# 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

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

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. $e^{j10\omega }\; \frac{2sin\omega}{\omega}$

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

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

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. $\frac{4}{\omega ^2}\: sin^2\omega$

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)
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. $sinc\left ( \frac{\omega }{2\pi } \right )sin\left ( \frac{\omega }{2} \right )$