Question

Let a continuous time signal be defined as $g\left(t\right)=rect\left(4t\right)\ast 4\delta (-2t)$ then the Fourier transform of the signal g(f) is given as

(where $sinc\left(f\right)=\frac{sin\pi f}{\pi f}$)

• A. $2sinc\left(4f\right)$
• B. $\frac{1}{2}sinc\left(\frac{f}{4}\right)$
• C. $4sinc\left(\frac{f}{4}\right)$
• D. $4sinc\left(4f\right)$

Answer 1

The correct option is B $\frac{1}{2}sinc\left(\frac{f}{4}\right)$

$g\left(t\right)=rect\left(4t\right)\ast 4\delta (-2t)$

$=4rect\left(4t\right)\ast \delta (-2t)(\because \delta (-t)=\delta (t\left)\right)$

$=2rect\left(4t\right)$ $(\because \delta (at)=\frac{1}{|a|}\delta (t\left)\right)$

thus g(t) is given as

now, $rect\left(t\right)\stackrel{F.T}{⟷}sinc\left(f\right)$

$2rect\left(t\right)\stackrel{F.T}{⟷}2sinc\left(f\right)$

$2rect\left(4t\right)\stackrel{F.T}{⟷}2.\frac{1}{4}sinc\left(\frac{f}{4}\right)$ (scaling property)

$2rect\left(4t\right)\stackrel{F.T}{⟷}\frac{1}{2}sinc\left(\frac{f}{4}\right)$