An analytical expression of FIR filters with derivative constraints at two frequencies

In this paper, a novel structure of finite impulse response (FIR) filter is proposed. We solve the problem of building an universal structure that allows a direct control of the derivatives of the frequency response at two given frequencies. Accordingly any frequency response can be approximated. The structure consists of FIR subfllters whose frequency responses are independent of the desired one. We use method of transformation to derive these subfllters analytically. Synthesis of variable fractional delay FIR filters with almost all-pass frequency responses is presented as an example. In conclusion, the proposed structure is useful for implementing derivative-constrained FIR filters.