On digital smoothing filters: A brief review of closed form solutions and two new filter approaches

The concept of smoothing noisy data using appropriate polynomials turns out to be equivalent to the application of suitable nonrecursive digital filters having the following properties: They process the data in such a way that the moments are conserved up to a desired order while the energy of their impulse response is minimum. Flatness constraints of their frequency response at =0 are equivalent to the moment condition. By using orthogonal polynomials, an explicit solution is known from the literature. A second approach which uses a special decomposition also yields closed form solutions. The realization is simplified, especially in the case where a large number of moments is supposed to be conserved.