  We present a data analysis methodology for a model-independent reconstruction
of the spectral shape of a stochastic gravitational wave background with LISA.
We improve a previously proposed reconstruction algorithm that relied on a
single Time-Delay-Interferometry (TDI) channel by including a complete set of
TDI channels. As in the earlier work, we assume an idealized equilateral
configuration. We test the improved algorithm with a number of case studies,
including reconstruction in the presence of two different astrophysical
foreground signals. We find that including additional channels helps in
different ways: it reduces the uncertainties on the reconstruction; it makes
the global likelihood maximization less prone to falling into local extrema;
and it efficiently breaks degeneracies between the signal and the instrumental
noise.
