Singular spectrum analysis (SSA) has been verified to be an effective method for decomposing non-stationary signals. The decomposition and reconstruction stages can be interpreted as a zero-phase filtering process where reconstructed components are obtained by inputting a signal through moving average filters. However, mathematical analysis indicates that the use of a default rectangular window in the embedding stage would corrupt the frequency characteristics of the trajectory matrix, resulting in spectral leakage. To attenuate the effect of spectral leakage and to obtain more concentrated SSA components, this study introduces a windowing technique in SSA, called generalized singular spectrum analysis (GSSA). In GSSA, the default rectangular window is replaced with adjustable taper windows, which are widely used for attenuating spectral leakage. Through windowing, GSSA can achieve less spectral leakage, and produce more energy-concentrated reconstructed components compared with conventional SSA. Grouped spectral entropy (GSE) is used as the metric for evaluating the performance of the proposed GSSA algorithm. Results from experiments, which were conducted on a synthetic signal and two real electroencephalogram signals, show that GSSA outperforms the conventional SSA and the baseline in the reduction of spectral leakage. Compared with the baseline, the proposed GSSA achieves a lower averaged GSE, resulting in reduction of 0.4 for eigenfilters and 0.11 for reconstructed components, respectively. Our results reveal the effectiveness of GSSA in the decomposition and analysis of non-stationary signals.