Fourier transform (got right answer, but not matching graph)

[jaus tail]

Homework Statement

Homework Equations

Scaling property and property of dual. I got the answer.

The Attempt at a Solution

I got the answer using scaling property and using property of dual.
x₁(t)---> X₂(W)----(another Fourier transform)--->2(3.14) x₁(-w)
But I think the final answer should be A.
1/4 rect (w/8(pie)) has time period of 8
Even when I use area under frequency domain = 2(3.14)x(t) at t = 0, I get A as answer.
I tried google but couldn't find in the expression rect (k w) where k is constant, what will the time period be.

 

[DrClaude]

Check out equation 3 at http://www.thefouriertransform.com/transform/properties.php

 

[jaus tail]

I got the answer in terms of equation or rect value. But I don't think (1/4) rect (w/8π) is D. I think it's A.
I'm searching on google how to represent: rect (w) but am failing.
Does rect (w) mean it spans from -w to +w or does it mean it spans from -2w to +2w?

 

[jaus tail]

This link also has the answer. But again they haven't plotted the graph. So I'm struggling in last stage.
Math Stack Exchange
https://math.stackexchange.com/questions/27463/fourier-transform-of-mathrmsinc4t

 

[jaus tail]

Even as per this formula:

in Time domain, the coefficient of 't' is k
and in frequency domain the rect pulse spans from -K to +K
Do you think A is right answer instead of D?

 

[jaus tail]

Can somebody please help me with this question? I get the matching answer in rect form but I think (1/4) rect (W/8 pie) is option A and not Sure as they've marked.

Thanks in advance.

 

[Orodruin]

But I don't think (1/4) rect (w/8π) is D. I think it's A.

This is correct. The rect function is defined according to
$$
\operatorname{rect}(x) = \begin{cases} 1, & |x| < 1/2 \\ 1/2, & |x| = 1/2 \\ 0, & |x| > 1/2\end{cases}
$$
In your case, this means that it is one when
$$
\left\lvert \frac{\omega}{8\pi}\right\rvert < \frac 12 \quad \Longleftrightarrow \quad \lvert \omega \rvert < 4\pi
$$

 

[haruspex]

Does rect (w) mean it spans from -w to +w or does it mean it spans from -2w to +2w?

I don't think that question makes sense.
rect(f(x)) spans from f(x)=-½ to f(x)=+½.
rect$\left(\frac{\omega}{8\pi}\right)$ spans from $\frac{\omega}{8\pi}=-\frac 12$, so from $\omega=-4\pi$.
So yes, I agree it should be A.

Took too long typing... Orodruin got there first.

 

[jaus tail]

Thanks a lot. I was freaked out thinking i was wrong somewhere. So A is right answer. Thanks.