1
GATE EE 2008
+1
-0.3
A signal $${e^{ - \alpha t}}\,\sin \left( {\omega t} \right)$$ is the input to a real Linear Time Invariant system. Given $$K$$ and $$\phi$$ are constants, the output of the system will be of the form $$K{e^{ - \beta t}}\,\sin \,\left( {\upsilon t + \phi } \right)$$ where
A
$$\beta$$ need not be equal to $$\alpha$$ but $$\upsilon$$ equal to
B
$$\upsilon$$ need not be equal to $$\omega$$ but $$\beta$$ equal to $$\alpha$$
C
$$\beta$$ equal to $$\alpha$$ and $$\upsilon$$ equal to $$\omega$$
D
$$\beta$$ need not be equal to $$\alpha$$ and $$\upsilon$$ need not be equal to $$\omega$$
2
GATE EE 2008
+1
-0.3
The impulse response of a causal linear time-invariant system is given as $$h(t)$$. Now consider the following two statements:

Statement-$$\left( {\rm I} \right)$$: Principle of superposition holds
Statement-$$\left( {\rm II} \right)$$: $$h\left( t \right) = 0$$ for $$t < 0$$

Which one of the following statements is correct?

A
Statement $$\left( {\rm I} \right)$$ is correct and Statement $$\left( {\rm II} \right)$$ is wrong
B
Statement $$\left( {\rm II} \right)$$ is correct and Statement $$\left( {\rm I} \right)$$ is wrong
C
Both Statement $$\left( {\rm I} \right)$$ and Statement $$\left( {\rm II} \right)$$ are wrong
D
Both Statement $$\left( {\rm I} \right)$$ and Statement $$\left( {\rm II} \right)$$ are correct
3
GATE EE 2007
+1
-0.3
Let a signal $${a_1}\,\sin \left( {{\omega _1}t + {\phi _1}} \right)$$ be applied to a stable linear time-invariant system. Let the corresponding steady state output be represented as $${a_2}F\left( {{\omega _2}t + {\phi _2}} \right).$$ Then which of the following statements is true?
A
$$F$$ is not necessarily a ''sine'' or ''cosine'' function but must be periodic with $${\omega _1} = {\omega _2}.$$
B
$$F$$ must be a ''sine'' or ''cosine'' function with $${a_1} = {a_2}.$$
C
$$F$$ must be a ''sine'' function with $${\omega _1} = {\omega _2}.$$ and $${\phi _1} = {\phi _2}.$$
D
$$F$$ must be a ''sine'' or ''cosine'' function with $${\omega _1} = {\omega _2}.$$
4
GATE EE 2002
+1
-0.3
$$s(t)$$ is step response and $$h(t)$$ is impulse response of a system. Its response $$y(t)$$ for any input $$u(t)$$ is given by
A
$${d \over {d\,t}}\int\limits_0^t s \left( {t - \tau } \right)\,u\left( \tau \right)\,d\,\tau$$
B
$$\int\limits_0^t s \left( {t - \tau } \right)\,u\left( \tau \right)\,d\,\tau$$
C
$$\int\limits_0^t {\int\limits_0^\tau s \left( {t - {\tau _1}} \right)\,u\left( {{\tau _1}} \right)\,d{\tau _1}} \,d\tau$$
D
$${d \over {d\,t}}\int\limits_0^t h \left( {t - \tau } \right)\,u\left( \tau \right)\,d\,\tau$$
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