Fourier Series MCQ

1. Consider g(t)=\begin{cases} t-\left \lfloor t \right \rfloor, & \text{} t\geq 0 \\ t-\left \lceil t \right \rceil, & \text{} otherwise \end{cases}, where t ∈ R. Here \left \lfloor t \right \rfloor represents the largest integer less than or equal to t and \left \lceil t \right \rceil denotes the samllest integer greater than or equal to t. The coefficient of the second harmonic component of the fourier series representing g(t) is

  1. 1
  2. 0
  3. -1
  4. 2
Answer. b

2. Let g: [0,∞) → [0,∞) be a function defined by g(x) = x – [x], where [x] represents the integer part of x. (i.e., it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of g(x) is

  1. 0.5
  2. 1
  3. 2
  4. 4
Answer. a

3.  Let x(t) be a periodic signal with time period T, Let y(t) = x(t – to) + x(t + to) for some to. The fourier series coefficients of y(t) are denoted by bk. If bk = 0 for all odd K. Then to can be equal to

  1. T/8
  2. T/4
  3. T/2
  4. 2T
Answer. b

4. The fourier series for the function f(x) = sin2x is

  1. sin x + sin 2x
  2. 1 – cos 2x
  3. sin 2x + cos 2x
  4. 0.5 – 0.5cos 2x
Answer. d

5. If an a.c. voltage wave is corrupted with an arbitrary number of harmonics, then the overall voltage waveform differs from its fundamental frequency component in terms of

  1. only the peak values
  2. only the rms values
  3. only the average values
  4. all the three measures (peak, rms and average values)
Answer. d

6. The signum function is given by

sgn(x)=\begin{cases} \frac{x}{\left | x \right |}; & \text{} x\neq 0 \\ 0\: ; & \text{} x= 0 \end{cases}

The Fourier series expansion of sgn(cos(t)) has

  1. only sine terms with all harmonics
  2. only cosine terms with all harmonics
  3. only sine terms with even numbered harmonics
  4. only cosine terms with odd numbered harmonics
Answer. d

7. The Fourier series coefficients of a periodic signal x(t) expressed as

x(t)=\sum_{k=-\infty }^{\infty}a_ke^{j2\pi kt/T} are given by

a-2 = 2 – j1; a-1 = 0.5 + j0.2; a0 = j2; a1 = 0.5 – j0.2; a2 = 2 + j1;

and ak = 0; for |k| > 2

Which of the following is true?

  1. x(t) has finite energy because only finitely many coefficients are non-zero
  2. x(t) has zero average value because it is periodic
  3. the imaginary part of x(t) is constant
  4. the real part of x(t) is even
Answer. c

8. x(t) is a real valued function of a real variable with period T. Its trigonometric Fourier series expansion contains no terms of frequency ω = 2π (2k)/T; k = 1, 2, …… Also, no sine terms are present. Then x(t) satisfies the equation

  1. x(t) = -x(t-T)
  2. x(t) = x(T-t) = -x(-t)
  3. x(t) = x(T-t) = -x(t-T/2)
  4. x(t) = x(t-T) = x(t-T/2)
Answer. c

9. Let f(x) be a real, periodic function satisfying f(-x) = -f(x). The general form of its Fourier series representation would be

  1. f(x)=a_0+\sum_{k=1}^{\infty }a_k\: cos(kx)
  2. f(x)=\sum_{k=1}^{\infty }b_{k}\: sin(kx)
  3. f(x)=a_0+\sum_{k=1}^{\infty }a_{2k}\: cos(kx)
  4. f(x)=\sum_{k=0}^{\infty }a_{2k+1}\: sin(2k+1)x
Answer. b

10. For a periodic signal

v(t) = 30sin100t + 10cos300t + 6sin(500t + π/4),

the fundamental frequency in rad/s is

  1. 100
  2. 300
  3. 500
  4. 1500
Answer. a
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