1
GATE EE 2007
MCQ (Single Correct Answer)
+2
-0.6
If u(t), r(t) denote the unit step and unit ramp functions respectively and u(t)*r(t) their convolution, then the function u(t+1)*r(t-2) is given by
A
(1/2)(t-1)(t-2)
B
(1/2)(t-1)(t-2)
C
(1/2)(t-1)2u(t-1)
D
none of the above
2
GATE EE 2007
MCQ (Single Correct Answer)
+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 $${\omega _1} = {\omega _2}.$$
3
GATE EE 2007
MCQ (Single Correct Answer)
+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.
4
GATE EE 2007
MCQ (Single Correct Answer)
+2
-0.6
A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must
A
provide zero phase shift for all frequencies
B
provide constant phase shift for all frequencies
C
provide linear phase shift that is proportional to frequency
D
provide a phase shift that is inversely proportional to frequency
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