This is related to waves & optics.

I am given the fourier transform of a function (the spectrum of frequences for a pulse of sound)

$$\hat f(w) = sinc((w-w_0)\tau ) $$

And now, a filter is attached so that it only allows waves with frequency $w_0$ to passthrough, and am asked to find what f(t) is like after we put the filter.

The question hints at using Fourier series to solve, but I am unaware of how I can use the Fourier transform along with the series to solve. (or any relations between them).

The solution gives that f(t) written as a fourier series is:

$$f(t) = \sum \hat f(w)cos(wt)$$

i.e. that the transform is the coefficients for this series. Why is this true?


Ok this became too long for a comment.

The sinc is perhaps the most popular "reconstruction" of a function inverted from the coefficients in the discrete fourier transform. It results from the assumption that the coefficient in the fourier domain is uniformly sampled (integrated) locally over the resolution.

You can check this if you want by checking that the function which has a rectangular box in the Fourier domain is the sinc function in the time domain.

If the samples in the Fourier domain on the other hand really are instantaneous "Dirac impulses" then they are "true" sines & cosines and that's more like what a Fourier series is.

I can try and give you some example plots later.

Something that matters in cases like this is whether the filter is applied before digitizing or after and how the actual sampling is done. If you just get a bunch of values, well then it's difficult to say how you should apply any subsequent filters or reconstruct.

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  • $\begingroup$ Thank you! So basically this will always hold true and is usable? $\endgroup$ – RonaldB Mar 23 '17 at 20:46
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    $\begingroup$ It is important to know how the function (signal) has been measured (sampled) if we are just handed a bunch of discrete values / Fourier coefficients and told to make sense of them. It is difficult to answer a question without knowing that. I will try and expand on my answer later. But as your question stands now (quite broad and unclear) maybe reading in a book about signal processing would be even better for in depth understanding. $\endgroup$ – mathreadler Mar 23 '17 at 21:03

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