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Signal System: Topic-test(Fourier series) Date : 30.07.2014 Time 30 Minute Marks-20

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1. This question paper contains printed 2 pages including pages for rough work. Please check all pages and report discrepancy, if any.

2. All the questions in this question paper are of objective type.

3. Questions must be answered on answer sheet itself by making tick against question number itself. Each question has only one correct answer. In case you wish to change an answer, erase the old answer

completely. More than one answer marked against a question will be treated as a wrong answer.

4. Questions 1 through 8 are 1-mark questions and questions 9 through 14 are 2-mark questions.

5. Un-attempted questions will carry zero marks.

6. NEGATIVE MARKING: For Q.1 to Q.8, 0.33 marks will be deducted for each wrong answer. For Q.9 to Q.14, 0.67 marks will be deducted for each wrong answer

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1. Consider the following statements about Fourier Series:

1. All periodic signals can be represented by Fourier series

2. It expresses the given periodic waveform as combination of D.C component, sine and cosine

waveforms of different harmonic frequencies

3. Both amplitude and phase spectra of Fourier series are discrete in nature

4. Amplitude spectra of F.S is continuous while phase spectra of F.S is discrete in nature.

(A) 1, 2 & 3 (B) 1, 2 & 4 (C) Only 1 & 2 (D) 1, 2, 3 & 4

2. Consider the following statements about Fourier Series:

1. The necessary condition for Fourier-series to exists is that signal must be periodic

2. Dirchilets conditions are sufficient condition to determine the Fourier-series.

3. For Fourier-series to exists, x(t) should have finite number of discontinuities in an given

interval T0, where discontinuities may be finite or infinite.

4. Fourier-series is linear function of Non orthogonal function of frequencies of a periodic

signal.

Which of the following statements are correct:

(A) 1, 2, 3 (B) 1, 2, 4 (C) 1, 2 (D) 3, 4

3. For signal x(t) to be even the complex exponential Fourier-series coefficient kC is

(A) Real and Continuous (B) Real and Discrete

(C) Imaginary and Continuous (D) Imaginary and Discrete

4. For the signal x(t) to be odd, the correct option that suits for Ck, ak, bk respectively is :

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1

2

3 |C |k

3 2 1 0 1 2 3k

60

2 1 0 1 2k

30

12

|C |k

3 2 1 1 2 3k

60

2 1 0 1 2k

30

3

60

1

2

|C |k

2 1 1 2k

60

2 1 0 1 2k

30

3

60

30

(A) Imaginary, Nonzero, Zero (B) Real, Nonzero, Zero

(C) Real, Zero, Nonzero (D) Imaginary, Zero, Nonzero

5. The Fourier-series expansion of a real periodic signal of fundamental frequency f0 is given as

0j2 nf tp nn

x t C e

It is given that *3C 4 j5 ; Then valve of C3 is

(A) 4 + j5 (B) 4 j5 (C) 5 j4 (D) 0

6. If the signal x(t) has complex exponential Fourier-series coefficient. Choose the correct statement.

X(t) Ck

St.1. Real and odd Imaginary and even

St.2. Real and even Real and odd

St.3. Imaginary and odd Real and even

St.4. Imaginary and even Imaginary and odd

(A) 1, 2 (B) 2, 3 (C) 1, 4 (D) None of these

7. Fourier series coeff. For a square wave of period 2, which is odd in nature and has amplitude +2 &

2 are:

2 2 2 2

( ) (1 ( 1) ) ( ) (1 ( 1) ) ( ) (1 ( 1) ) ( ) (1 ( 1) )k k k ka b c dk jk jk k

8. In case of Fourier series Power can be expressed as:

2 2 2 2 2 2

1 1 1

11. 2 2. 2 3. ( )

2

N N

o k k o k k

K K K

P c c P c P a a b

(a) 1, 2 (b) 2, 3 (c) 1, 3 (d) 1, 2, 3

9. Which of following given amplitude and phase spectrum shows that signal is real.

(A)

(B)

(C)

(D) None of the above

10. Calculate the value 0T .x 0 , where T0 is the fundamental period. X t is represented by complex

exponential Fourier-series coefficient of periodic signal X(t),

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4 3 2 0 1 2 4 5 6

3

1

t

k

k

0.5 , k 0,C

0, k 0

Where fundamental frequency is 4 radian per second:

(A) 1 (B) 2 (C) 1/2 (D) 1/4

11. If complex exponential Fourier-series of signal x(t) and y(t) are given by Cn and Dn respectively.

Given that n

1, n 0C

0, n 0

. Also given that

n n"C D " . Then Fourier-series expansion (complex

exponential) of z(t), where z t x t .y t is

(A) n u n (B) n 1 u n (C) u(n) (D) 2u(n)

12. Calculate the value of 2

kC

of the following given waveform.

(A) 3 (B) 3/4 (C) 3/2 (D) 6

13. Calculate trigonometric Fourier-series of signal x(t)

(A) DC value 2 / , k k2

4a , b 0

1 4k

(B) DC value 4/, k k2

4a , b 0

1 4k

(C) DC value = 1/ 2 , k k2

1a , b 0

1 4k

(D) DC value = 1

2,

k k21

a , b 01 4k

14. Fourier-series of a periodic function given by

0, x 0

f xx, 0 x

is 21

1 1cos n 1 cos nx cos n sin nx

4 n n

fundamental time period is 2, Then sum of series 2 2 2

1 1 11

3 5 7 Given : f 2

(A) 2

4 (B)

2

8 (C)

2

16 (D)

2

2