Question

Fourier transform of the signal is
$x\left(t\right)=3[1+\frac{sin30\pi t}{\pi t}]cos\left(200\pi t\right)$

• A. $x\left(f\right)=\frac{3}{2}\left[\delta \right(f-100)+\delta (f+100\left)\right]+\frac{3}{2}[rect\left(\frac{f-100}{30}\right)+rect\left(\frac{f+100}{30}\right)]$
• B. $x\left(f\right)=3\left[\delta \right(f-100)+\delta (f+100\left)\right]+3[rect\left(\frac{f-100}{30}\right)+rect\left(\frac{f+100}{30}\right)]$
• C. $x\left(f\right)=\frac{3}{2}\left[\delta \right(f-100)+\delta (f+100\left)\right]+\frac{3}{2}[rect(\frac{f}{30}-100)+rect(\frac{f}{30}+100)]$
• D. $X\left(f\right)=3\left[\delta \right(f-100)+\delta (f+100\left)\right]+3[rect(\frac{f}{30}-100)+rect(\frac{f}{30}+100)]$

Answer 1

The correct option is C $x\left(f\right)=\frac{3}{2}\left[\delta \right(f-100)+\delta (f+100\left)\right]+\frac{3}{2}[rect(\frac{f}{30}-100)+rect(\frac{f}{30}+100)]$

$x\left(t\right)=3[1+\frac{sin30\pi t}{\pi t}]cos\left(200\pi t\right)$

$x\left(t\right)=3cos\left(200\pi t\right)+\frac{3sin30\pi t}{\pi t}cos\left(200\pi t\right)$

Fourier transform of above signal x(t) is X(f)

$X\left(f\right)=\frac{3}{2}\left[\delta \right(f-100)+\delta (f+100\left)\right]+\left[\frac{3}{2}\left(rect\left(\frac{f}{30}\right)\right)\right]\ast \left[\delta \right(f-100)+\delta (f+100\left)\right]$

$X\left(f\right)=\frac{3}{2}\left[\delta \right(f-100)+\delta (f+100)+\frac{3}{2}[rect\left(\frac{f-100}{30}\right)+rect\left(\frac{f+100}{30}\right)]$