# Distinguishing signals with same frequency but different phase

Fourier decomposition of a mixed signal can straightforwardly give me the frequencies of the different components and their relative amplitudes, but how can I extract the components of a mixed signal if they have the same frequency and differ only in phase?

I have a data series consisting of a metric of global activity in a certain domain (e.g. volume of Google searches). There is a strong diurnal rhythm in the data, i.e. activity is high during certain hours and low during other hours, on a 24-hour cycle. The peaks and troughs of the activity occur respectively during the day and night periods in North America.

I suspect that, while the North American contribution dominates, there are smaller contributions to the overall global activity from other time-zones. These would follow the same 24-hour cycle but would be shifted by a certain number of hours and have smaller amplitudes.

My problem is to identify the amplitude of the signal centred on each time-zone. I know that the relative amplitudes of the different phased signals affect the balance between real and imaginary components of the complex Fourier coefficient, but can I go backwards from the Fourier analysis to extract the amplitudes of the different-phased contributions? Or is there something similar to Fourier analysis that will do this?

To put it another way, I have a signal of the form $S(t)=\sum_{i=1}^{24}a_i cos⁡(ωt+2πi/24)+\epsilon(t)$ (where $\epsilon(t)$ is noise), and my problem is to find the $a_i$. I have thousands of data points covering multiple cycles, so if there is nothing like Fourier analysis that can do it, instead I could treat this as an overdetermined set of simultaneous equations in 24 unknowns. Is that the best or only way to approach the problem?

• That's pretty much impossible without additional assumptions even if the noise is absent since $\cos(x+a)+\cos(x-a)=2\cos a\cos x$, say, so how on Earth can you hope to distinguish the combination of two signals from North America and Asia from one from Europe? – fedja Sep 25 '16 at 14:29
• $\cos(\omega t + 2\pi i / 24) = \cos(\omega t) \cos(2\pi i / 24) - \sin(\omega t) \sin(2\pi i / 24)$. Your Fourier transform should give you both the cosine series and the sine series. You have now 24 variables ($a_1 \ldots a_{24}$) and 2 equations, which is vastly underdetermined. Now: it is absolutely necessary that the equations are underdetermined, no sort of signal analysis can ever reveal to you how to separate two signals that are exactly 180-degrees out of phase. – Willie Wong Sep 25 '16 at 14:29
• Right, so you seem to be saying that, given a waveform known to be composed of 24 phase-shifted signals, there is no unique solution for the amplitudes of the 24 signals, and in fact there are 22 degrees of freedom available. – Parsifal Sep 26 '16 at 15:28