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11013 Pages: 3

PART ‏تا‎
‎Answer any two full questions, each carries 15 marks.
Derive the relation between Laplace transform and Continuous Time Fourier
transform.
Evaluate the Fourier Transform of x(t) = sgn(t). Plot magnitude and phase

response.
T ⋅ ⋅ ⋅ ⋅ 5+5 [ग
An LTI system is characterized with the transfer function H(s) = न्त ind the

response of the system to the input x(t) = cos2t u(t).

State Sampling theorem. Compute the Nyquist rate of the signal x(t).

x(t) = cos (5) — sin (=) + cos + ड)
Determine the Fourier Series Representation for x(t)}= 2sin(2zt-3) + sin(6zt).

Show that the spectrum of the sampled signal is the infinite sum of shifted replicas
of the spectrum of original signal.

d(te~2" sin(t)u(t))

Evaluate the Fourier Transform of x(t) = न

4८ u(t). Using Fourier

A causal LTI system has an impulse response h(t) = 07
transform find,
(i) Frequency response of the system.
(ii) Output of the system for an input x(t) = 3९ ८ u(t).
State and prove the following properties of Laplace Transform
(i) Time domain differentiation
(ii) Final value theorem
Find the Inverse Fourier transform of the following signals
(i)
(ii) 276 (0) + 766೧ — 47೫) + 76(೧ + 47೫)

1
1೧(1೧--1)

+ 276 (0)

PART C
Answer any two full questions, each carries20 marks.

Find the Z - transform of x(n) = 2(3)"u(—n)

Compute the DTFT of the signal x(n).

_ (10 ; ॥| ۷
५०४) = [0 ; [| > ۷

Prove that, for a BIBO stable discrete time LTI system the ROC of system

function includes unit circle.

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