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I'm currently studying wavelets and had an interesting thought experiment: If you were to calculate the wavelet transform of a signal using a wavelet of a fixed frequency, you would get time varying wavelet coefficient. Say this fixed frequency wavelet is 1000Hz. You essentially yield a signal that shows how a 1000Hz frequency component varies with time.

If you were to run a Fourier transform on this new signal, what is the maximum frequency content you could find in it?

My intuition says that there is some upper limit to this (i.e. you couldn't find 10kHz frequency content), but I'm curious if there is a way to prove this limit.

After thinking about it some more, it seems like what I'm asking is: what the maximum modulation frequency that can be encoder into a carrier frequency?

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In the frequency domain, an AM modulated sinusoid looks like a carrier plus two side-bands in complex conjugate symmetry above and below the carrier. If you permit the lower sideband to cross into the negative frequency spectrum, then there is no limit to maximum modulating frequency. Of course that may not look like your typical AM modulated signal in the time domain.

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  • $\begingroup$ So in the context of my question, it still doesn't "feel right" that the time varying amplitude of some low frequency component could change at high frequencies (i.e. could a 1Hz signal vary in amplitude at a frequency of 1MHz?). Why doesn't this "feel" right? $\endgroup$ – Izzo Apr 22 '19 at 18:19

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