The maximally flat fractional-delay FIR filter is obtained by equating
to zero all
leading terms in the Taylor (Maclaurin) expansion of
the frequency-response error at dc:
This is a linear system of equations of the form
Making this substitution in the solution obtained by Cramer's rule
yields that the impulse response of the order
, maximally flat,
fractional-delay FIR filter may be written in closed form as
which is the formula for Lagrange-interpolation coefficients (Eq.(4.6)) adapted to this problem (in which abscissae are equally spaced on the integers from 0 to
Further details regarding the theory of Lagrange interpolation can be found in [506, Ch. 3, Pt. 2, pp. 82-84].5.5