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?