1

GATE ECE 2002

Subjective

+5

-0

A deterministic signal x(t) = $$\cos (2\pi t)$$ is passed through a differentiator as shown in
Figure.

(a) Determine the autocorrelation R

(b) Find the output power spectral density S

(c) Evaluate R

(a) Determine the autocorrelation R

_{xx }($$\tau $$) and the power spectral density S_{xx}(f).(b) Find the output power spectral density S

_{yy}( f ).(c) Evaluate R

_{xy}(0) and R_{xy}(1/4).2

GATE ECE 2000

Subjective

+5

-0

For the linear, time-invariant system whose block diagram is shown in Fig.(a),
with input x(t) and output y(t).

(a) Find the transfer function.

(b) For the step response of the system [i.e. find y(t) when x(t) is a unit step function and the initial conditions are zero]

(c) Find y(t), if x(t) is as shown in Fig.(b), and the initial conditions are zero.

(a) Find the transfer function.

(b) For the step response of the system [i.e. find y(t) when x(t) is a unit step function and the initial conditions are zero]

(c) Find y(t), if x(t) is as shown in Fig.(b), and the initial conditions are zero.

3

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.

$$$\left[ {\matrix{ {Note:u(t)\, \equiv \,0} & {t\, \le \,0} \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \equiv 1} & {t\, > \,0} \cr } } \right]$$$

4

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

Signals and Systems

Representation of Continuous Time Signal Fourier Series Fourier Transform Continuous Time Signal Laplace Transform Discrete Time Signal Fourier Series Fourier Transform Discrete Fourier Transform and Fast Fourier Transform Discrete Time Signal Z Transform Continuous Time Linear Invariant System Discrete Time Linear Time Invariant Systems Transmission of Signal Through Continuous Time LTI Systems Sampling Transmission of Signal Through Discrete Time Lti Systems Miscellaneous

Network Theory

Control Systems

Digital Circuits

General Aptitude

Electronic Devices and VLSI

Analog Circuits

Engineering Mathematics

Microprocessors

Communications

Electromagnetics