# Questions & Answers of Applications of Fourier Transforms

## Weightage of Applications of Fourier Transforms

Total 13 Questions have been asked from Applications of Fourier Transforms topic of Signals and Systems subject in previous GATE papers. Average marks 1.77.

The Fourier transform of a continuous-time signal $x(t)$ is given by $X(w)=\frac1{(10+jw)^2}\;,-\infty<w<\infty,\;$ where $j=\sqrt{-1}$ and  $w$ denotes frequency. Then the value of | ln $x(t) |$ at $t=1$ is ___________ (up to 1 decimal place). ( ln denotes the logarithm to base $e$ )

Suppose  have the Fourier transforms as shown below.

Which one of the following statements is TRUE?

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 ________.

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

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

Consider a signal defined by

Its Fourier Transform is

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 $g\left(t\right)$?

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

The Fourier transform F(ω) of f(t) is

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

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?

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
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