Lowpass delay filters with flat magnitude and group delay constraints

This paper describes the design of finite impulse response (FIR) delay filters that minimize a squared error and have prescribed number of zeros at ω=π and prescribed magnitude and group delay flatness at ω=0. An important special case is the design of least squared error lowpass filters with prescribed flatness constraints and zeros at ω=π. Even though the flatness constraints are in general nonlinear functions of the filter coefficients, we show the remarkable fact that for a subclass of the filters a simple orthogonal projection of least squared error filters onto a special linear subspace determined via Baher (1982) filters gives the solution. The paper also introduces the notion of delay filters that are high-order approximations to the ideal delay and establishes their equivalence to Baher filters. This connection gives novel elementary derivations of Baher filters and their properties. Matlab programs are provided at the end of the paper for the design of filters described in this paper.