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Signal averaging is often used to extract a useful signal embedded in noise. This method is especially useful for biomedical signals, where the spectra of the signal and noise significantly overlap. In this case, traditional filtering techniques introduce unacceptable signal distortion. In averaging methods, constancy of the noise power is usually assumed, but in reality noise features a variable power. In this case, it is more appropriate to use a weighted averaging. The main problem in this method is the estimation of the noise power in order to obtain the weight values. Additionally, biomedical signals often contain outliers. This requires robust averaging methods. This paper shows that signal averaging can be formulated as a problem of minimization of a criterion function. Based on this formulation new weighted averaging methods are introduced, including weighted averaging based on criterion function minimization (WACFM) and robust ε-insensitive WACFM. Performances of these new methods are experimentally compared with the traditional averaging and other weighted averaging methods using electrocardiographic signal with the muscle noise, impulsive noise, and time-misalignment of cycles. Finally, an application to the late potentials extraction is shown.