1
GATE ECE 1997
Subjective
+5
-0
Fig.1, shows the block diagram representation of a control system. The system in block A has an impulse response $${h_A}(t) = {e^{ - t}}\,u(t)$$. The system in block B has an impulse response $${h_B}(t) = {e^{ - 2t}}\,u(t)$$. The block 'k' amplifies its input by a factor k. For the overall system with input x(t) and output y(t)

(a) Find the transfer function $${{Y(s)} \over {X(s)}}$$, when k=1
(b) Find the impulse response, when k = 0
(c) Find the value of k for which the system becomes unstable.
$$$\left[ {\matrix{ {Note:u(t)\, \equiv \,0} & {t\, \le \,0} \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \equiv 1} & {t\, > \,0} \cr } } \right]$$$
2
GATE ECE 1993
Subjective
+5
-0
Consider the following interconnection of the three LTI systems (Fig.1). $${h_1}(t)$$ , $${h_2}(t)$$ and $${h_3}(t)$$ are the impulse responses of these three LTI systems with $${H_1}(\omega )$$, $${H_2}(\omega )$$, and $${H_3}(\omega )$$ as their respective Fourier transforms. Given that $${h_1}\,(t)\, = \,{d \over {dt}}\left[ {{{\sin ({\omega _0}t)} \over {2\,\pi \,t}}} \right],{H_2}(\omega ) = \exp \left( {{{ - j2\pi \omega } \over {{\omega _0}}}} \right)$$
$${h_3}\,(t)\, = u(t)\,and\,x(t)\, = \,\sin \,2\,{\omega _0}t\, + \,\cos \,({\omega _0}t/2),$$ find the output y(t).
$${h_3}\,(t)\, = u(t)\,and\,x(t)\, = \,\sin \,2\,{\omega _0}t\, + \,\cos \,({\omega _0}t/2),$$ find the output y(t).

Questions Asked from Continuous Time Linear Invariant System (Marks 5)
Number in Brackets after Paper Indicates No. of Questions
GATE ECE Subjects
Network Theory
Control Systems
Electronic Devices and VLSI
Analog Circuits
Digital Circuits
Microprocessors
Signals and Systems
Representation of Continuous Time Signal Fourier Series
Discrete Time Signal Fourier Series Fourier Transform
Discrete Time Signal Z Transform
Continuous Time Linear Invariant System
Transmission of Signal Through Continuous Time LTI Systems
Discrete Time Linear Time Invariant Systems
Sampling
Continuous Time Signal Laplace Transform
Discrete Fourier Transform and Fast Fourier Transform
Transmission of Signal Through Discrete Time Lti Systems
Miscellaneous
Fourier Transform
Communications
Electromagnetics
General Aptitude