Median filters are especially used when processing signal which exhibit smooth behaviour with unknown switching points (e.g. blood pressure signals). They work on the basis of the so-called Standard Median (SM) filter algorithm. A better solution has been found by the introduction of finite-impulse-response (FIR) median hybrid filters (FMH) which allow the rejection of the impulsive noise. In the paper the FMH are generalized so that they act as optimal predictors for k-th order polynomial signals corrupted by Gaussian white noise. It is interesting to remark that for the SM filters there exist the so-called root signals which are invariants under this kind of filtering. Calculations are made for FIR coefficients when considering polynomials of different orders. Finally, some examples of smoothing blood pressure signals are presented which show the efficiency of the proposed method. We think the paper represents a basic contribution in the area of signal processing in the presence of noise.