# GATE Questions & Answers of Applications of Fourier Transforms

## What is the Weightage of Applications of Fourier Transforms in GATE Exam?

Total 12 Questions have been asked from Applications of Fourier Transforms topic of Signals and Systems subject in previous GATE papers. Average marks 1.75.

Suppose  have the Fourier transforms as shown below. Which one of the following statements is TRUE?

##### Show Answer

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 ________.

##### Show Answer

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

##### Show Answer

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

##### Show Answer

Consider a signal defined by

Its Fourier Transform is

##### Show Answer

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 $g\left(t\right)$?

##### Show Answer

A function f(t) is shown in the figure. The Fourier transform F(ω) of f(t) is

##### Show Answer

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

##### Show Answer

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?

##### Show Answer

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

##### Show Answer

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 ##### Show Answer

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