  The radial velocity (RV) detection of exoplanets is challenged by stellar
spectroscopic variability that can mimic the presence of planets and by
instrumental instability that can further obscure the detection. Both stellar
and instrumental changes can distort the spectral line profiles and be
misinterpreted as apparent RV shifts.
  We present an improved FourIEr \textit{phase} SpecTrum Analysis (FIESTA
a.k.a. $\mathit{\phi}$ESTA) to disentangle apparent velocity shifts due to a
line deformation from a true Doppler shift. $\mathit{\phi}$ESTA projects
stellar spectrum's cross correlation function (CCF) onto a truncated set of
Fourier basis functions. Using the amplitude and phase information from each
Fourier mode, we can trace the line variability at different CCF width scales
to robustly identify and mitigate multiple sources of RV contamination. For
example, in our study of the 3 years of HARPS-N solar data, $\mathit{\phi}$ESTA
reveals the solar rotational effect, the long-term trend due to solar magnetic
cycle, instrumental instability and apparent solar rotation rate changes.
Applying a multiple linear regression model on $\mathit{\phi}$ESTA metrics, we
reduce the weighted rms noise from 1.89 m/s to 0.98 m/s. In addition, we
observe a $\sim$3 days lag in the $\mathit{\phi}$ESTA metrics, similar to the
findings from previous studies on BIS and FWHM.
