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If Doppler searches for earth-mass, habitable planets are to succeed,observers must be able to identify and model out stellar activity signals. Herewe demonstrate how to diagnose activity signals by calculating themagnitude-squared coherence $\hat{C}^2_{xy}(f)$ between an activity indicatortime series $x_t$ and the radial velocity (RV) time series $y_t$. Since planetsonly cause modulation in RV, not in activity indicators, a high value of$\hat{C}^2_{xy}(f)$ indicates that the signal at frequency $f$ has a stellarorigin. We use Welch's method to measure coherence between activity indicatorsand RVs in archival observations of GJ 581, alpha Cen B, and GJ 3998. HighRV-H$\alpha$ coherence at the frequency of GJ 3998 b, and high RV-S indexcoherence at the frequency of GJ 3998 c, indicate that the planets may actuallybe stellar signals. We also replicate previous results showing that GJ 581 dand g are rotation harmonics and demonstrate that alpha Cen B has activitysignals that are not associated with rotation. Welch's power spectrum estimateshave cleaner spectral windows than Lomb-Scargle periodograms, improving ourability to estimate rotation periods. We find that the rotation period of GJ581 is 132 days, with no evidence of differential rotation. Welch's method mayyield unacceptably large bias for datasets with $N < 75$ observations and worksbest on datasets with $N > 100$. Tapering the time-domain data can reduce thebias of the Welch's power spectrum estimator, but observers should not applytapers to datasets with extremely uneven observing cadence. A software packagefor calculating magnitude-squared coherence and Welch's power spectrumestimates is available on github.

Magnitude-squared coherence: A powerful tool for disentangling Doppler planet discoveries from stellar activity