This paper proposes exponential type nonlinearities in order to blindly separate instantaneous mixtures of signals with mixed kurtosis signs. These nonlinear functions are applied only in a certain range around zero in order to ensure that the relative gradient algorithm remains locally stable. The proposed truncated nonlinearities neutralize the effect of outliers while the higher order terms inherently present in the exponential function result in fast convergence especially for signals with bounded support. By varying the truncation threshold, signals with both sub-Gaussian and super-Gaussian probability distributions can be separated. Furthermore, when the sources consist of signals with mixed kurtosis signs we propose to estimate the characteristic function online in order to classify the signals as sub-Gaussian or super-Gaussian and consequently choose an adequate value of the truncation threshold. Some computer simulations are presented to demonstrate the effectiveness of the proposed idea.