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I have a component of a signal $$\sin (k\omega_1t + \ell\omega_2t)$$ with wavenumbers $k, \ell \in \mathbb{Z}$, frequencies $\omega_1, \omega_2 \in \mathbb{R^+}$ and time $t \in \mathbb{R^+}$. (This would be one term of many in sums over $k$ and $\ell$.)

I would like to disentangle $\omega_1$ and $\omega_2$ from each other into their own respective terms and express the result as a linear combination. Is this possible?

Perhaps more specifically, I would like to represent this as a sum of sines for some constant vectors $\mathbf a$ and $\boldsymbol \phi$ like so: $$a_1 \sin(k\omega_1t + \phi_1) + a_2 \sin(\ell\omega_2t + \phi_2) + \ldots$$ (The dots are there because perhaps there would this would only work with many more terms of that same kind of form.) Is this possible? If not, how can I be sure of it?

I know that $$\sin(kw_1t + \ell w_2t) = \sin(k\omega_1t)\cos(\ell\omega_2t) + \cos(k\omega_1t)\sin(\ell\omega_2t)$$ but I'm not sure this is helpful for me (except maybe in convincing me that what I'm trying to do is impossible.)

Thanks for any insights or pushes in the right direction.

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    $\begingroup$ indeed, it's impossible: just take $\omega_1=0$ to find the dependence on $\omega_2$, then take $\omega_2=0$ to find the dependence on $\omega_1$ and conclude that no such decomposition exists. $\endgroup$
    – Carlo Beenakker
    Commented Jan 21, 2015 at 7:41
  • $\begingroup$ Thanks for your reply, that is what I had suspected. I'm wondering: is the $\sin(k\omega_1t+\ell\omega_2t)$ formulation a "generalization" of sorts of $\sin(k\omega_1t)+\sin(\ell\omega_2t)+\dots$, or do they describe different systems altogether? $\endgroup$
    – rhombidodecahedron
    Commented Jan 21, 2015 at 8:50
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    $\begingroup$ one is periodic in time, the other is not (at least not if $\omega_1$ and $\omega_2$ are incommensurate) $\endgroup$
    – Carlo Beenakker
    Commented Jan 21, 2015 at 10:11
  • $\begingroup$ To be clear, the first is the one that is periodic in time, right? Its angular frequency would be $(k\omega_1 + \ell\omega_2)$? $\endgroup$
    – rhombidodecahedron
    Commented Jan 21, 2015 at 11:24

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