# wave separation

Say there is a wave of sines and cosines. (<- one can think of Fourier theory.)

A) There is a wave that has the same frequency all the time. However, amplitude (- shape) of each period differs. Is it possible to separate this wave into combinations of waves that have the same amplitude for each wave?

B) Say there is a added combination of two waves that are same in frequency but different in amplitude. The amplitude of each wave is equal all the time. Can the signal be decomposed into two signals that were combined?

Thanks.

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Your concept of "wave" is a little vague to me. But:

A) A "wave with constant frequency but varying amplitude" is (informally) what one have in AM : amplitude modulation. It can be shown that if the amplitud varies "slowly" (as compared with the main period), the signal can be expressed as a (in general, complicated) combination of sinusoids of frequencies near the main frequency.

In the wikipedia article it's shown the simplest case, $y(t)=[1 + M \cdot \cos(\omega_m t + \phi)]\cdot \sin(\omega_c t)$ Here we have a sinuosid of "central frequency" $\omega_c$ and its amplitude varies by a sinusoid of frequency $\omega_m$ ("modulation frequency"). It's easy to show (oops) that this signal can be expressed as the sum of three sinusoids of frequencies $\omega_c$, $\omega_c+\omega_m$ and $\omega_c -\omega_m$.

B) The sum of two sinusoids of same frequency and distinct amplitude (and perhaps phase) results in another sinusoid of the same frequency - this is easy, and it's fundamental property of sinusoids.

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For A, what you probably mean by the same frequency all the time is that the wave passes through zeros (maybe also maxima) at the right time spacing. It would be something like $f(t)=a_n\cos t$ where $a_n$ is the amplitude for period $n$. This is a well defined function (given the list of $a_n$) but it has energy at frequencies other than $\frac 1{2\pi}$ as you can see from an FFT. In particular, the derivative is not continuous when you change between the $a$'s

For B, yes, you can split out the time variation and use the usual formulas for the sum or difference of sinusoids

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