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. The system having input x (n)
related to output y(n) as 
is:
(A) nonlinear, causal, stable. (B) linear, noncausal, stable.
(C) nonlinear, causal, not stable. (D) linear, noncausal, not stable.
b. To obtain 
from the given signal x (n), the
following precedence (or priority) rule is used for operations on the
independent variable n:
(A)
Time
scaling 
Time
shifting Reflection.
(B)
Reflection
Time
scaling Time
shifting.
(C)
Time
scaling Reflection
Time
shifting.
(D)
Time
shiftingTime
scaling Reflection.
c. The unit
step-response of a system with impulse response 
is:
(A)
. (B)
.
(C) 
. (D)
u (n).
d. If the notation *
is used to denote convolution, and 
then, x (t). 
F
given by:
(A)
. (B)
.
(C) 
. (D)
.
e. For a nonperiodic
discrete-time signal, the frequency-shift property states that if the DTFT of x
(n) is 
,
then the DTFT of 
is 
, where is given by
(A)
(B)
(C) 
(D)
f. If 
is the
phase-response of a communication channel and 
is the channel frequency, then 
represents:
(A) Phase delay (B) Carrier delay
(C) Group delay (D) None of these
g. Zero-order hold used in practical reconstruction of continuous-time signals is mathematically represented as a weighted-sum of rectangular pulses shifted by:
(A) Any multiples of the sampling interval.
(B) Integer multiples of the sampling interval.
(C) One sampling interval.
(D) 1 second intervals.
h. If 
then 
is given by:
(A)
. (B)
.
(C) 
. (D)
.
i. The region of convergence of the z-transform of the signal
is
(A)
all z, except z=0 and z=
(B)
all z, except z=0.
(C) all z, except z=. (D)
all z.
j. When two honest coins are simultaneously tossed, the probability of two heads on any given trial is:
(A) 1 (B)
(C) 
(D)
Answer any FIVE Questions out of EIGHT Questions.
Each question carries 16 marks.
Q.2 a. Distinguish between power and energy signals. Determine the total energy of the raised-cosine pulse x(t), shown in Fig.1 defined by:
. (8)
b. Compute the following convolution and sketch the output:
(8)
Q.3 a. Find
the Fourier series representation for the signal 
for all t. Sketch the magnitude and
phase spectra. (8)
b. State the sampling
theorem, given 
.
For the spectrum of the continuous-time signal, shown in Fig.2, consider the
three cases 
and
draw the spectra, indicating aliasing. (8)
|
Q.4 a. Consider a continuous-time signal x(t).
(i) Show that 
using duality
(or similarity) property of FTs.
(ii)
Find x (t) from 
, using the convolution property of
FTs. (8)
b. Find the difference equation describing the system represented by the block-diagram shown in Fig.3, where D stands for unit delay. (8)
|
Q.5 a. For the simple
continuous-time RC frequency-selective filter shown in Fig.4, obtain the
frequency response 
. Sketch its magnitude and phase for
. (8)
|
b. Consider
the signal 
.
Express its Laplace Transform in the form: 
K = system constant. Identify the
region of convergence. Indicate poles and zeros in the s-plane. (8)
Q.6 a. Given input x (n) and impulse response h (n), as shown
in Fig.5, evaluate 
, using DTFTs. (8)
|
|
||||
b. Determine the inverse
DTFT, by partial fraction expansion, of 
. (8)
Q.7 a. State
the initial-value and final-value theorems of Laplace Transforms. Compute the
initial-value and final-value for 
where 
(8)
b. Find, by Laplace Transform
method, the output y(t) of the system described by the differential equation: 
where input 
and the initial
condition is 
.
(8)
Q.8 a. An
LTI system is characterised by the difference equation: 
with initial conditions 
and 
. Find x(n) by
using z-transform and state the properties of z-transform used in your
calculation. (8)
b. Determine
the discrete-time sequence x (n), given that 
. (8)
Q.9 a. Explain the meaning of the following terms with respect to random variables/processes:
(i) Wide-sense stationary process.
(ii) Ergodic process.
(iii) White noise.
(iv) Cross power spectral density. (8)
b. A random variable X is
characterised by the probability density function shown in Fig.6:
Compute its: Probability distribution function;
Probability in the range 
;
Mean
value between 
;
and
Mean-square
value 
. (8)