We propose the Stationary spectrum Plus Low-rank Iterative TransmiTtance EstimatoR (SPLITTER) for removing wideband atmospheric noise from observations of high-redshift galaxies. This algorithm has specifically been developed for the DEep Spectroscopic HIgh-redshift MApper (DESHIMA) 2.0, a spectrometer that is designed to observe the waveband from 220 GHz to 440 GHz in 347 spectral channels. This octave bandwidth poses a challenge, due to the spectrotemporal changes in the atmosphere column between the instrument and the target source. Removing the time-varying nonlinear interference and distortion caused by the atmosphere is a difficult task, as the atmospheric emission is much stronger than a typical galaxy signal. The goal of this thesis is to develop a method that can estimate both narrow spectral lines and the broad continuum emission with a higher sensitivity than the currently used method of directly subtracting noisy on- and off-source spectra. We develop a logarithmic data model for separating atmospheric noise from the galaxy signal in position switching-observations. Because the atmospheric transmittance appears as a multiplicative term in both the atmospheric interference and the signal modulation, the logarithmic model allows for an additive decomposition of the data. The atmospheric transmittance behaves as a low-rank component in this model. Using the model, we develop an optimization algorithm (SPLITTER) to perform the separation of the signal and the low-rank atmospheric transmittance. Several implementations are discussed. The final algorithm uses a Singular Value Decomposition (SVD) to estimate the atmosphere component and the Alternating Directions Method of Multipliers (ADMM) for estimating the source signal. Instead of subtracting the noisy estimate of the source from the data directly, a denoised model is used in this step, such that we can trade some spectral resolution for a higher sensitivity. SPLITTER is tested on simulated data using the Time-dependent End-to-end Model for Post-process Optimization of the DESHIMA spectrometer (TiEMPO), a dedicated software package for simulating DESHIMA observations. We show that SPLITTER is able to estimate the spectrum with a higher sensitivity than the conventional method. The improvement factor in our weighted root mean squared error is up to ~1.7 for the full spectrum and up to ~1.3 for the spectral lines only compared to the conventional method. The larger improvement for the full spectrum is achieved by trading spectral resolution for a higher sensitivity in the smooth continuum. With these results, we have an indication that a statistically driven method for DESHIMA observations can provide better estimates than the current method with the same amount of observing time. More work is needed to create a robust version of the algorithm, because although the sensitivity benefit of SPLITTER is larger in the continuum regions, there are also situations where the continuum is overestimated. The conditions for this to occur are not yet clear. A more robust version could make SPLITTER a reliable new method that can replace current data reduction methods for wideband atmospheric noise removal. In this way, it can be used to make background-limited direct detection spectrometers on both existing and future telescopes observe more efficiently.