  We propose a non-parametric method to denoise 1D stellar spectra based on
wavelet shrinkage followed by adaptive Kalman thresholding. Wavelet shrinkage
denoising involves applying the Discrete Wavelet Transform (DWT) to the input
signal, `shrinking' certain frequency components in the transform domain, and
then applying inverse DWT to the reduced components. The performance of this
procedure is influenced by the choice of base wavelet, the number of
decomposition levels, and the thresholding function. Typically, these
parameters are chosen by `trial and error', which can be strongly dependent on
the properties of the data being denoised. We here introduce an adaptive
Kalman-filter-based thresholding method that eliminates the need for choosing
the number of decomposition levels. We use the `Haar' wavelet basis, which we
found to be the best-suited for 1D stellar spectra. We introduce various levels
of Poisson noise into synthetic PHOENIX spectra, and test the performance of
several common denoising methods against our own. It proves superior in terms
of noise suppression and peak shape preservation. We expect it may also be of
use in automatically and accurately filtering low signal-to-noise galaxy and
quasar spectra obtained from surveys such as SDSS, Gaia, LSST, PESSTO, VANDELS,
LEGA-C, and DESI.
