequency such that $$0 \le \left| f \right| \le {f_1},{{\left| {H({f_1})} \right|} \over {H(0)}} \ge 0.95$$. Then $${f_1}$$ (in Hz) is
GATE ECE 2003
11
In Fig. m(t) = $$ = {{2\sin 2\pi t} \over t}$$, $$s(t) = \cos \,200\pi t\,\,andn(t) = {{\sin 199\pi t} \over t}$$.
The output y(t) will be
GATE ECE 2002
12
A system has a phase response given by $$\phi \,(\omega )$$ where $$\omega $$ is the angular frequency. The phase delay and group delay at $$\omega $$ = $${\omega _0}$$ are respectively given by
GATE ECE 2000
13
The input to a matched filter is given by $$s(t) = \left\{ {\matrix{
{10\sin (2\pi \times {{10}^6}t),} & {0 < \left| t \right| < {{10}^{ - 4}}\sec } \cr
0 & {Otherwise} \cr
} } \right.$$
The peak amplitude of the filter output is
GATE ECE 1999
14
Sketch the waveform (with properly marked axes) at the output of a matched filter matched for a signal s(t), of duration T, given by $$s(t) = \left\{ {\matrix{
{A\,\,\,\,for} & {0 \le t < {2 \over 3}T} \cr
{0\,\,\,\,\,\,for} & {{2 \over 3}T \le t < T} \cr
} } \right.$$
GATE ECE 1993
15
The pole-zero pattern of a certain filter is shown in the Fig. The filter must be of the following type.
GATE ECE 1991
16
The magnitude and phase transfer functions for a distortionless filter should respectively be:
GATE ECE 1990
17
Specify the filter type if its voltage transfer function H(s) is given by
H(s) = $${{K({s^2} + {\omega _0}^2)} \over {{s^2} + ({\omega _0}/Q)s + {\omega _0}^2}}$$
GATE ECE 1988
Marks 5
1
A base band signal g(t) band limited to 100 Hz modulates a carrier of frequency $${{f_0}}$$ Hz. The modulated signal g(t) $$\cos 2\,\pi \,{f_0}t$$ is transmitted over a channel whose input x and output y are related by $$y = 2x + {x^2}$$. The spectrum of g(t) is shown in Fig. Sketch the spectrum of the transmitted signal and the spectrum of the received signal.
GATE ECE 2001
2
An input signal A exp $$\left( { - \alpha \,t} \right)$$ u(t) with $$\alpha > 0$$ is applied to a causal filter, the impulse response of which is A exp $$\,( - \alpha \,\,t)$$. Determine the filter output, sketch it as a function of time and lable the important points.
GATE ECE 1996
3
A signal 3 sin $$\left( {\pi \,\,{f_0}t} \right) + \,5\,\,\cos \,\,\,(3\pi \,\,{f_0}t)$$ is applied to an RC low pass filter of 3 dB cutoff frequency $${f_0}$$. Determine and plot the output power spectrum and aslo calculate the total input and output normalized power.
GATE ECE 1996
4
A signal, f(t) = $${e^{ - at}}$$ u(t), where u(t) is the unit step function, is applied to the input of a low-pass filter having $$\left| {H(\omega )} \right| = {b \over {\sqrt {{\omega ^2} + {b^2}} }}$$.
Calculate the value of the ratio, $${a \over b}$$, for which 50% of the input signal energy is transferred to the output.
GATE ECE 1994
5
Obtain an expression for the signal in figure, for the signal $${v_3}(t)$$ in Fig for $${v_1}(t) = 100\cos (2000\pi t) + 4\sin (200\pi t)$$. Assume that $${v_2}(t)$$=$${v_1}(t)$$+0.1 $$v_1^2(t)$$ and that the BPF is an ideal unity gain filter with pass band from 800 Hz to 1200 Hz.
GATE ECE 1993