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So I have function

$$f(t)=\frac{sin(7t)}{\pi t}+\frac{\sin(\pi t)}{7t}$$

and I figured I should turn it into

$$f(t)=\frac{7t}{\pi t}sinc(7t)+\frac{\pi t}{7t}sinc(\pi t)$$

But I'm not quite sure how the fourier transformation/integration acts with sinc.

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    Correct typos.. – 2017-02-14
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    What means $\dfrac{7t}{\pi t}$ and $\dfrac{\pi t}{7t}$.? – 2017-02-14
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    7 times t divided by pi times t and its opposite in the latter. What I did to the initial function was that I expanded the terms with their sine functions innards so they could be transformed into sinc functions. It seems like I actually messed up with the symbolism, not sure if I edited it into bad form or what but both sine functions were supposed to be dividends as they are now. – 2017-02-14
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    Your $f(t)$ has two terms but they are the same. Write $f(t)$ in its original form that is given. – 2017-02-14
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    I'm sorry, but what do you mean exactly? The first term with sine-functions was the original form/function given and the second was the one I figured I should use for the transformation, made by me. – 2017-02-15
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    $f(t)=\frac{\sin(7t)}{\pi t}+\frac{\sin(7t)}{\pi t}$ means $f(t)=2\frac{\sin(7t)}{\pi t}$. Are you sure that $f(t)$ is like that? – 2017-02-15
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    Ah, I keep messing it up, let me fix it one last time. But I did find a crude solution to it, it won't lead to the exact answer, but the gist of it will be revealed. – 2017-02-15

1 Answers 1

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Alright, this is not the perfect answer, but how sinc basicly works here is:

$$f(t)=\frac{sin(7t)}{\pi t}+\frac{\sin(\pi t)}{7t}$$

$$F(w)=7rect_7(w)+\frac{pi^2}{7}rect_7(w)$$

The exact answer might be wrong, but the logic here is:

$$f(t)=sinc(t)$$

$$F(w)=\pi rect_1(w)$$

and

$$f(t)=sinc(at)$$

$$F(w)=\frac{1}{|a|}F(\frac{w}{a})$$