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Code: A-06/C-04/T-04 Subject: SIGNALS & SYSTEMS
Time: 3 Hours Max. Marks: 100
NOTE: There are 9 Questions in all.
· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.
· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.
· Any required data not explicitly given, may be suitably assumed and stated.
Q.1 Choose the correct or best alternative in the following: (2x10)
a.
Given a
unit step function u(t), its time-derivative is :
(A) a unit impulse.
(B) another step function.
(C) a unit ramp function.
(D) a sine function.
b. The impulse response of a system described by the differential equation
will be
(A) a constant. (B) an impulse function.
(C) a sinusoid. (D) an exponentially decaying
function.
c. The function 
is denoted by:
(A)
. (B)
sinc (u).
(C) signum. (D) none of these.
d. The frequency
response of a system with 
is given by
(A)
. (B)
.
(C) 
. (D)
.
e. The order of a linear constant-coefficient differential equation representing a system refers to the number of
(A) active devices. (B) elements including sources.
(C) passive devices. (D) none of these.
f. z-transform converts convolution of time-signals to
(A) addition. (B) subtraction.
(C) multiplication. (D) division.
g. Region of convergence of a causal LTI system
(A) is the entire s-plane. (B) is the right-half of s-plane
(C) is the left-half of s-plane. (D) does not exist.
h. The DFT of a signal x (n) of length N is X(k). When X (k) is given and x(n) is computed from it, the length of x(n)
(A) is increased to infinity (B) remains N
(C) becomes 2N-1 (D) becomes N2
i. The Fourier transform of u (t) is
(A)
(B)
.
(C) 
. (D)
none of these.
j. For the
probability density function of a random variable X given by 
, where 
is the unit step
function, the value of K is
(A) 
(B)
(C) 25 (D) 5
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Find
the complex Fourier coefficients for the periodic 
signal : 
. Using Parseval’s theorem,
evaluate the average power.
(8)
b. Determine :
(i)
the odd
and even parts of the signal: 
.
(ii)
the output
, given 
. (4+4)
Q.3 a. Find the FT of the continuous-time signal shown in Fig.1. (8)
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b. Find :
(i)
the convolution of 
, given
.
(ii)
whether the system with impulse response 
is causal and stable. (4+4)
Q.4 a. A continuous-time signal is shown in Fig.2. Find its FT. (8)
b. Determine:
(i)
the FT of 
.
(ii) the time-signal x (n), whose DFT is {1,1} (4+4)
Q.5 a. The
input to a discrete-time system is given by: 
. The impulse response of the system
is specified as follows:
(in
period)
Obtain the output y(n) using the convolution property of DTFT. (12)
b. Show that 
if 
. (4)
Q.6 a. Use z-transform to determine the impulse response of the
system described by the difference equation: 
. (8)
b. Find:
(i)
the value of the DTFT for 
for the signal: 
.
(ii)
the Laplace transform and Fourier transform of
the function 
,
and the z-transform and DTFT for 
. (4+4)
Q.7 a. Use
the Laplace transform to determine the output 
of the system represented by the
differential equation:
. (12)
b. Show that:
. (4)
Q.8 a. From
the pole-zero plot in z-plane (Fig.5), identify all valid RoCs for X(z) and
specify the characteristics of the time-signal corresponding to each RoC. (8)
b. (i) Given 
, show that n 
.
(ii)
A signal 
is
ideally sampled at intervals of 
seconds to obtain 
. Find the maximum
allowable value for . To reconstruct the signal without
distortion, find the minimum filter bandwidth required when 
is passed through a
rectangular low-pass filter. (4+4)
Q.9 a. Explain the terms: continuous and discrete random processes; deterministic and non-deterministic random processes; stationary and non-stationary random processes; and ergodic and non-ergodic random processes. (8)
b. Consider a stationary
random process that has a spectral density : 
. Express 
as 
and then determine the
mean-square value of this process. (8)