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Proof of Maximum Flatness at DC

The maximumally 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:

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

Further details regarding the theory of Lagrange interpolation can be found (online) in [502, Ch. 3, Pt. 2, pp. 82-84].

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