1
GATE EE 2008
+2
-0.6
A signal $$x\left( t \right) = \sin c\left( {\alpha t} \right)$$ where $$\alpha$$ is a real constant $$\left( {\sin \,c\left( x \right) = {{\sin \left( {\pi x} \right)} \over {\pi x}}} \right)$$ is the input to a linear Time invariant system whose impulse response $$h\left( t \right) = \sin c\left( {\beta t} \right)$$ where $$\beta$$ is a real constant. If $$\min \left( {\alpha ,\,\,\beta } \right)$$ denotes the minimum of $$\alpha$$ and $$\beta$$, and similarly $$\max \left( {\alpha ,\,\,\beta } \right)$$ denotes the maximum of $$\alpha$$ and $$\beta$$, and $$K$$ is a constant, which one of the following statements is true about the output of the system?
A
It will be of the form $$K$$ $$sinc$$$$\left( {\gamma t} \right)$$ where $$\gamma = \,\min \left( {\alpha ,\,\,\beta } \right)$$
B
It will be of the form $$K$$ $$sinc$$$$\left( {\gamma t} \right)$$ where $$\gamma = \,\max \left( {\alpha ,\,\,\beta } \right)$$
C
It will be of the form $$K$$ $$\sin c\left( {\alpha t} \right)$$
D
It cannot be a $$sinc$$ type of signal
2
GATE EE 2008
+2
-0.6
The transfer function of a linear time invariant system is given as $$G\left( s \right) = {1 \over {{s^2} + 3s + 2}}.$$ The steady state value of the output of this system for a unit impulse input applied at time instant $$t=1$$ will be
A
$$0$$
B
$$0.5$$
C
$$1$$
D
$$2$$
3
GATE EE 2008
+2
-0.6
A system with input $$x(t)$$ and output $$y(t)$$ is defined by the input $$-$$ output relation:
$$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t} {x\left( \tau \right)} d\tau .$$ The system will be
A
causal, time $$-$$ invariant and unstable
B
causal, time $$-$$ invariant and stable
C
non $$-$$ causal, time $$-$$ invariant and unstable
D
non $$-$$ causal, time $$-$$ variant and unstable
4
GATE EE 2007
+2
-0.6
$$X\left( z \right) = 1 - 3\,\,{z^{ - 1}},\,\,Y\left( z \right) = 1 + 2\,\,{z^{ - 2}}$$ are $$Z$$-transforms of two signals $$x\left[ n \right],\,\,y\left[ n \right]$$ respectively. A linear time invariant system has the impulse response $$h\left[ n \right]$$ defined by these two signals as $$h\left[ n \right] = x\left[ {n - 1} \right] * y\left[ n \right]$$ where $$*$$ denotes discrete time convolution. Then the output of the system for the input $$\delta \left[ {n - 1} \right]$$
A
has $$Z$$-transforms $${z^{ - 1}}X\left( z \right)Y\left( z \right)$$
B
equals
$$\delta \left[ {n - 2} \right] - 3\delta \left[ {n - 3} \right] + 2\delta \left[ {n - 4} \right] - 6\delta \left[ {n - 5} \right]$$
C
has $$Z$$-transform $$1 - 3\,{z^{ - 1}} + 2\,{z^{ - 2}} - 6\,{z^{ - 3}}$$
D
does not satisfy any of the above three.
EXAM MAP
Medical
NEET