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A general framework for the design of an FIR filter based on the cascade of an FIR pre-filter with quantized coefficients followed by an all pole IIR stage is proposed. A simple design method results by designing the pre-filter stage coefficients via a differential coding scheme. The IIR stage is then simply a linear predictor reconstruction stage. The differential coding method eliminates the redundancies of the filter coefficient sequence and makes it possible to quantize the coefficients into a few levels. A linear prediction algorithm is applied to optimize the differential coding method using the statistical characteristics of the given filter. Relatively wideband filters as well as narrowband filters have been sucessfully designed without interpolating the original filters. Overall design procedures and design examples for a variety of filters are included, and the theoretical aspects of this approach are studied.