Design of Narrow Band FIR Filters

A new concept for the design of finite impulse response (FIR) filters is introduced here. This analytical approach has led to three fundamental and new results. First, a differential equation for the approximating polynomial of the filter is developed. Second, the linear differential equation is solved by iteration, yielding an algorithm for recursive evaluation of the impulse response of a filter. Finally, the degree equation of the filter is introduced. We present the design of 4 types of narrow band FIR filters: the maximally at notch FIR filter, the equiripple notch FIR filter, the equiripple DC-notch FIR filter and the equiripple comb FIR filter. Equiripple filters are optimal in the Chebyshev sense. The design procedure starts with frequency specifications of the filter and ends with a recursive evaluation of the impulse response coefficients refraining from other numerical procedures.