The equations for the optimum zero memory filter are derived using the mean absolute error criterion. It is shown that the characteristic of this filter is antisymmetric about a certain point when the signal and noise densities have symmetry properties. The coordinates of this point are found. The "direct connection" filter (no filter at all) is found to be nonoptimal in all cases of interest: consequently the need for filtering is established. The optimum zero memory filter is shown to be a quantizer for quantized signals and a method of obtaining the characteristic of this filter is developed when the signal has a finite number of levels. The only case considered in this paper is the common one for which the signal and noise are statistically independent.