# Questions & Answers of Applications of Fourier Transforms

Question No. 44

Suppose  have the Fourier transforms as shown below.

Which one of the following statements is TRUE?

Question No. 115

Suppose the maximum frequency in a band-limited signal $x\left(t\right)$ is 5 kHz. Then, the maximum frequency in $x\left(t\right)\mathrm{cos}\left(2000\mathrm{\pi t}\right)$, in kHz, is ________.

Question No. 138

Let ${x}_{1}\left(t\right)↔{X}_{1}\left(\omega \right)$ and ${x}_{2}\left(t\right)↔{X}_{2}\left(\omega \right)$ be two signals whose Fourier Transforms are as shown in the figure below. In the figure, $h\left(t\right)={e}^{-2\left|t\right|}$ denotes the impulse response.
For the system shown above, the minimum sampling rate required to sample y(t), so that y(t) can be uniquely reconstructed from its samples, is

Question No. 139

The value of the integral $2{\int }_{-\infty }^{\infty }\left(\frac{sin\mathit{2}\pi t}{\pi t}\right)dt$ is equal to

Question No. 139

Consider a signal defined by

Its Fourier Transform is

Question No. 45

Let $f\left(t\right)$ be a continuous time signal and let $F\left(\omega \right)$ be its Fourier Transform defined by

 Define $g\left(t\right)$ by $F\left(\omega\right)=\int\limits_{-\infty}^\infty f\left(t\right)e^{-j\omega t}dt$ $g\left(t\right)=\int\limits_{-\infty}^\infty F\left(u\right)e^{-jut}du$

What is the relationship between $f\left(t\right)$ and $g\left(t\right)$?

Question No. 215

A function f(t) is shown in the figure.

The Fourier transform F(ω) of f(t) is

Question No. 219

A signal is represented by

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

The Fourier transform of the convolved signal $\mathrm{y}\left(\mathrm{t}\right)=\mathrm{x}\left(2\mathrm{t}\right)*\mathrm{x}\left(\mathrm{t}/2\right)$ is

Question No. 245

A differentiable non constant even function x(t) has a derivative y(t), and their respective Fourier Transforms are $X\left(\omega \right)$ and $Y\left(\omega \right)$. Which of the following statements is TRUE?

Question No. 42

The Fourier transform of a signal h(t) is $H\left(j\omega \right)=\left(2\mathrm{cos}\omega \right)\left(\mathrm{sin}2\omega \right)/\omega .$The value of h(0) is

x(t) is a positive rectangular pulse from t = -1 to t = +1 with unit height as shown in the figure. The value of $\int\limits_{-\infty}^\infty\left|X\left(\omega\right)\right|^2d\omega$ {where X(ω) is the Fourier transform of x(t)} is
Let $x\left(t\right)=rect\left(t-\frac{1}{2}\right)$ (where rect(x)=1 for $-\frac{1}{2}\le x\le \frac{1}{2}$ and zero otherwise). Then if $\mathrm{sin}c\left(x\right)=\frac{\mathrm{sin}\left(\mathrm{\pi x}\right)}{\mathrm{\pi x}},$, the Fourier Transform of x(t)+x(-t) will be given by