In this paper, we present a successful implementation of a subtraction-noise projection method into a simple, simulated data analysis pipeline of a gravitational-wave search. We investigate the problem to reveal a weak stochastic background signal which is covered by a strong foreground of compact-binary coalescences. The foreground, which is estimated by matched filters, has to be subtracted from the data. Even an optimal analysis of foreground signals will leave subtraction noise due to estimation errors of template parameters which may corrupt the measurement of the background signal. The subtraction noise can be removed by a noise projection. We apply our analysis pipeline to the proposed future-generation space-borne Big Bang Observer mission which seeks for a stochastic background of primordial gravitational waves in the frequency range ~0.1 Hz—1 Hz covered by a foreground of black-hole and neutron-star binaries. Our analysis is based on a simulation code which provides a dynamical model of a time-delay interferometer network. It generates the data as time series and incorporates the analysis pipeline together with the noise projection. Our results confirm previous ad hoc predictions which say that the Big Bang Observer will be sensitive to backgrounds with fractional energy densities below Omega=10-16

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