AMIETE – ET/CS/IT (NEW SCHEME) – Code: AE57/AC57/AT57

Subject: SIGNALS AND 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 the best alternative in the following: (210)

a. Odd signal satisfies

(A) (B)

(C) (D)

b. Which system is non-causal system

(A) y(t) = x(t + 1) (B) y(t) = x(t - 1)

(C) y(t) = x(t)+ c (D) y(t) = x(t - 1)+c

c. The output of a linear system for a step input is t²e^-t, then transfer function is

(A) (B)

(C) (D)

d. The discrete LTI system is represented by impulse response Then, the system is

(A) noncausal and stable (B) noncausal and unstable

(C) causal and unstable (D) causal and stable

e. Laplace transform of is

(A) (B)

(C) (D)

f. Inverse z-transform of X[z/a]

(A) (B)

(C) (D)

g. ROC of the z-transform of unit step sequence is

(A) (B)

(C) Real part of z > 0 (D)

h. Fourier transform of x(t)=1 is

(A) zero (B)

(C) (D) 1

i. For distortionless transmission through LTI system phase of H(ω) is

(A) constant (B) one

(C) zero (D) linearly dependent on ω

j. A random process X(t) is called wide sense stationary if its

(A) first order moment is constant

(B) second order moment is constant

(C) autocorrelation function is independent of time

(D) all the above

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Evaluate the following integrals:

(i)

(ii) ₍₂₎

b. Given x(t) as shown in Fig. 2(b)

Sketch the following

(i) x(-t)

(ii)

(iii) x(2t-1)

(iv) x(4-t) (4)

c. For each of the following systems determine whether the system is

(a) Linear (b) Causal (c) Stable (d) Time- invariant (e) Memory less

(i) T[x[n]] = ax[n] + b (ii) y(t) = e^x(t) (10)

Q.3 a. Given

and

Find for and. (8)

b. Derive the condition for the stability of an LTI discrete system. (4)

c. Find the convolution of x₁(t)=U(t+1) and x₂(t)=U(t-2) where U(t) is a unit step function. (4)

Q.4 a. Determine the Fouriers Series representation for signal;

(i) (ii) (8)

b. Find the Fourier Series representation of the signal x(t) shown in Fig 4(b) (6)

c. For the system equation y(n)= 3x(n) +0.5y(n-1), find the transfer function and the impulse response. (2)

Q.5 a. State and prove the following properties of continuous signal Fourier Transform.

(i) Time shifting property (ii) Scaling property

(iii) Convolution property (9)

b. Find the Fourier Transform of the

signal x(t) shown in the Fig. 5(b). (7)

Q.6 a. Find the frequency response of an LTI system having impulse response h(t)=2(1-2t)e^-2t u(t). (6)

b. State and prove sampling theorem for Low pass signal. (6)

c. Determine the differential equation for the following system with frequency response

(i) (ii) (4)

Q.7 a. Find the Laplace transform of the following signals.

(i) x(t)=t²e^-2tu(t) (ii) x(t) =e^-3t sin(2t) u(t) (6)

b. Find the Inverse Laplace transform of the following X(s)

(i) ₍₈₎

c. State initial and final value theorem in Laplace transform. (2)

Q.8 a. Find the Z-transform of the following sequence and also, determine its ROC

(i)

(6)

b. State and prove the following properties of Z- transform

(i) Conjugate property (ii) Scaling property (6)

c. Find the Inverse Z-transform of with ROC

(i)

₍₄₎

Q.9 a. Write short note on the following:-

_{(i) Ergodic processes (ii) Wide sense stationary process}

_{(iii) Strict sense stationary process (iv) Power spectral density} (10)

b. For a stationary Ergodic process X(t) if autocorrelation function is given by Find the mean and variance of X(t) (6)