The objective of this paper is to present a new technique for jointly decomposing two sets of signals. The proposed method is a modified version of Canonical Correlation Analysis (CCA), which automatically identifies from the two (a priori noisy) data-sets, having the same number of samples but potentially different number of variables (measurements), an approximate bisector common subspace and its complementary specific subspaces. Within these subspaces, common and specific parts of the signals can be reconstructed and analysed separately. The method we propose here can also be seen as an extension of other joint decomposition methods based on "stacking" the analysed data sets, but, unlike these methods, we propose a "stacked basis" approach and we show its relationship with the CCA. The proposed method is validated with convincing results on simulated data and applied successfully on (stereo-)electroencephalographic signals, either for artefact cancelling or for identifying common and specific activities for two different physiological conditions (sleep-wake).