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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. Any signal x(t) can be represented in terms of its odd and even components as.

(A) (B)

(C) (D)

 b. Find the type of system described by

 

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(A) Linear and dynamic (B) Linear and static

(C) Non linear and dynamic (D) Non linear and static

 

c. The discrete LTI system is represented by impulse response

h(n) = u(n). Then the system is

(A) Anti-causal and Stable (B) Causal and Stable

 

(C) Causal and Unstable (D) Anti-causal and Unstable

 

d. Laplace transform of is

 

(A) (B)

(C) (D)

e. The impulse response of the system having transfer function H(s) = is

(A) () u(t) (B) u(t)

(C) (D) t u(t)

 

f. If X() = the x(t) is

 

(A) (B)

(C) (D)

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g. Fourier transform of x(-n) is

(A) –X() (B)

(C) (D)

h. The condition for events A and B to be statistically independent.

(A) P(A/B)=P(AB)P(A) (B) P(AB)=P(A)P(B)

(C) P(A/B)=P(A) and P(B/A)=P(B) (D) P(A/B)=P(AB)

 

i. Inverse Z transform of X(z)= is

 

(A) x(n)=a2u(n) (B) x(n)=anu(n)

(C) x(n)=2a2u(n) (D) x(n)=nanu(n)

 

 j. System function H(z) for the system described by difference equationy(n)=2x(n)+3x(n-1)-4y(n-1) is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

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Q.2 a. Evaluate the following integrals.

(i) (ii)

(iii) (23 = 6)

 

  b. Compute the convolution sum of . (4)

 

c. Determine whether the systems are Linear, Causal, Time-invariant, Stable

and Memoryless (6)

(i) T[x(n)]=x(-n) (ii) y(t)=x(t) cos

Q.3 a. State and prove the scaling and duality property of continuous time fourier 

transform. (8)

 

 b. Find the fourier transform of the following signals

(i) x(t) = (ii) x(t) =

(iii) x(t) = u(t) (iv) x(t) = u(-t + 2) (8)

Q.4 a. Determine the discrete fourier series representation for each of the following

sequence.

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(a) x(n) = 2cos (b) x(n) = cos

(c) x(n) = 1+ 2cos (6)

 

 b. For the signal shown in Fig. 4(b) Find the fourier series co-efficient.

(10)

 

Q.5 a. Determine the frequency response and impulse response of the systems

described by the following equations.

(i)

(ii) 3y(n)-4y(n-1) + y(n-2) = 3x(n)

(8)

 

 b. State and prove the Sampling theorem for Lowpass signals and also explain

the reconstruction of the signal from its sample value. (8)

 

Q.6 a. Find the Z Transform of the following sequences and mention their ROC.

(i) x(n) =n u(n) (ii) x(n)=

(iii) (9)

 

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b. Find the inverse Z transform of the following X(z)

(i) X(z) = log , >

(ii) X(z) = , >2

(iii) X(z) = , >1 (2+2+3)

Q.7 a. Find the Laplace transform of the following signals and the associated ROCin each case.

(i) x(t) = (ii) x(t) =

(iii) x(t) = u(t) (iv) x(t) = (8)

 

 b. State and prove the initial and final value theorems in Laplace transform.

(8)

 

Q.8 a. If the probability density function of a random variable X is given by

,

find the mean, variance and standard deviation.

(8)

 

 b. Consider a sinusoidal signal with random phase designed by

x(t)= A cos()

Where A and fc are constant and is a random variable that is uniformlydistributed over the interval [, ], i.e

 

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Find (i) Auto correlation function of x(t).

(ii) Power spectral density of x(t). (8)

 

Q.9 a. Let x[n] and h[n] be signals with the following Fourier transforms

 

Determine . (8)

 

 b. Find the discrete Time fourier Transform of the following:

(i) (ii)

(iii) (iv) (8)