Download GTU BE/B.Tech 2019 Summer 4th Sem New 2141005 Signals And Systems Question Paper

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Seat No.: Enrolment No.

GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-IV(NEW) - EXAMINATION - SUMMER 2019
Subject Code:2141005 Date:28/05/2019
Subject Name: Signals and Systems

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Instructions:

1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

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Q.1 (a) Find whether given signal is periodic or not? If yes, give its fundamental 03 period.

(i) x(t) = 3cos(10pt) + 5sin(6pt)

(ii) x[n] = ?

(b) Decompose following signals into their even and odd parts. 04

(i) x(t) = 3t²+2t+1

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(ii) x[n] = {1,2,1}

(c) Explain following property for the system y(t) = x(t)+ 2. 07

(i) Linearity (ii) Time-invariance (iii) Causality (iv) Dynamicity

(v) Stability.

Q.2 (a) Let x[n] be a signal with x[n] =0 for n < -2 and n > 4, For each of the 03 following signal, determine the values of n for which it is guaranteed to be zero.

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(i) x[n-3]

(ii) x[-n— 2]

(b) Prove associativity property of convolution sum. 03

(c) For,

x[n] = d[n] + 2d[n— 1] - d[n - 3] and

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h[n] = 2d[n+1] +2d[n—1],

Compute (i) y₁[n] = x[n] * h[n] and (ii) y₂[n] = x[n] * h[n +2]. 07

(d) Sketch each of the following signals for a signal shown in Figure 1. 07

(i) x(2-t)

(ii) x(2t + 1)

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(iii) x(t) + x(-t)u(t) [2+2+3 Marks]

x(t)

Figure 1

Q.3 (a) Find the convolution x[n] * h[n], where x[n] ={1,2,3} and 03 h[n] = {1,2,1}.

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(b) Given that y[n] = x[n] * h[n], x[n] = {1,0,1}, y[n]=0 for n < -1 and 04 y[n] = 2 for n = —1,0,3. Find h[n). Given y[n] is of finite duration signal with length of 5.

(c) Consider periodic signal x(t) with fundamental frequency w₀ = p, determine 07 its complex exponential Fourier series representation. Where,

x(t):{1.5, 0<t<1
—1.5, 1<t<2

OR

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Q.3 (a) Find the convolution x(t) * h(t), where x(t) = h(t) = e^-atu(t). 03

(b) Given that x[n] has Fourier transform X (e^j?), express the Fourier transform of 04 the w[n] = (n — 1)x[n] in terms of X(e^j?). [hint: Use Fourier transform property.]

(c) Let x(t) be a periodic signal with fundamental frequency w₀ and Fourier 07 coefficients a_k. Given that y(t) = x(1— t) +x(t— 1), how is the fundamental frequency w₀ of y(t) related to w₀? Also, find a relationship between the Fourier series coefficients b_k of y(t) and the coefficients a_k.

Q.4 (a) State and prove Time scaling property of Fourier transform. 03

(b) Given the relationships y(t) = x(t) * h(t) and g(t) = x(3t) * h(3t). Also 04 given that x(t)and h(t) have Fourier transform X(jw) and H(jw) respectively. Using Fourier transform property show that g(t) = Ay(Bt) and determine the values of A and B.

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(c) Find the response of an LTI system with impulse response 07 h(t) = e^-atu(t), a > 0 to the input signal x(t) = e^-btu(t), b>0,a? b, using Fourier transform.

OR

Q.4 (a) State and prove Duality property of Fourier transform. 03

(b) A stable LTI system characterized by the differential equation 04

d/dt y(t) + ay(t) = x(t), a > 0.

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Find the impulse response of the system.

(c) Find the Fourier transform of x(t) = e^-|t| Using property find Fourier 07 transform of 1/(1+t²)

Q.5 (a) Find the inverse z-transform of X(z) = z+1+2z^-1, 0<|z|<8. 03

(b) Discuss causality and stability of LTI system using z-transforms. 04

(c) Using partial fraction expansion find the inverse z-transform of 07

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X(z) = (3z² - 3/2 z)/(z² - 3/4 z + 1/8)

OR

Q.5 (a) Find DTFT of x[n] = {1,0,4,2}. 03

(b) State and prove differentiation property of z-transform. 04

(c) Consider the following algebraic expression for the z-transform X(z) of 07 signal x[n]:

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X(z) = (1+2z^-1)/(1+ 1/3 z^-1)

(i) Assuming the ROC to be |z| > 1/3, use long division to determine the values of x[0], x[1], and x[2].

(ii) Assuming the ROC to be |z| < 1/3, use long division to determine the values of x[0], x[—1], and x[-2].

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