Chebyshev approximation of a constant group delay with constraints at the origin

This paper deals with low-pass filter functions approximating a constant delay in an equiripple manner which does not yield a standard delay error curve. This type of Chebyshev approximation is obtained by imposing a constraint on the error curve at \omega = 0 . It is shown that using the constrained approximation, the delay approximation bandwidth for n odd and a prescribed ripple factor \epsilon may be equal to, or even larger than, that obtained by Abele's polynomials; the latter solution is neither unique nor the best approximation. The magnitude characteristics of the constrained approximants are very much improved and the transient responses to a unit step input compare favorably with those for the other known systems including Schüssler's functions with equiripple step response. Tables are presented which include the pole locations of some selected constrained approximants of 3, 5, 7, and 9 degrees, the comparative stopband attenuation relative to the Abele case, and the most important quantities associated with a step response.