A unified framework for derivation and implementation of Savitzky–Golay filters

• A general definition for Savitzky–Golay filters.
• Explicit derivation of several Savitzky–Golay filters for various applications.
• A structure with a reduced computational load and its connection with the suggested definition.

The Savitzky–Golay (SG) filter design problem is posed as the minimum norm solution of an underdetermined equation system. A unified SG filter design framework encompassing several important applications such as smoothing, differentiation, integration and fractional delay is developed. In addition to the generality and flexibility of the framework, an efficient SG filter implementation structure, naturally emerging from the framework, is proposed. The structure is shown to reduce the number of multipliers in the smoothing application. More specifically, the smoothing application, where an L  th degree polynomial to the frame of 2N+12N+1 samples is fitted, can be implemented with N−L/2N−L/2 multiplications per output sample instead of N+1N+1 multiplications with the suggested structure.