An explicit expression for the impulse response coefficients of the predictive FIR digital filters is derived.The formula specifies a four-parameter family of smoothing FIR digital filters containing theSavitsky-Goaly filters, the Heinonen-Neuvo polynomial predictors, and the smoothing differentiators of arbitrary integer orders.The Hahn polynomials, which are orthogonal with respect to a discrete variable,are the main tool employed in the derivation of the formula.A recursive formula for the computation of the transfer function of the filters, which is the $z$-transform of a terminated sequence of polynomial ordinates, is also introduced.The formula can be used to design structures with low computational complexity for filters of any order.
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