What is the precise mathematical description of an incoherent single-frequency signal for any type of wave? The reason I'm asking is because of the following apparent paradox in which incoherent light cannot exist.
Consider sunlight, for example, which has passed through a polarizing filter and frequency filter, so that only waves with wave numbers very close to $k_0$ are allowed to pass through. Since sunlight is totally incoherent, it seems reasonable to model the signal as a sum of sine waves $E_\alpha(x,t)=A_\alpha\sin(k_0x-\omega_0t+\phi_\alpha),$ where $E$ is the electric field in the direction of the polarizing filter, $\omega_0=ck_0$, $A_\alpha$ is a random amplitude, and $\phi_\alpha$ is a random phase shift. If the light were coherent, then the $\phi_\alpha$'s would all be identical; so it seems reasonable that for "maximal incoherence" the $\phi$'s and $A_\alpha$'s would be different and uniformly distributed. But then for every component with phase shift $\phi$ and amplitude $A$, there exists a wave $A_\alpha\sin(k_0x-\omega t-\phi-\pi)$, which cancels the original. Hence all components cancel and there is no wave (spectrometer detects nothing).
So what's the flaw here? I'm guessing the model of incoherent light is where the problem lies, but maybe it's in the reasoning. I'm also curious whether or not the answer necessarily relies on quantum mechanics.
Since there are some votes to close based on the proposed duplicate, I'll just say that both questions get at the same idea, but I think mine (which could also have focused on polarization) is more specific, since I'm asking for a precise model and whether quantum physics is a necessary part of the explanation.
From what I can tell, the answers to the linked question do not address these points.