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If we have a rectangular pulse function, we know that after a Fourier transform we obtain a sinc:

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We know that the left part (negative frequencies) has no physical meaning and it's just specular. It's important from the mathematical point of view but not from a physical point of view.

Anyway, if we had multiplied the rectangular pulse function with a cosine of a certain frequency, we know that the sinc will be shifted of the cosine frequency. For example, we shift it at 3hz if cosine had 3hz frequency.

enter image description here

So now, even if the left part of sinc has also positive frequencies, we ignore them again since it's just a specular part.

Is it for the same physics reason that harmonics only create in multiples of the center frequency? Is the concept of "ignoring left part" linked in some way with the harmonics that only creates in multiples of the frequency and not in dividers/previous frequencies?

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  • $\begingroup$ The left part has no physical meaning? Both the left part and the right part is a mathematical description. The left part exists for a “physical” real sine wave when you decide the represent it with positive and negative frequencies (meaning terms of $e^{j\omega t}$ rather than terms of sinusoids.) Consider Euler’s formula $2\cos(\omega t) = e^{j\omega t} + e^{-j\omega t}$ $\endgroup$ Commented Mar 12 at 20:07
  • $\begingroup$ You can “ignore” the left part for real signals given the left would be the complex conjugate (so redundant). But you can’t for complex signals. We can represent physical processes using complex or all real representations- it’s just math in both cases. $\endgroup$ Commented Mar 12 at 20:11
  • $\begingroup$ Perhaps imaginary quantities have no physical meaning, but that doesn't mean that imaginary numbers have no mathematical meaning. Because $j=\sqrt{-1}$ has mathematical meaning, so also negative frequencies have mathematical meaning. $\endgroup$ Commented Mar 12 at 23:33

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Consider how with the Fourier Series of a periodic function (single values, analytic) that repeats over duration $T$ seconds, that it can be decomposed into an infinite sum of frequencies, including a DC offset (frequency = 0), a “fundamental frequency” with frequency $1/T$ and integer harmonics. Any one of those components could have zero weight, and no frequencies can exist at any other location. Why is this so? Since the waveform is periodic over period $T$, whatever occurs over the time duration from $t=0$ to $t=T$ must repeat exactly from $t=t$ to $t=2T$, and so on. If that waveform is the sum of components, the only way it can do this is if each of those components also repeat as such. The only frequencies that can do this are DC, the fundamental and integer harmonics.

Note that this applies to either complex or real signals. If the signal is complex, the individual frequencies are in the form of $e^{j2 \pi f t}$ with $f$ being a signed frequency. When the signal is real we can either do the same (and have positive and negative frequencies that are complex conjugate symmetric), or use real sines and cosines in which only positive frequency representation applies.

If the signal has no periodicity, there will not be any harmonics and the a signal power will be distributed over a range of frequencies.

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