Not sure if this should go into a math section, but I am trying to understand it in order to understand the Hesenberg uncertainy principle. I can't find a simple introduction to fourier series to answer this question.(adsbygoogle = window.adsbygoogle || []).push({});

In my modern physics book, it does a quick introduction to fourier series without proving any of it (fine with me), but I get a little confused. Any periodic waveform can be the sum of an infinite number of sines and cosines. Now, my understanding of infinite here needs to be defined, as off into infinity on the frequency axises. So we are summing sizes and cosines with larger and larger frequencies.

f = 0, 1/T, 2/T, 3/T, ..., presumably off into infinity. If we are apprioximating, the more terms we include the closer we get to the true answer.

It then says we can begin to let T grow to make are signal's period larger and larger (being influenced to model a single wave packet), and in this case, the fundlemental frequency gets smaller and all the harmonics move closer together. WE then determine we can move all the way to one single pulse by making T infinity large, and therefor using a continuous spread of frequencies.

The next section discusses that if we make our single pulse half as wide, it increases (perhaps doubles) the range of continues frequencies we need. Here is my confusion. Didn't we assume at the start the the frequencies needed to be exact went off into infinity. It seems now though, we have infinity many frequencies (since we are assuming a continious band), but the band itself is now finite. In other words, we may have a delta(f) centered around f at 100Mhz or so, with a delta of 10Mhz, (90-110Mhz), with infinitly many frequencies in between (to make the period T as large as we need). BUT(!), why don't we need to sum out the frequencies up to inifinity like we did at the start, 1000/T, 1000000/T, 50000343234/T, etc. What happen to all of these terms which would have frequencies greater then 110Mhz. My book makes no effort to explain this. I can't understand Heisenberg uncertainty principle until I understand why delta(f) is only a delta and does not have terms reaching out into infinite f.

I hope my question makes sense.

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# Fourier series

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