A basic problem in the study of binary stars is that of separating the two original stellar spectra which are embedded in the composite spectrum of the system.
Going beyond the most trivial cases when both sets of lines were directly recognized by sight, early attempts trying to isolate the spectral contribution of each component date back to analogical techniques and hand-made computations (Wright 1954).
Various digital methods were later developed, aiming to separate the spectra of particular binaries with favourable orbital configurations, allowing a straightforward mathematical decomposition of the systemic spectrum.
Totally-eclipsing binaries immediately make it possible to identify the spectrum of the uneclipsed component, so that the other can be simply obtained by a luminosity-weighted subtraction and renormalization (Griffin R.&R. 1986). The same approach has also been used in the absence of a real eclipse, by reproducing the known component with a simulated spectrum (synthetic or from a standard star), and then deriving the companion by difference.
It is the geometry of the system itself, that defines the mathematical
operation required for spectral separation. For example, in the case of an
atmospheric eclipse, as that of 
Aur, the spectrum of the
obscure eclipsing body has been obtained as the ratio between the
eclipsed and uneclipsed spectra of the system (Ferluga & Mangiacapra 1991).
Similarly, in 
Aur-type binaries, the chromospheric spectrum of the
supergiant can be extracted from limb-occultation observations, using a
computational procedure involving spectra taken at different eclipse-phases
(Griffin et al. 1993).
Partially-eclipsing binaries, at each phase of occultation, produce a composite spectrum which is given by a different linear combination of the two unknown spectra. This, in principle, should allow a complete spectral reconstruction, when starting from a suitable set of composite spectra, observed at different eclipse phases. Although conceptually simple, the implementation of this method has still to be fully developed.
The majority of spectroscopic binaries, being non-eclipsing systems, require special separation methods, based on the different radial velocity of the components. In the simplest situations, the two spectra may consist of a few non-overlapping lines; this may happen for detached early-type binaries, observed at opposite elongation phases. In such conditions, the lines of each component may be isolated from the continuum by a practical divide-and-cut method (Pilachowski & Sowell 1992).
Generally, however, more sophisticated separation and reconstruction techniques are required to treat the common cases when the two spectra are crowded with lines and very closely intertwined, and also when the Doppler splitting of input spectra is poor. This happens, for example, in the case of semidetached (sd) and contact (c) binaries, or when available observations concern only intermediate phases, instead of opposite elongations. Recently, different advanced methods have been independently presented.
Providing full spectral reconstruction under most general conditions, these new proposed methods are based on complex algorithms; they usually require a large number of composite spectra, observed at different rotational phases of the system, in order to extract the two components. In particular, the advanced disentangling method by Simon & Sturm (1993) is based on the inversion of a linear-transformation matrix to separate the two components; its precision depends on the dimension of the matrix, which corresponds to the number of composite spectra employed. And also the sophisticated tomographic method by Bagnuolo & Gies (1991), which is based on an iterative algorithm similar in principle to the tomography used in medical physics, has a precision depending on the number of composite spectra used in the reconstruction process.
With the purpose of conceptual and practical simplicity, we developed an original Doppler-shift method to separate the spectra of binary stars. It requires only a pair of observations at different orbital phases (elongations preferably) and elementary computations.
A full reconstruction of the secondary absorption spectrum is obtained after few iterations, by applying a simple sequence of algebraic operations and Doppler-shift compensations to the pair of observed spectra. Then the primary spectrum is trivially derived by difference, renormalization and shift-compensation, from the systemic spectrum and the reconstructed secondary.
A larger number of available spectra, at various orbital phases, will obviously increase the signal-to-noise quality of the result; and this may be useful for faint targets. The procedure works powerfully even when the secondary spectrum is very weak, and/or when the primary and secondary lines overlap. In fact, the earliest conception of our method (Ferluga et al. 1991) was aimed precisely at the detection of faint double-lined eclipsing binaries.