Fourier Transform MCQ

1. The value of the integral $2\int_{-\infty }^{\infty}\left ( \frac{sin2\pi t}{\pi t} \right )dt$ is equal to

  1. 0
  2. 0.5
  3. 1
  4. 2
Answer
Answer. d

2. Suppose the maximum frequency in a band limited signal x(t) is 5 kHz. Then, the maximum frequency in x(t) cos(2000πt), in kHz is

  1. 1
  2. 4
  3. 6
  4. 8
Answer
Answer. c

3. Consider a signal defined by

$x(t)=\begin{cases} e^{j10t} & \text{ for } \left |t \right |\leq 1 \\ 0 & \text{ for } \left |t \right |> 1 \end{cases}$

Its Fourier transform is

  1. $\frac{2sin(\omega -10)}{\omega -10}$
  2. $2e^{j10}\; \frac{sin(\omega -10)}{\omega -10}$
  3. $\frac{2sin\omega}{\omega -10}$
  4. $e^{j10\omega }\; \frac{2sin\omega}{\omega}$
Answer
Answer. a

4. A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier transforms are X(ω) and Y(ω). Which of the following statements is true?

  1. X(ω) and Y(ω) are both real
  2. X(ω) is real and Y(ω) is imaginary
  3. X(ω) and Y(ω) are both imaginary
  4. X(ω) is imaginary and Y(ω) is real
Answer
Answer. b

5. The Fourier transform of a signal h(t) is H(jω) = (2cosω)(sin2ω)/ω. The value of h(0) is

  1. 1/4
  2. 1/2
  3. 1
  4. 2
Answer
Answer. c

6. A signal is represented by

$x(t)=\begin{cases} 1 & \text{} \left | t \right |< 1 \\ 0 & \text{} \left | t \right |> 1 \end{cases}$

The Fourier transform of the convolved signal

y(t) = x(2t) ∗ x(t/2) is

  1. $\frac{4}{\omega ^2}sin\left ( \frac{\omega }{2} \right )sin(2\omega )$
  2. $\frac{4}{\omega ^2}sin\left ( \frac{\omega }{2} \right )$
  3. $\frac{4}{\omega ^2}sin(2\omega )$
  4. $\frac{4}{\omega ^2}\: sin^2\omega$
Answer
Answer. a

7. Let f(t) be a continous time signal and let F(ω) be its Fourier transform defined by

$F(\omega)=\int_{-\infty }^{\infty}f(t)e^{-j\omega t}\; dt$

and g(t) is defined by

$g(t)=\int_{-\infty }^{\infty}F(u)e^{-ju t}\; du$

What is the relationship between f(t) and g(t)?

  1. g(t) would always be proportional to f(t).
  2. g(t) would be proportional to f(t) if f(t) is an even function
  3. g(t) would be proportional to f(t) only if f(t) is a sinusoidal function.
  4. g(t) would never be proportional to f(t)
Answer
Answer. b

8. Let , x(t) = rect(t-1/2) (where, rect(t) = 1 for -1/2 ≤ x ≤ 1/2 and zero otherwise). Then sinc(x) = sin(πx)/πx, the Fourier transform of x(t) + x(-t) will be given by

  1. $sinc\left ( \frac{\omega }{2\pi } \right )$
  2. $2sinc\left ( \frac{\omega }{2\pi } \right )$
  3. $2sinc\left ( \frac{\omega }{2\pi } \right )cos\left ( \frac{\omega }{2} \right )$
  4. $sinc\left ( \frac{\omega }{2\pi } \right )sin\left ( \frac{\omega }{2} \right )$
Answer
Answer. c