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I have $m$ complex-valued time-series $z_m(t_n)$, which I am currently analyzing.

I started by calculating their PSDs and CSDs. These exhibit an unexpected peak at a given frequency, which is present for all PSDs and CSDs, signaling some kind of coherence between the time-series.

Interestingly this peak disappears, when I calculate the PSDs or CSDs of only the modulus $|z_m(t_n)|$ of the time-series, suggesting that the phase $\phi[z_m(t_n)]$ of the time-series is somehow related to the time-domain and thus is important for determining its the power.

My question is acutally two-fold:

1) What does the relevance of the phase of $z_m(t_n)$ vs. $|z_m(t_n)|$ tell me about $z_m(t_n)$? I.e. is there some mathematical relevance/relation that could explain this?

2) Could this be due to some kind of "principal phase" of the $z_m(t_n)$ that, by throwing it out when calculating the modulus, "decorrelates" the spectral information of my time-series in the frequency-domain?

I am a little puzzled by the behavior and hope somebody could suggest, where to look or if there is some way to unravel the relevance of the phase in this.

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    $\begingroup$ Imagine your time series is $z(t)=\exp{i \omega_0 t}$. The power spectrum of this time series will obviously have a large line at $\omega_0$, while the time series of the modulus $|z(t)|$ is just a constant function. $\endgroup$ – nibot Apr 10 '14 at 22:45

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