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In both the natural gradient algorithm and the FastICA algorithm for blind source separation (BSS), output nonlinearities for each extracted source must be selected, and the performance of each approach can be sensitive to the chosen output nonlinearity. In this paper, we propose to use simple piecewise-linear output nonlinearities for these algorithms and obtain a number of useful properties with such a choice. For the natural gradient BSS algorithm, nonlinearity- switching is easily achieved through a common stability criterion that guarantees local stability for all source distributions. For the FastICA algorithm, the chosen nonlinearities can be very close to linear, suggesting that simple (e.g. mu-law) output companding is sufficiently nonlinear to allow separation when used with this algorithm. Simulations are provided to verify the theoretical results.